jeff quadratic root 1

Percentage Accurate: 72.2% → 91.1%

Time: 21.9s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

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 = (-b - t_0) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + t_0);
	}
	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 = (-b - t_0) / (2.0d0 * a)
    else
        tmp = (2.0d0 * c) / (-b + t_0)
    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 = (-b - t_0) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + t_0);
	}
	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 = (-b - t_0) / (2.0 * a)
	else:
		tmp = (2.0 * c) / (-b + t_0)
	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(Float64(-b) - t_0) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_0));
	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 = (-b - t_0) / (2.0 * a);
	else
		tmp = (2.0 * c) / (-b + t_0);
	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[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Initial Program: 72.2% accurate, 1.0× speedup

Math

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

FPCore

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

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 = (-b - t_0) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + t_0);
	}
	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 = (-b - t_0) / (2.0d0 * a)
    else
        tmp = (2.0d0 * c) / (-b + t_0)
    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 = (-b - t_0) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + t_0);
	}
	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 = (-b - t_0) / (2.0 * a)
	else:
		tmp = (2.0 * c) / (-b + t_0)
	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(Float64(-b) - t_0) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_0));
	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 = (-b - t_0) / (2.0 * a);
	else
		tmp = (2.0 * c) / (-b + t_0);
	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[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 91.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\\ t_1 := \frac{-c}{b}\\ \mathbf{if}\;b \leq -3 \cdot 10^{+136}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{+116}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* 4.0 a))))) (t_1 (/ (- c) b)))
   (if (<= b -3e+136)
     (if (>= b 0.0) (/ c b) t_1)
     (if (<= b 8.2e+116)
       (if (>= b 0.0) (/ (- (- b) t_0) (* a 2.0)) (/ (* c 2.0) (- t_0 b)))
       (if (>= b 0.0) (- (/ b a)) t_1)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (4.0 * a))));
	double t_1 = -c / b;
	double tmp_1;
	if (b <= -3e+136) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c / b;
		} else {
			tmp_2 = t_1;
		}
		tmp_1 = tmp_2;
	} else if (b <= 8.2e+116) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_0) / (a * 2.0);
		} else {
			tmp_3 = (c * 2.0) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -(b / a);
	} 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 * (4.0d0 * a))))
    t_1 = -c / b
    if (b <= (-3d+136)) then
        if (b >= 0.0d0) then
            tmp_2 = c / b
        else
            tmp_2 = t_1
        end if
        tmp_1 = tmp_2
    else if (b <= 8.2d+116) then
        if (b >= 0.0d0) then
            tmp_3 = (-b - t_0) / (a * 2.0d0)
        else
            tmp_3 = (c * 2.0d0) / (t_0 - b)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = -(b / a)
    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 * (4.0 * a))));
	double t_1 = -c / b;
	double tmp_1;
	if (b <= -3e+136) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c / b;
		} else {
			tmp_2 = t_1;
		}
		tmp_1 = tmp_2;
	} else if (b <= 8.2e+116) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_0) / (a * 2.0);
		} else {
			tmp_3 = (c * 2.0) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -(b / a);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (c * (4.0 * a))))
	t_1 = -c / b
	tmp_1 = 0
	if b <= -3e+136:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = c / b
		else:
			tmp_2 = t_1
		tmp_1 = tmp_2
	elif b <= 8.2e+116:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (-b - t_0) / (a * 2.0)
		else:
			tmp_3 = (c * 2.0) / (t_0 - b)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = -(b / a)
	else:
		tmp_1 = t_1
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a))))
	t_1 = Float64(Float64(-c) / b)
	tmp_1 = 0.0
	if (b <= -3e+136)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c / b);
		else
			tmp_2 = t_1;
		end
		tmp_1 = tmp_2;
	elseif (b <= 8.2e+116)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(-b) - t_0) / Float64(a * 2.0));
		else
			tmp_3 = Float64(Float64(c * 2.0) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(-Float64(b / a));
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) - (c * (4.0 * a))));
	t_1 = -c / b;
	tmp_2 = 0.0;
	if (b <= -3e+136)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = c / b;
		else
			tmp_3 = t_1;
		end
		tmp_2 = tmp_3;
	elseif (b <= 8.2e+116)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (-b - t_0) / (a * 2.0);
		else
			tmp_4 = (c * 2.0) / (t_0 - b);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = -(b / a);
	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[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-c) / b), $MachinePrecision]}, If[LessEqual[b, -3e+136], If[GreaterEqual[b, 0.0], N[(c / b), $MachinePrecision], t$95$1], If[LessEqual[b, 8.2e+116], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], (-N[(b / a), $MachinePrecision]), t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.99999999999999979e136

