jeff quadratic root 2

Percentage Accurate: 72.1% → 90.1%

Time: 18.8s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

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

real(8) function code(a, b, c)
    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: 72.1% accurate, 1.0× speedup

Math

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

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

real(8) function code(a, b, c)
    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: 90.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot -2}{2 \cdot a}\\ \mathbf{if}\;b \leq -1.8 \cdot 10^{+93}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;0 \cdot \left(c \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{+92}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \frac{1}{{\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)}^{-0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;0 \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* b -2.0) (* 2.0 a))))
   (if (<= b -1.8e+93)
     (if (>= b 0.0) (* 0.0 (* c -2.0)) t_0)
     (if (<= b 3.5e+92)
       (if (>= b 0.0)
         (* c (/ -2.0 (+ b (/ 1.0 (pow (+ (* b b) (* c (* a -4.0))) -0.5)))))
         (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a)))
       (if (>= 0.0 0.0) (/ (* c 2.0) (* b -2.0)) t_0)))))

C

double code(double a, double b, double c) {
	double t_0 = (b * -2.0) / (2.0 * a);
	double tmp_1;
	if (b <= -1.8e+93) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 0.0 * (c * -2.0);
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= 3.5e+92) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = c * (-2.0 / (b + (1.0 / pow(((b * b) + (c * (a * -4.0))), -0.5))));
		} else {
			tmp_3 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (0.0 >= 0.0) {
		tmp_1 = (c * 2.0) / (b * -2.0);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    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 = (b * (-2.0d0)) / (2.0d0 * a)
    if (b <= (-1.8d+93)) then
        if (b >= 0.0d0) then
            tmp_2 = 0.0d0 * (c * (-2.0d0))
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b <= 3.5d+92) then
        if (b >= 0.0d0) then
            tmp_3 = c * ((-2.0d0) / (b + (1.0d0 / (((b * b) + (c * (a * (-4.0d0)))) ** (-0.5d0)))))
        else
            tmp_3 = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (2.0d0 * a)
        end if
        tmp_1 = tmp_3
    else if (0.0d0 >= 0.0d0) then
        tmp_1 = (c * 2.0d0) / (b * (-2.0d0))
    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 = (b * -2.0) / (2.0 * a);
	double tmp_1;
	if (b <= -1.8e+93) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 0.0 * (c * -2.0);
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= 3.5e+92) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = c * (-2.0 / (b + (1.0 / Math.pow(((b * b) + (c * (a * -4.0))), -0.5))));
		} else {
			tmp_3 = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (0.0 >= 0.0) {
		tmp_1 = (c * 2.0) / (b * -2.0);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (b * -2.0) / (2.0 * a)
	tmp_1 = 0
	if b <= -1.8e+93:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = 0.0 * (c * -2.0)
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b <= 3.5e+92:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = c * (-2.0 / (b + (1.0 / math.pow(((b * b) + (c * (a * -4.0))), -0.5))))
		else:
			tmp_3 = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)
		tmp_1 = tmp_3
	elif 0.0 >= 0.0:
		tmp_1 = (c * 2.0) / (b * -2.0)
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * -2.0) / Float64(2.0 * a))
	tmp_1 = 0.0
	if (b <= -1.8e+93)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(0.0 * Float64(c * -2.0));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b <= 3.5e+92)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(c * Float64(-2.0 / Float64(b + Float64(1.0 / (Float64(Float64(b * b) + Float64(c * Float64(a * -4.0))) ^ -0.5)))));
		else
			tmp_3 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (0.0 >= 0.0)
		tmp_1 = Float64(Float64(c * 2.0) / Float64(b * -2.0));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = (b * -2.0) / (2.0 * a);
	tmp_2 = 0.0;
	if (b <= -1.8e+93)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = 0.0 * (c * -2.0);
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b <= 3.5e+92)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = c * (-2.0 / (b + (1.0 / (((b * b) + (c * (a * -4.0))) ^ -0.5))));
		else
			tmp_4 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
		end
		tmp_2 = tmp_4;
	elseif (0.0 >= 0.0)
		tmp_2 = (c * 2.0) / (b * -2.0);
	else
		tmp_2 = t_0;
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * -2.0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.8e+93], If[GreaterEqual[b, 0.0], N[(0.0 * N[(c * -2.0), $MachinePrecision]), $MachinePrecision], t$95$0], If[LessEqual[b, 3.5e+92], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[(b + N[(1.0 / N[Power[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[0.0, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[(b * -2.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.8e93

   1. Initial program 55.4%

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

   3. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   4. Step-by-step derivation

   5. Simplified 55.4%

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

   6. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(-2 \cdot b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(b \cdot -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. *-lowering-*.f64 98.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   8. Simplified 98.0%

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

   10. Applied egg-rr 98.0%

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

   if -1.8e93 < b < 3.49999999999999986e92

   1. Initial program 90.0%

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

   4. Applied egg-rr 90.0%

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

      1. remove-double-div N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \left(\frac{1}{\frac{1}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}}\right)\right)\right), c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \left(\frac{1}{\frac{\sqrt{1}}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}}\right)\right)\right), c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. sqrt-div N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \left(\frac{1}{\sqrt{\frac{1}{b \cdot b + c \cdot \left(a \cdot -4\right)}}}\right)\right)\right), c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{/.f64}\left(1, \left(\sqrt{\frac{1}{b \cdot b + c \cdot \left(a \cdot -4\right)}}\right)\right)\right)\right), c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      5. inv-pow N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{/.f64}\left(1, \left(\sqrt{{\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)}^{-1}}\right)\right)\right)\right), c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      6. sqrt-pow1 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{/.f64}\left(1, \left({\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)}^{\left(\frac{-1}{2}\right)}\right)\right)\right)\right), c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      7. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{/.f64}\left(1, \left({\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)}^{\frac{-1}{2}}\right)\right)\right)\right), c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      8. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{/.f64}\left(1, \left({\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)}^{\left(\frac{1}{2} \cdot -1\right)}\right)\right)\right)\right), c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right), \left(\frac{1}{2} \cdot -1\right)\right)\right)\right)\right), c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(c \cdot \left(a \cdot -4\right)\right)\right), \left(\frac{1}{2} \cdot -1\right)\right)\right)\right)\right), c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(c \cdot \left(a \cdot -4\right)\right)\right), \left(\frac{1}{2} \cdot -1\right)\right)\right)\right)\right), c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(a \cdot -4\right)\right)\right), \left(\frac{1}{2} \cdot -1\right)\right)\right)\right)\right), c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right), \left(\frac{1}{2} \cdot -1\right)\right)\right)\right)\right), c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      14. metadata-eval 90.0%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right), \frac{-1}{2}\right)\right)\right)\right), c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   6. Applied egg-rr 90.0%

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

   if 3.49999999999999986e92 < b

   1. Initial program 50.2%

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

   3. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   4. Step-by-step derivation

   5. Simplified 94.6%

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

   6. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(-2 \cdot b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(b \cdot -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. *-lowering-*.f64 94.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   8. Simplified 94.6%

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

   10. Applied egg-rr 94.6%

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

       1. /-lowering-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\left(2 \cdot c\right), \color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) - b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       2. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\left(c \cdot 2\right), \left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} - b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} - b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       4. sub-neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(\left(\mathsf{neg}\left(b\right)\right) + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       5. neg-mul-1 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(-1 \cdot b + \left(\mathsf{neg}\left(\color{blue}{b}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       6. neg-mul-1 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(-1 \cdot b + -1 \cdot \color{blue}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       7. distribute-rgt-out N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(b \cdot \color{blue}{\left(-1 + -1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       8. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(b \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       9. *-lowering-*.f64 94.6%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \mathsf{*.f64}\left(b, \color{blue}{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   12. Applied egg-rr 94.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;0 \geq 0:\\ \;\;\;\;\color{blue}{\frac{c \cdot 2}{b \cdot -2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{+93}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;0 \cdot \left(c \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{+92}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \frac{1}{{\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)}^{-0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;0 \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - \left(b + b\right)\\ t_1 := \frac{c \cdot 2}{t\_0}\\ \mathbf{if}\;b \leq -1.02 \cdot 10^{-27}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a}}{2}\\ \end{array}\\ \mathbf{elif}\;b \leq 1700:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(0 - b\right) - \sqrt{-4 \cdot \left(c \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;0 \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- 0.0 (+ b b))) (t_1 (/ (* c 2.0) t_0)))
   (if (<= b -1.02e-27)
     (if (>= b 0.0) t_1 (- (/ c b) (/ b a)))
     (if (<= b -5e-310)
       (if (>= b 0.0) t_1 (/ (/ (- (sqrt (* c (* a -4.0))) b) a) 2.0))
       (if (<= b 1700.0)
         (if (>= b 0.0)
           (/ (* c 2.0) (- (- 0.0 b) (sqrt (* -4.0 (* c a)))))
           (/ t_0 (* 2.0 a)))
         (if (>= 0.0 0.0)
           (/ (* c 2.0) (* b -2.0))
           (/ (* b -2.0) (* 2.0 a))))))))

C

double code(double a, double b, double c) {
	double t_0 = 0.0 - (b + b);
	double t_1 = (c * 2.0) / t_0;
	double tmp_1;
	if (b <= -1.02e-27) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = (c / b) - (b / a);
		}
		tmp_1 = tmp_2;
	} else if (b <= -5e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = ((sqrt((c * (a * -4.0))) - b) / a) / 2.0;
		}
		tmp_1 = tmp_3;
	} else if (b <= 1700.0) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c * 2.0) / ((0.0 - b) - sqrt((-4.0 * (c * a))));
		} else {
			tmp_4 = t_0 / (2.0 * a);
		}
		tmp_1 = tmp_4;
	} else if (0.0 >= 0.0) {
		tmp_1 = (c * 2.0) / (b * -2.0);
	} else {
		tmp_1 = (b * -2.0) / (2.0 * a);
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    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 = 0.0d0 - (b + b)
    t_1 = (c * 2.0d0) / t_0
    if (b <= (-1.02d-27)) then
        if (b >= 0.0d0) then
            tmp_2 = t_1
        else
            tmp_2 = (c / b) - (b / a)
        end if
        tmp_1 = tmp_2
    else if (b <= (-5d-310)) then
        if (b >= 0.0d0) then
            tmp_3 = t_1
        else
            tmp_3 = ((sqrt((c * (a * (-4.0d0)))) - b) / a) / 2.0d0
        end if
        tmp_1 = tmp_3
    else if (b <= 1700.0d0) then
        if (b >= 0.0d0) then
            tmp_4 = (c * 2.0d0) / ((0.0d0 - b) - sqrt(((-4.0d0) * (c * a))))
        else
            tmp_4 = t_0 / (2.0d0 * a)
        end if
        tmp_1 = tmp_4
    else if (0.0d0 >= 0.0d0) then
        tmp_1 = (c * 2.0d0) / (b * (-2.0d0))
    else
        tmp_1 = (b * (-2.0d0)) / (2.0d0 * a)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = 0.0 - (b + b);
	double t_1 = (c * 2.0) / t_0;
	double tmp_1;
	if (b <= -1.02e-27) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = (c / b) - (b / a);
		}
		tmp_1 = tmp_2;
	} else if (b <= -5e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = ((Math.sqrt((c * (a * -4.0))) - b) / a) / 2.0;
		}
		tmp_1 = tmp_3;
	} else if (b <= 1700.0) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c * 2.0) / ((0.0 - b) - Math.sqrt((-4.0 * (c * a))));
		} else {
			tmp_4 = t_0 / (2.0 * a);
		}
		tmp_1 = tmp_4;
	} else if (0.0 >= 0.0) {
		tmp_1 = (c * 2.0) / (b * -2.0);
	} else {
		tmp_1 = (b * -2.0) / (2.0 * a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = 0.0 - (b + b)
	t_1 = (c * 2.0) / t_0
	tmp_1 = 0
	if b <= -1.02e-27:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_1
		else:
			tmp_2 = (c / b) - (b / a)
		tmp_1 = tmp_2
	elif b <= -5e-310:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = t_1
		else:
			tmp_3 = ((math.sqrt((c * (a * -4.0))) - b) / a) / 2.0
		tmp_1 = tmp_3
	elif b <= 1700.0:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = (c * 2.0) / ((0.0 - b) - math.sqrt((-4.0 * (c * a))))
		else:
			tmp_4 = t_0 / (2.0 * a)
		tmp_1 = tmp_4
	elif 0.0 >= 0.0:
		tmp_1 = (c * 2.0) / (b * -2.0)
	else:
		tmp_1 = (b * -2.0) / (2.0 * a)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(0.0 - Float64(b + b))
	t_1 = Float64(Float64(c * 2.0) / t_0)
	tmp_1 = 0.0
	if (b <= -1.02e-27)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(Float64(c / b) - Float64(b / a));
		end
		tmp_1 = tmp_2;
	elseif (b <= -5e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = Float64(Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / a) / 2.0);
		end
		tmp_1 = tmp_3;
	elseif (b <= 1700.0)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(c * 2.0) / Float64(Float64(0.0 - b) - sqrt(Float64(-4.0 * Float64(c * a)))));
		else
			tmp_4 = Float64(t_0 / Float64(2.0 * a));
		end
		tmp_1 = tmp_4;
	elseif (0.0 >= 0.0)
		tmp_1 = Float64(Float64(c * 2.0) / Float64(b * -2.0));
	else
		tmp_1 = Float64(Float64(b * -2.0) / Float64(2.0 * a));
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = 0.0 - (b + b);
	t_1 = (c * 2.0) / t_0;
	tmp_2 = 0.0;
	if (b <= -1.02e-27)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = (c / b) - (b / a);
		end
		tmp_2 = tmp_3;
	elseif (b <= -5e-310)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = t_1;
		else
			tmp_4 = ((sqrt((c * (a * -4.0))) - b) / a) / 2.0;
		end
		tmp_2 = tmp_4;
	elseif (b <= 1700.0)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = (c * 2.0) / ((0.0 - b) - sqrt((-4.0 * (c * a))));
		else
			tmp_5 = t_0 / (2.0 * a);
		end
		tmp_2 = tmp_5;
	elseif (0.0 >= 0.0)
		tmp_2 = (c * 2.0) / (b * -2.0);
	else
		tmp_2 = (b * -2.0) / (2.0 * a);
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(0.0 - N[(b + b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * 2.0), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[b, -1.02e-27], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -5e-310], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] / 2.0), $MachinePrecision]], If[LessEqual[b, 1700.0], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[(N[(0.0 - b), $MachinePrecision] - N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[0.0, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[(b * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(b * -2.0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.02000000000000002e-27

   1. Initial program 72.2%

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

   3. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   4. Step-by-step derivation

   5. Simplified 72.2%

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

   6. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(\mathsf{neg}\left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right) \cdot b\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(\left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \left(-2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      6. associate-*r/ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{-2 \cdot \left(a \cdot c\right)}{{b}^{2}}\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(-2 \cdot \left(a \cdot c\right)\right), \left({b}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(a \cdot c\right)\right), \left({b}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      9. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(c \cdot a\right)\right), \left({b}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), \left({b}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      11. unpow2 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), \left(b \cdot b\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      13. neg-sub0 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(0 - b\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      14. --lowering--.f64 87.0%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{\_.f64}\left(0, b\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   8. Simplified 87.0%

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

   9. Taylor expanded in c around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a} + \frac{c}{b}\\ \end{array} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + -1 \cdot \frac{b}{a}\\ \end{array} \]

       2. mul-1-neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right)\\ \end{array} \]

       3. unsub-neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

       4. --lowering--.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(\left(\frac{c}{b}\right), \left(\frac{b}{a}\right)\right)\\ \end{array} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(\mathsf{/.f64}\left(c, b\right), \left(\frac{b}{a}\right)\right)\\ \end{array} \]

       6. /-lowering-/.f64 88.3%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(\mathsf{/.f64}\left(c, b\right), \mathsf{/.f64}\left(b, a\right)\right)\\ \end{array} \]

   11. Simplified 88.3%

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

   if -1.02000000000000002e-27 < b < -4.999999999999985e-310

   1. Initial program 86.2%

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

   3. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   4. Step-by-step derivation

   5. Simplified 86.2%

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

   6. Taylor expanded in b around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\left(-4 \cdot \left(a \cdot c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\left(\left(a \cdot c\right) \cdot -4\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a \cdot c\right), -4\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(c \cdot a\right), -4\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      4. *-lowering-*.f64 71.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -4\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   8. Simplified 71.7%

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

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(c \cdot a\right) \cdot -4}}{a \cdot 2}\\ \end{array} \]

      2. associate-/r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(c \cdot a\right) \cdot -4}}{a}}{2}\\ \end{array} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(c \cdot a\right) \cdot -4}}{a}\right), 2\right)\\ \end{array} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(c \cdot a\right) \cdot -4}\right), a\right), 2\right)\\ \end{array} \]

      5. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{\left(c \cdot a\right) \cdot -4} + \left(\mathsf{neg}\left(b\right)\right)\right), a\right), 2\right)\\ \end{array} \]

      6. unsub-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{\left(c \cdot a\right) \cdot -4} - b\right), a\right), 2\right)\\ \end{array} \]

      7. --lowering--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{\left(c \cdot a\right) \cdot -4}\right), b\right), a\right), 2\right)\\ \end{array} \]

      8. associate-*r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{c \cdot \left(a \cdot -4\right)}\right), b\right), a\right), 2\right)\\ \end{array} \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(c \cdot \left(a \cdot -4\right)\right)\right), b\right), a\right), 2\right)\\ \end{array} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(a \cdot -4\right)\right)\right), b\right), a\right), 2\right)\\ \end{array} \]

      11. *-lowering-*.f64 71.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right), b\right), a\right), 2\right)\\ \end{array} \]

   10. Applied egg-rr 71.8%

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

   if -4.999999999999985e-310 < b < 1700

   1. Initial program 87.8%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \left(-1 \cdot b\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. neg-sub0 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \left(0 - b\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. --lowering--.f64 87.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{\_.f64}\left(0, b\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   5. Simplified 87.8%

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

   6. Taylor expanded in b around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{\_.f64}\left(0, b\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{\_.f64}\left(0, b\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a \cdot c\right), -4\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{\_.f64}\left(0, b\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(c \cdot a\right), -4\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{\_.f64}\left(0, b\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      4. *-lowering-*.f64 76.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -4\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{\_.f64}\left(0, b\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   8. Simplified 76.5%

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

   if 1700 < b

   1. Initial program 59.4%

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

   3. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   4. Step-by-step derivation

   5. Simplified 93.9%

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

   6. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(-2 \cdot b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(b \cdot -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. *-lowering-*.f64 93.9%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   8. Simplified 93.9%

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

   10. Applied egg-rr 93.9%

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

       1. /-lowering-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\left(2 \cdot c\right), \color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) - b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       2. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\left(c \cdot 2\right), \left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} - b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} - b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       4. sub-neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(\left(\mathsf{neg}\left(b\right)\right) + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       5. neg-mul-1 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(-1 \cdot b + \left(\mathsf{neg}\left(\color{blue}{b}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       6. neg-mul-1 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(-1 \cdot b + -1 \cdot \color{blue}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       7. distribute-rgt-out N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(b \cdot \color{blue}{\left(-1 + -1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       8. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(b \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       9. *-lowering-*.f64 93.9%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \mathsf{*.f64}\left(b, \color{blue}{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   12. Applied egg-rr 93.9%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{-27}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{0 - \left(b + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{0 - \left(b + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a}}{2}\\ \end{array}\\ \mathbf{elif}\;b \leq 1700:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(0 - b\right) - \sqrt{-4 \cdot \left(c \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - \left(b + b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;0 \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\\ t_1 := \frac{b \cdot -2}{2 \cdot a}\\ \mathbf{if}\;b \leq -3.3 \cdot 10^{+91}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;0 \cdot \left(c \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{+92}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;0 \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* b b) (* c (* a -4.0)))))
        (t_1 (/ (* b -2.0) (* 2.0 a))))
   (if (<= b -3.3e+91)
     (if (>= b 0.0) (* 0.0 (* c -2.0)) t_1)
     (if (<= b 2.45e+92)
       (if (>= b 0.0) (* c (/ -2.0 (+ b t_0))) (/ (- t_0 b) (* 2.0 a)))
       (if (>= 0.0 0.0) (/ (* c 2.0) (* b -2.0)) t_1)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) + (c * (a * -4.0))));
	double t_1 = (b * -2.0) / (2.0 * a);
	double tmp_1;
	if (b <= -3.3e+91) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 0.0 * (c * -2.0);
		} else {
			tmp_2 = t_1;
		}
		tmp_1 = tmp_2;
	} else if (b <= 2.45e+92) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = c * (-2.0 / (b + t_0));
		} else {
			tmp_3 = (t_0 - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (0.0 >= 0.0) {
		tmp_1 = (c * 2.0) / (b * -2.0);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    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 = sqrt(((b * b) + (c * (a * (-4.0d0)))))
    t_1 = (b * (-2.0d0)) / (2.0d0 * a)
    if (b <= (-3.3d+91)) then
        if (b >= 0.0d0) then
            tmp_2 = 0.0d0 * (c * (-2.0d0))
        else
            tmp_2 = t_1
        end if
        tmp_1 = tmp_2
    else if (b <= 2.45d+92) then
        if (b >= 0.0d0) then
            tmp_3 = c * ((-2.0d0) / (b + t_0))
        else
            tmp_3 = (t_0 - b) / (2.0d0 * a)
        end if
        tmp_1 = tmp_3
    else if (0.0d0 >= 0.0d0) then
        tmp_1 = (c * 2.0d0) / (b * (-2.0d0))
    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 = Math.sqrt(((b * b) + (c * (a * -4.0))));
	double t_1 = (b * -2.0) / (2.0 * a);
	double tmp_1;
	if (b <= -3.3e+91) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 0.0 * (c * -2.0);
		} else {
			tmp_2 = t_1;
		}
		tmp_1 = tmp_2;
	} else if (b <= 2.45e+92) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = c * (-2.0 / (b + t_0));
		} else {
			tmp_3 = (t_0 - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (0.0 >= 0.0) {
		tmp_1 = (c * 2.0) / (b * -2.0);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) + (c * (a * -4.0))))
	t_1 = (b * -2.0) / (2.0 * a)
	tmp_1 = 0
	if b <= -3.3e+91:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = 0.0 * (c * -2.0)
		else:
			tmp_2 = t_1
		tmp_1 = tmp_2
	elif b <= 2.45e+92:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = c * (-2.0 / (b + t_0))
		else:
			tmp_3 = (t_0 - b) / (2.0 * a)
		tmp_1 = tmp_3
	elif 0.0 >= 0.0:
		tmp_1 = (c * 2.0) / (b * -2.0)
	else:
		tmp_1 = t_1
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0))))
	t_1 = Float64(Float64(b * -2.0) / Float64(2.0 * a))
	tmp_1 = 0.0
	if (b <= -3.3e+91)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(0.0 * Float64(c * -2.0));
		else
			tmp_2 = t_1;
		end
		tmp_1 = tmp_2;
	elseif (b <= 2.45e+92)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(c * Float64(-2.0 / Float64(b + t_0)));
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (0.0 >= 0.0)
		tmp_1 = Float64(Float64(c * 2.0) / Float64(b * -2.0));
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) + (c * (a * -4.0))));
	t_1 = (b * -2.0) / (2.0 * a);
	tmp_2 = 0.0;
	if (b <= -3.3e+91)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = 0.0 * (c * -2.0);
		else
			tmp_3 = t_1;
		end
		tmp_2 = tmp_3;
	elseif (b <= 2.45e+92)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = c * (-2.0 / (b + t_0));
		else
			tmp_4 = (t_0 - b) / (2.0 * a);
		end
		tmp_2 = tmp_4;
	elseif (0.0 >= 0.0)
		tmp_2 = (c * 2.0) / (b * -2.0);
	else
		tmp_2 = t_1;
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * -2.0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.3e+91], If[GreaterEqual[b, 0.0], N[(0.0 * N[(c * -2.0), $MachinePrecision]), $MachinePrecision], t$95$1], If[LessEqual[b, 2.45e+92], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[(b + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[0.0, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[(b * -2.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.30000000000000017e91

   1. Initial program 55.4%

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

   3. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   4. Step-by-step derivation

   5. Simplified 55.4%

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

   6. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(-2 \cdot b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(b \cdot -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. *-lowering-*.f64 98.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   8. Simplified 98.0%

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

   10. Applied egg-rr 98.0%

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

   if -3.30000000000000017e91 < b < 2.4500000000000001e92

   1. Initial program 90.0%

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

   4. Applied egg-rr 90.0%

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

      1. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right)\right), c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. sub-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right)\right), c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(\sqrt{b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)} + \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right)\right), c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(\sqrt{b \cdot b + \left(\mathsf{neg}\left(c \cdot \left(4 \cdot a\right)\right)\right)} + \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right)\right), c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(\sqrt{b \cdot b + c \cdot \left(\mathsf{neg}\left(4 \cdot a\right)\right)} + \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right)\right), c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(\sqrt{b \cdot b + c \cdot \left(\left(\mathsf{neg}\left(4\right)\right) \cdot a\right)} + \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      6. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right)\right), c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(\sqrt{b \cdot b + c \cdot \left(-4 \cdot a\right)} + \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      7. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right)\right), c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} + \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      8. remove-double-div N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right)\right), c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(\frac{1}{\frac{1}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}} + \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      9. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right)\right), c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(\frac{1}{\frac{\sqrt{1}}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}} + \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      10. sqrt-div N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right)\right), c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(\frac{1}{\sqrt{\frac{1}{b \cdot b + c \cdot \left(a \cdot -4\right)}}} + \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      11. unsub-neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right)\right), c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(\frac{1}{\sqrt{\frac{1}{b \cdot b + c \cdot \left(a \cdot -4\right)}}} - b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      12. --lowering--.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right)\right), c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{\sqrt{\frac{1}{b \cdot b + c \cdot \left(a \cdot -4\right)}}}\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   6. Applied egg-rr 90.0%

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

   if 2.4500000000000001e92 < b

   1. Initial program 50.2%

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

   3. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   4. Step-by-step derivation

   5. Simplified 94.6%

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

   6. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(-2 \cdot b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(b \cdot -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. *-lowering-*.f64 94.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   8. Simplified 94.6%

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

   10. Applied egg-rr 94.6%

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

       1. /-lowering-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\left(2 \cdot c\right), \color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) - b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       2. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\left(c \cdot 2\right), \left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} - b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} - b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       4. sub-neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(\left(\mathsf{neg}\left(b\right)\right) + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       5. neg-mul-1 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(-1 \cdot b + \left(\mathsf{neg}\left(\color{blue}{b}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       6. neg-mul-1 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(-1 \cdot b + -1 \cdot \color{blue}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       7. distribute-rgt-out N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(b \cdot \color{blue}{\left(-1 + -1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       8. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(b \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       9. *-lowering-*.f64 94.6%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \mathsf{*.f64}\left(b, \color{blue}{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   12. Applied egg-rr 94.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;0 \geq 0:\\ \;\;\;\;\color{blue}{\frac{c \cdot 2}{b \cdot -2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{+91}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;0 \cdot \left(c \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{+92}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;0 \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-27}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{0 - \left(b + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 370:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;0 \geq 0:\\ \;\;\;\;\frac{2}{\frac{b - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;0 \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.15e-27)
   (if (>= b 0.0) (/ (* c 2.0) (- 0.0 (+ b b))) (- (/ c b) (/ b a)))
   (if (<= b 370.0)
     (if (>= 0.0 0.0)
       (/ 2.0 (/ (- b (sqrt (+ (* b b) (* c (* a -4.0))))) c))
       (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a)))
     (if (>= 0.0 0.0) (/ (* c 2.0) (* b -2.0)) (/ (* b -2.0) (* 2.0 a))))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1.15e-27) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c * 2.0) / (0.0 - (b + b));
		} else {
			tmp_2 = (c / b) - (b / a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 370.0) {
		double tmp_3;
		if (0.0 >= 0.0) {
			tmp_3 = 2.0 / ((b - sqrt(((b * b) + (c * (a * -4.0))))) / c);
		} else {
			tmp_3 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (0.0 >= 0.0) {
		tmp_1 = (c * 2.0) / (b * -2.0);
	} else {
		tmp_1 = (b * -2.0) / (2.0 * a);
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    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
    real(8) :: tmp_3
    if (b <= (-1.15d-27)) then
        if (b >= 0.0d0) then
            tmp_2 = (c * 2.0d0) / (0.0d0 - (b + b))
        else
            tmp_2 = (c / b) - (b / a)
        end if
        tmp_1 = tmp_2
    else if (b <= 370.0d0) then
        if (0.0d0 >= 0.0d0) then
            tmp_3 = 2.0d0 / ((b - sqrt(((b * b) + (c * (a * (-4.0d0)))))) / c)
        else
            tmp_3 = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (2.0d0 * a)
        end if
        tmp_1 = tmp_3
    else if (0.0d0 >= 0.0d0) then
        tmp_1 = (c * 2.0d0) / (b * (-2.0d0))
    else
        tmp_1 = (b * (-2.0d0)) / (2.0d0 * a)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1.15e-27) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c * 2.0) / (0.0 - (b + b));
		} else {
			tmp_2 = (c / b) - (b / a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 370.0) {
		double tmp_3;
		if (0.0 >= 0.0) {
			tmp_3 = 2.0 / ((b - Math.sqrt(((b * b) + (c * (a * -4.0))))) / c);
		} else {
			tmp_3 = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (0.0 >= 0.0) {
		tmp_1 = (c * 2.0) / (b * -2.0);
	} else {
		tmp_1 = (b * -2.0) / (2.0 * a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if b <= -1.15e-27:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (c * 2.0) / (0.0 - (b + b))
		else:
			tmp_2 = (c / b) - (b / a)
		tmp_1 = tmp_2
	elif b <= 370.0:
		tmp_3 = 0
		if 0.0 >= 0.0:
			tmp_3 = 2.0 / ((b - math.sqrt(((b * b) + (c * (a * -4.0))))) / c)
		else:
			tmp_3 = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)
		tmp_1 = tmp_3
	elif 0.0 >= 0.0:
		tmp_1 = (c * 2.0) / (b * -2.0)
	else:
		tmp_1 = (b * -2.0) / (2.0 * a)
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -1.15e-27)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(c * 2.0) / Float64(0.0 - Float64(b + b)));
		else
			tmp_2 = Float64(Float64(c / b) - Float64(b / a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 370.0)
		tmp_3 = 0.0
		if (0.0 >= 0.0)
			tmp_3 = Float64(2.0 / Float64(Float64(b - sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0))))) / c));
		else
			tmp_3 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (0.0 >= 0.0)
		tmp_1 = Float64(Float64(c * 2.0) / Float64(b * -2.0));
	else
		tmp_1 = Float64(Float64(b * -2.0) / Float64(2.0 * a));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	tmp_2 = 0.0;
	if (b <= -1.15e-27)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (c * 2.0) / (0.0 - (b + b));
		else
			tmp_3 = (c / b) - (b / a);
		end
		tmp_2 = tmp_3;
	elseif (b <= 370.0)
		tmp_4 = 0.0;
		if (0.0 >= 0.0)
			tmp_4 = 2.0 / ((b - sqrt(((b * b) + (c * (a * -4.0))))) / c);
		else
			tmp_4 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
		end
		tmp_2 = tmp_4;
	elseif (0.0 >= 0.0)
		tmp_2 = (c * 2.0) / (b * -2.0);
	else
		tmp_2 = (b * -2.0) / (2.0 * a);
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.15e-27], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[(0.0 - N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 370.0], If[GreaterEqual[0.0, 0.0], N[(2.0 / N[(N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[0.0, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[(b * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(b * -2.0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.15e-27

   1. Initial program 72.2%

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

   3. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   4. Step-by-step derivation

   5. Simplified 72.2%

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

   6. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(\mathsf{neg}\left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right) \cdot b\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(\left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \left(-2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      6. associate-*r/ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{-2 \cdot \left(a \cdot c\right)}{{b}^{2}}\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(-2 \cdot \left(a \cdot c\right)\right), \left({b}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(a \cdot c\right)\right), \left({b}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      9. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(c \cdot a\right)\right), \left({b}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), \left({b}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      11. unpow2 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), \left(b \cdot b\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      13. neg-sub0 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(0 - b\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      14. --lowering--.f64 87.0%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{\_.f64}\left(0, b\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   8. Simplified 87.0%

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

   9. Taylor expanded in c around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a} + \frac{c}{b}\\ \end{array} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + -1 \cdot \frac{b}{a}\\ \end{array} \]

       2. mul-1-neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right)\\ \end{array} \]

       3. unsub-neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

       4. --lowering--.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(\left(\frac{c}{b}\right), \left(\frac{b}{a}\right)\right)\\ \end{array} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(\mathsf{/.f64}\left(c, b\right), \left(\frac{b}{a}\right)\right)\\ \end{array} \]

       6. /-lowering-/.f64 88.3%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(\mathsf{/.f64}\left(c, b\right), \mathsf{/.f64}\left(b, a\right)\right)\\ \end{array} \]

   11. Simplified 88.3%

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

   if -1.15e-27 < b < 370

   1. Initial program 86.9%

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

   4. Applied egg-rr 80.4%

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

   6. Applied egg-rr 71.1%

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

   if 370 < b

   1. Initial program 59.4%

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

   3. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   4. Step-by-step derivation

   5. Simplified 93.9%

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

   6. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(-2 \cdot b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(b \cdot -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. *-lowering-*.f64 93.9%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   8. Simplified 93.9%

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

   10. Applied egg-rr 93.9%

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

       1. /-lowering-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\left(2 \cdot c\right), \color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) - b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       2. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\left(c \cdot 2\right), \left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} - b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} - b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       4. sub-neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(\left(\mathsf{neg}\left(b\right)\right) + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       5. neg-mul-1 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(-1 \cdot b + \left(\mathsf{neg}\left(\color{blue}{b}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       6. neg-mul-1 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(-1 \cdot b + -1 \cdot \color{blue}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       7. distribute-rgt-out N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(b \cdot \color{blue}{\left(-1 + -1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       8. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(b \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       9. *-lowering-*.f64 93.9%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \mathsf{*.f64}\left(b, \color{blue}{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   12. Applied egg-rr 93.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;0 \geq 0:\\ \;\;\;\;\color{blue}{\frac{c \cdot 2}{b \cdot -2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-27}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{0 - \left(b + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 370:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;0 \geq 0:\\ \;\;\;\;\frac{2}{\frac{b - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;0 \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot 2}{0 - \left(b + b\right)}\\ \mathbf{if}\;b \leq -2.6 \cdot 10^{-25}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* c 2.0) (- 0.0 (+ b b)))))
   (if (<= b -2.6e-25)
     (if (>= b 0.0) t_0 (- (/ c b) (/ b a)))
     (if (>= b 0.0) t_0 (/ (/ (- (sqrt (* c (* a -4.0))) b) a) 2.0)))))

C

double code(double a, double b, double c) {
	double t_0 = (c * 2.0) / (0.0 - (b + b));
	double tmp_1;
	if (b <= -2.6e-25) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = (c / b) - (b / a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = ((sqrt((c * (a * -4.0))) - b) / a) / 2.0;
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    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 = (c * 2.0d0) / (0.0d0 - (b + b))
    if (b <= (-2.6d-25)) then
        if (b >= 0.0d0) then
            tmp_2 = t_0
        else
            tmp_2 = (c / b) - (b / a)
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = t_0
    else
        tmp_1 = ((sqrt((c * (a * (-4.0d0)))) - b) / a) / 2.0d0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (c * 2.0) / (0.0 - (b + b));
	double tmp_1;
	if (b <= -2.6e-25) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = (c / b) - (b / a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = ((Math.sqrt((c * (a * -4.0))) - b) / a) / 2.0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (c * 2.0) / (0.0 - (b + b))
	tmp_1 = 0
	if b <= -2.6e-25:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_0
		else:
			tmp_2 = (c / b) - (b / a)
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = t_0
	else:
		tmp_1 = ((math.sqrt((c * (a * -4.0))) - b) / a) / 2.0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * 2.0) / Float64(0.0 - Float64(b + b)))
	tmp_1 = 0.0
	if (b <= -2.6e-25)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(Float64(c / b) - Float64(b / a));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = t_0;
	else
		tmp_1 = Float64(Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / a) / 2.0);
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = (c * 2.0) / (0.0 - (b + b));
	tmp_2 = 0.0;
	if (b <= -2.6e-25)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_0;
		else
			tmp_3 = (c / b) - (b / a);
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = t_0;
	else
		tmp_2 = ((sqrt((c * (a * -4.0))) - b) / a) / 2.0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * 2.0), $MachinePrecision] / N[(0.0 - N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.6e-25], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.6e-25

   1. Initial program 72.2%

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

   3. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   4. Step-by-step derivation

   5. Simplified 72.2%

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

   6. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(\mathsf{neg}\left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right) \cdot b\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(\left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \left(-2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      6. associate-*r/ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{-2 \cdot \left(a \cdot c\right)}{{b}^{2}}\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(-2 \cdot \left(a \cdot c\right)\right), \left({b}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(a \cdot c\right)\right), \left({b}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      9. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(c \cdot a\right)\right), \left({b}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), \left({b}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      11. unpow2 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), \left(b \cdot b\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      13. neg-sub0 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(0 - b\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      14. --lowering--.f64 87.0%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{\_.f64}\left(0, b\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   8. Simplified 87.0%

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

   9. Taylor expanded in c around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a} + \frac{c}{b}\\ \end{array} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + -1 \cdot \frac{b}{a}\\ \end{array} \]

       2. mul-1-neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right)\\ \end{array} \]

       3. unsub-neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

       4. --lowering--.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(\left(\frac{c}{b}\right), \left(\frac{b}{a}\right)\right)\\ \end{array} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(\mathsf{/.f64}\left(c, b\right), \left(\frac{b}{a}\right)\right)\\ \end{array} \]

       6. /-lowering-/.f64 88.3%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(\mathsf{/.f64}\left(c, b\right), \mathsf{/.f64}\left(b, a\right)\right)\\ \end{array} \]

   11. Simplified 88.3%

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

   if -2.6e-25 < b

   1. Initial program 74.3%

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

   3. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   4. Step-by-step derivation

   5. Simplified 75.6%

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

   6. Taylor expanded in b around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\left(-4 \cdot \left(a \cdot c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\left(\left(a \cdot c\right) \cdot -4\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a \cdot c\right), -4\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(c \cdot a\right), -4\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      4. *-lowering-*.f64 71.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -4\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   8. Simplified 71.2%

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

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(c \cdot a\right) \cdot -4}}{a \cdot 2}\\ \end{array} \]

      2. associate-/r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(c \cdot a\right) \cdot -4}}{a}}{2}\\ \end{array} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(c \cdot a\right) \cdot -4}}{a}\right), 2\right)\\ \end{array} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(c \cdot a\right) \cdot -4}\right), a\right), 2\right)\\ \end{array} \]

      5. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{\left(c \cdot a\right) \cdot -4} + \left(\mathsf{neg}\left(b\right)\right)\right), a\right), 2\right)\\ \end{array} \]

      6. unsub-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{\left(c \cdot a\right) \cdot -4} - b\right), a\right), 2\right)\\ \end{array} \]

      7. --lowering--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{\left(c \cdot a\right) \cdot -4}\right), b\right), a\right), 2\right)\\ \end{array} \]

      8. associate-*r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{c \cdot \left(a \cdot -4\right)}\right), b\right), a\right), 2\right)\\ \end{array} \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(c \cdot \left(a \cdot -4\right)\right)\right), b\right), a\right), 2\right)\\ \end{array} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(a \cdot -4\right)\right)\right), b\right), a\right), 2\right)\\ \end{array} \]

      11. *-lowering-*.f64 71.2%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right), b\right), a\right), 2\right)\\ \end{array} \]

   10. Applied egg-rr 71.2%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-25}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{0 - \left(b + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{0 - \left(b + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot 2}{0 - \left(b + b\right)}\\ \mathbf{if}\;b \leq -2.4 \cdot 10^{-25}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{-4 \cdot \left(c \cdot a\right)}}{a}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* c 2.0) (- 0.0 (+ b b)))))
   (if (<= b -2.4e-25)
     (if (>= b 0.0) t_0 (- (/ c b) (/ b a)))
     (if (>= b 0.0) t_0 (/ (/ (sqrt (* -4.0 (* c a))) a) 2.0)))))

C

double code(double a, double b, double c) {
	double t_0 = (c * 2.0) / (0.0 - (b + b));
	double tmp_1;
	if (b <= -2.4e-25) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = (c / b) - (b / a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = (sqrt((-4.0 * (c * a))) / a) / 2.0;
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    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 = (c * 2.0d0) / (0.0d0 - (b + b))
    if (b <= (-2.4d-25)) then
        if (b >= 0.0d0) then
            tmp_2 = t_0
        else
            tmp_2 = (c / b) - (b / a)
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = t_0
    else
        tmp_1 = (sqrt(((-4.0d0) * (c * a))) / a) / 2.0d0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (c * 2.0) / (0.0 - (b + b));
	double tmp_1;
	if (b <= -2.4e-25) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = (c / b) - (b / a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = (Math.sqrt((-4.0 * (c * a))) / a) / 2.0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (c * 2.0) / (0.0 - (b + b))
	tmp_1 = 0
	if b <= -2.4e-25:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_0
		else:
			tmp_2 = (c / b) - (b / a)
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = t_0
	else:
		tmp_1 = (math.sqrt((-4.0 * (c * a))) / a) / 2.0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * 2.0) / Float64(0.0 - Float64(b + b)))
	tmp_1 = 0.0
	if (b <= -2.4e-25)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(Float64(c / b) - Float64(b / a));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = t_0;
	else
		tmp_1 = Float64(Float64(sqrt(Float64(-4.0 * Float64(c * a))) / a) / 2.0);
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = (c * 2.0) / (0.0 - (b + b));
	tmp_2 = 0.0;
	if (b <= -2.4e-25)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_0;
		else
			tmp_3 = (c / b) - (b / a);
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = t_0;
	else
		tmp_2 = (sqrt((-4.0 * (c * a))) / a) / 2.0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * 2.0), $MachinePrecision] / N[(0.0 - N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.4e-25], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.40000000000000009e-25

   1. Initial program 72.2%

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

   3. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   4. Step-by-step derivation

   5. Simplified 72.2%

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

   6. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(\mathsf{neg}\left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right) \cdot b\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(\left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \left(-2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      6. associate-*r/ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{-2 \cdot \left(a \cdot c\right)}{{b}^{2}}\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(-2 \cdot \left(a \cdot c\right)\right), \left({b}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(a \cdot c\right)\right), \left({b}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      9. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(c \cdot a\right)\right), \left({b}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), \left({b}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      11. unpow2 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), \left(b \cdot b\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      13. neg-sub0 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(0 - b\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      14. --lowering--.f64 87.0%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{\_.f64}\left(0, b\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   8. Simplified 87.0%

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

   9. Taylor expanded in c around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a} + \frac{c}{b}\\ \end{array} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + -1 \cdot \frac{b}{a}\\ \end{array} \]

       2. mul-1-neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right)\\ \end{array} \]

       3. unsub-neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

       4. --lowering--.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(\left(\frac{c}{b}\right), \left(\frac{b}{a}\right)\right)\\ \end{array} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(\mathsf{/.f64}\left(c, b\right), \left(\frac{b}{a}\right)\right)\\ \end{array} \]

       6. /-lowering-/.f64 88.3%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(\mathsf{/.f64}\left(c, b\right), \mathsf{/.f64}\left(b, a\right)\right)\\ \end{array} \]

   11. Simplified 88.3%

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

   if -2.40000000000000009e-25 < b

   1. Initial program 74.3%

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

   3. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   4. Step-by-step derivation

   5. Simplified 75.6%

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

      1. pow1/2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \left({\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{\frac{1}{2}}\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. sub-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \left({\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)}^{\frac{1}{2}}\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \left({\left(b \cdot b + \left(\mathsf{neg}\left(c \cdot \left(4 \cdot a\right)\right)\right)\right)}^{\frac{1}{2}}\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \left({\left(b \cdot b + c \cdot \left(\mathsf{neg}\left(4 \cdot a\right)\right)\right)}^{\frac{1}{2}}\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \left({\left(b \cdot b + c \cdot \left(\left(\mathsf{neg}\left(4\right)\right) \cdot a\right)\right)}^{\frac{1}{2}}\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      6. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \left({\left(b \cdot b + c \cdot \left(-4 \cdot a\right)\right)}^{\frac{1}{2}}\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      7. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \left({\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)}^{\frac{1}{2}}\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      8. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \left({\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)}^{\left(-1 \cdot \frac{-1}{2}\right)}\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      9. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \left({\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)}^{\left(-1 \cdot \left(\frac{1}{2} \cdot -1\right)\right)}\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      10. pow-pow N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \left({\left({\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)}^{-1}\right)}^{\left(\frac{1}{2} \cdot -1\right)}\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      11. inv-pow N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \left({\left(\frac{1}{b \cdot b + c \cdot \left(a \cdot -4\right)}\right)}^{\left(\frac{1}{2} \cdot -1\right)}\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{pow.f64}\left(\left(\frac{1}{b \cdot b + c \cdot \left(a \cdot -4\right)}\right), \left(\frac{1}{2} \cdot -1\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)\right), \left(\frac{1}{2} \cdot -1\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(b \cdot b\right), \left(c \cdot \left(a \cdot -4\right)\right)\right)\right), \left(\frac{1}{2} \cdot -1\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(c \cdot \left(a \cdot -4\right)\right)\right)\right), \left(\frac{1}{2} \cdot -1\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(a \cdot -4\right)\right)\right)\right), \left(\frac{1}{2} \cdot -1\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right), \left(\frac{1}{2} \cdot -1\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      18. metadata-eval 75.2%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   7. Applied egg-rr 75.2%

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

   9. Applied egg-rr 70.5%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-25}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{0 - \left(b + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{0 - \left(b + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{-4 \cdot \left(c \cdot a\right)}}{a}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.4% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot -2}{2 \cdot a}\\ \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;0 \cdot \left(c \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;0 \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* b -2.0) (* 2.0 a))))
   (if (<= b -5e-310)
     (if (>= b 0.0) (* 0.0 (* c -2.0)) t_0)
     (if (>= 0.0 0.0) (/ (* c 2.0) (* b -2.0)) t_0))))

C

double code(double a, double b, double c) {
	double t_0 = (b * -2.0) / (2.0 * a);
	double tmp_1;
	if (b <= -5e-310) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 0.0 * (c * -2.0);
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (0.0 >= 0.0) {
		tmp_1 = (c * 2.0) / (b * -2.0);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    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 = (b * (-2.0d0)) / (2.0d0 * a)
    if (b <= (-5d-310)) then
        if (b >= 0.0d0) then
            tmp_2 = 0.0d0 * (c * (-2.0d0))
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (0.0d0 >= 0.0d0) then
        tmp_1 = (c * 2.0d0) / (b * (-2.0d0))
    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 = (b * -2.0) / (2.0 * a);
	double tmp_1;
	if (b <= -5e-310) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 0.0 * (c * -2.0);
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (0.0 >= 0.0) {
		tmp_1 = (c * 2.0) / (b * -2.0);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (b * -2.0) / (2.0 * a)
	tmp_1 = 0
	if b <= -5e-310:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = 0.0 * (c * -2.0)
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif 0.0 >= 0.0:
		tmp_1 = (c * 2.0) / (b * -2.0)
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * -2.0) / Float64(2.0 * a))
	tmp_1 = 0.0
	if (b <= -5e-310)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(0.0 * Float64(c * -2.0));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (0.0 >= 0.0)
		tmp_1 = Float64(Float64(c * 2.0) / Float64(b * -2.0));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = (b * -2.0) / (2.0 * a);
	tmp_2 = 0.0;
	if (b <= -5e-310)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = 0.0 * (c * -2.0);
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (0.0 >= 0.0)
		tmp_2 = (c * 2.0) / (b * -2.0);
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * -2.0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -5e-310], If[GreaterEqual[b, 0.0], N[(0.0 * N[(c * -2.0), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[0.0, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[(b * -2.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 77.9%

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

   3. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   4. Step-by-step derivation

   5. Simplified 77.9%

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

   6. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(-2 \cdot b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(b \cdot -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. *-lowering-*.f64 60.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   8. Simplified 60.6%

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

   10. Applied egg-rr 60.6%

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

   if -4.999999999999985e-310 < b

   1. Initial program 69.0%

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

   3. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   4. Step-by-step derivation

   5. Simplified 70.9%

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

   6. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(-2 \cdot b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(b \cdot -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. *-lowering-*.f64 70.9%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   8. Simplified 70.9%

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

   10. Applied egg-rr 70.9%

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

       1. /-lowering-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\left(2 \cdot c\right), \color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) - b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       2. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\left(c \cdot 2\right), \left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} - b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} - b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       4. sub-neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(\left(\mathsf{neg}\left(b\right)\right) + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       5. neg-mul-1 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(-1 \cdot b + \left(\mathsf{neg}\left(\color{blue}{b}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       6. neg-mul-1 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(-1 \cdot b + -1 \cdot \color{blue}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       7. distribute-rgt-out N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(b \cdot \color{blue}{\left(-1 + -1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       8. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(b \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       9. *-lowering-*.f64 70.9%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \mathsf{*.f64}\left(b, \color{blue}{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   12. Applied egg-rr 70.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;0 \geq 0:\\ \;\;\;\;\color{blue}{\frac{c \cdot 2}{b \cdot -2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;0 \cdot \left(c \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;0 \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.6% accurate, 8.6× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (c * 2.0) / (0.0 - (b + b));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    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) / (0.0d0 - (b + b))
    else
        tmp = (c / b) - (b / 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) / (0.0 - (b + b));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.7%

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

3. Taylor expanded in b around inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
4. Step-by-step derivation

5. Simplified 74.6%

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

6. Taylor expanded in b around -inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(\mathsf{neg}\left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   2. *-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right) \cdot b\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

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

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(\left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right) \cdot \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \left(-2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   6. associate-*r/ N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{-2 \cdot \left(a \cdot c\right)}{{b}^{2}}\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(-2 \cdot \left(a \cdot c\right)\right), \left({b}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(a \cdot c\right)\right), \left({b}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   9. *-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(c \cdot a\right)\right), \left({b}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), \left({b}^{2}\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   11. unpow2 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), \left(b \cdot b\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(\mathsf{neg}\left(b\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   13. neg-sub0 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \left(0 - b\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   14. --lowering--.f64 64.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{\_.f64}\left(0, b\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

8. Simplified 64.7%

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

9. Taylor expanded in c around 0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a} + \frac{c}{b}\\ \end{array} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + -1 \cdot \frac{b}{a}\\ \end{array} \]

    2. mul-1-neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right)\\ \end{array} \]

    3. unsub-neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

    4. --lowering--.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(\left(\frac{c}{b}\right), \left(\frac{b}{a}\right)\right)\\ \end{array} \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(\mathsf{/.f64}\left(c, b\right), \left(\frac{b}{a}\right)\right)\\ \end{array} \]

    6. /-lowering-/.f64 65.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(\mathsf{/.f64}\left(c, b\right), \mathsf{/.f64}\left(b, a\right)\right)\\ \end{array} \]

11. Simplified 65.8%

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

12. Final simplification 65.8%

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

Alternative 9: 35.1% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0 \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= 0.0 0.0) (/ (* c 2.0) (* b -2.0)) (/ (* b -2.0) (* 2.0 a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (0.0 >= 0.0) {
		tmp = (c * 2.0) / (b * -2.0);
	} else {
		tmp = (b * -2.0) / (2.0 * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (0.0d0 >= 0.0d0) then
        tmp = (c * 2.0d0) / (b * (-2.0d0))
    else
        tmp = (b * (-2.0d0)) / (2.0d0 * a)
    end if
    code = tmp
end function

Java

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

Python

def code(a, b, c):
	tmp = 0
	if 0.0 >= 0.0:
		tmp = (c * 2.0) / (b * -2.0)
	else:
		tmp = (b * -2.0) / (2.0 * a)
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.7%

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

3. Taylor expanded in b around inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
4. Step-by-step derivation

5. Simplified 74.6%

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

6. Taylor expanded in b around -inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(-2 \cdot b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(b \cdot -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   2. *-lowering-*.f64 65.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

8. Simplified 65.5%

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

10. Applied egg-rr 35.0%

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

    1. /-lowering-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\left(2 \cdot c\right), \color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) - b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

    2. *-commutative N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\left(c \cdot 2\right), \left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} - b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} - b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

    4. sub-neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(\left(\mathsf{neg}\left(b\right)\right) + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

    5. neg-mul-1 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(-1 \cdot b + \left(\mathsf{neg}\left(\color{blue}{b}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

    6. neg-mul-1 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(-1 \cdot b + -1 \cdot \color{blue}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

    7. distribute-rgt-out N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(b \cdot \color{blue}{\left(-1 + -1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

    8. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(b \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

    9. *-lowering-*.f64 35.0%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \mathsf{*.f64}\left(b, \color{blue}{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

12. Applied egg-rr 35.0%

    \[\leadsto \begin{array}{l} \mathbf{if}\;0 \geq 0:\\ \;\;\;\;\color{blue}{\frac{c \cdot 2}{b \cdot -2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \]
13. Add Preprocessing

Alternative 10: 35.0% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0 \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= 0.0 0.0) (* c (/ 2.0 (* b -2.0))) (/ (* b -2.0) (* 2.0 a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (0.0 >= 0.0) {
		tmp = c * (2.0 / (b * -2.0));
	} else {
		tmp = (b * -2.0) / (2.0 * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (0.0d0 >= 0.0d0) then
        tmp = c * (2.0d0 / (b * (-2.0d0)))
    else
        tmp = (b * (-2.0d0)) / (2.0d0 * a)
    end if
    code = tmp
end function

Java

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

Python

def code(a, b, c):
	tmp = 0
	if 0.0 >= 0.0:
		tmp = c * (2.0 / (b * -2.0))
	else:
		tmp = (b * -2.0) / (2.0 * a)
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.7%

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

3. Taylor expanded in b around inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
4. Step-by-step derivation

5. Simplified 74.6%

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

6. Taylor expanded in b around -inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(-2 \cdot b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(b \cdot -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   2. *-lowering-*.f64 65.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

8. Simplified 65.5%

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

10. Applied egg-rr 35.0%

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

    1. *-commutative N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\frac{c \cdot 2}{\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} - b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

    2. associate-/l* N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;c \cdot \color{blue}{\frac{2}{\left(\mathsf{neg}\left(b\right)\right) - b}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(c, \color{blue}{\left(\frac{2}{\left(\mathsf{neg}\left(b\right)\right) - b}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(2, \color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) - b\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

    5. sub-neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(2, \left(\left(\mathsf{neg}\left(b\right)\right) + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

    6. neg-mul-1 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(2, \left(-1 \cdot b + \left(\mathsf{neg}\left(\color{blue}{b}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

    7. neg-mul-1 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(2, \left(-1 \cdot b + -1 \cdot \color{blue}{b}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

    8. distribute-rgt-out N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(2, \left(b \cdot \color{blue}{\left(-1 + -1\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

    9. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(2, \left(b \cdot -2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

    10. *-lowering-*.f64 34.9%

        \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f32}\left(0, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{-2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

12. Applied egg-rr 34.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;0 \geq 0:\\ \;\;\;\;\color{blue}{c \cdot \frac{2}{b \cdot -2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(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))))