   1. Initial program 43.9%

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

   2. Taylor expanded in b around -inf 95.6%

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

      1. associate-*r/ 95.6%

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

      2. neg-mul-1 95.6%

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

   4. Simplified 95.6%

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

   5. Taylor expanded in b around inf 95.6%

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

   6. Taylor expanded in b around 0 95.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

   if -2.99999999999999979e136 < b < 8.1999999999999996e116

   1. Initial program 84.3%

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

   if 8.1999999999999996e116 < b

   1. Initial program 43.7%

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

   2. Taylor expanded in b around -inf 43.7%

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

      1. associate-*r/ 43.7%

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

      2. neg-mul-1 43.7%

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

   4. Simplified 43.7%

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

   5. Taylor expanded in b around inf 94.7%

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

   6. Taylor expanded in b around inf 98.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
   7. Step-by-step derivation

      1. associate-*r/ 98.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-1 \cdot b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

      2. mul-1-neg 98.3%

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

   8. Simplified 98.3%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{+136}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{+116}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2: 90.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\\ t_1 := \frac{-c}{b}\\ \mathbf{if}\;b \leq -1.3 \cdot 10^{+85}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{+117}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t_0 - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* 4.0 (* c a))))) (t_1 (/ (- c) b)))
   (if (<= b -1.3e+85)
     (if (>= b 0.0) (/ c b) t_1)
     (if (<= b 1e+117)
       (if (>= b 0.0) (/ (- (- b) t_0) (* a 2.0)) (/ 2.0 (/ (- t_0 b) c)))
       (if (>= b 0.0) (- (/ b a)) t_1)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (4.0 * (c * a))));
	double t_1 = -c / b;
	double tmp_1;
	if (b <= -1.3e+85) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c / b;
		} else {
			tmp_2 = t_1;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1e+117) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_0) / (a * 2.0);
		} else {
			tmp_3 = 2.0 / ((t_0 - b) / c);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -(b / a);
	} 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) - (4.0d0 * (c * a))))
    t_1 = -c / b
    if (b <= (-1.3d+85)) then
        if (b >= 0.0d0) then
            tmp_2 = c / b
        else
            tmp_2 = t_1
        end if
        tmp_1 = tmp_2
    else if (b <= 1d+117) then
        if (b >= 0.0d0) then
            tmp_3 = (-b - t_0) / (a * 2.0d0)
        else
            tmp_3 = 2.0d0 / ((t_0 - b) / c)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = -(b / a)
    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) - (4.0 * (c * a))));
	double t_1 = -c / b;
	double tmp_1;
	if (b <= -1.3e+85) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c / b;
		} else {
			tmp_2 = t_1;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1e+117) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_0) / (a * 2.0);
		} else {
			tmp_3 = 2.0 / ((t_0 - b) / c);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -(b / a);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (4.0 * (c * a))))
	t_1 = -c / b
	tmp_1 = 0
	if b <= -1.3e+85:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = c / b
		else:
			tmp_2 = t_1
		tmp_1 = tmp_2
	elif b <= 1e+117:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (-b - t_0) / (a * 2.0)
		else:
			tmp_3 = 2.0 / ((t_0 - b) / c)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = -(b / a)
	else:
		tmp_1 = t_1
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a))))
	t_1 = Float64(Float64(-c) / b)
	tmp_1 = 0.0
	if (b <= -1.3e+85)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c / b);
		else
			tmp_2 = t_1;
		end
		tmp_1 = tmp_2;
	elseif (b <= 1e+117)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(-b) - t_0) / Float64(a * 2.0));
		else
			tmp_3 = Float64(2.0 / Float64(Float64(t_0 - b) / c));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(-Float64(b / a));
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) - (4.0 * (c * a))));
	t_1 = -c / b;
	tmp_2 = 0.0;
	if (b <= -1.3e+85)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = c / b;
		else
			tmp_3 = t_1;
		end
		tmp_2 = tmp_3;
	elseif (b <= 1e+117)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (-b - t_0) / (a * 2.0);
		else
			tmp_4 = 2.0 / ((t_0 - b) / c);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = -(b / a);
	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[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-c) / b), $MachinePrecision]}, If[LessEqual[b, -1.3e+85], If[GreaterEqual[b, 0.0], N[(c / b), $MachinePrecision], t$95$1], If[LessEqual[b, 1e+117], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$0 - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], (-N[(b / a), $MachinePrecision]), t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.30000000000000005e85

   1. Initial program 51.6%

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

   2. Taylor expanded in b around -inf 96.2%

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

      1. associate-*r/ 96.2%

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

      2. neg-mul-1 96.2%

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

   4. Simplified 96.2%

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

   5. Taylor expanded in b around inf 96.2%

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

   6. Taylor expanded in b around 0 96.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

   if -1.30000000000000005e85 < b < 1.00000000000000005e117

   1. Initial program 83.6%

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

   3. Simplified 83.5%

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

   if 1.00000000000000005e117 < b

   1. Initial program 43.7%

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

   2. Taylor expanded in b around -inf 43.7%

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

      1. associate-*r/ 43.7%

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

      2. neg-mul-1 43.7%

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

   4. Simplified 43.7%

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

   5. Taylor expanded in b around inf 94.7%

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

   6. Taylor expanded in b around inf 98.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
   7. Step-by-step derivation

      1. associate-*r/ 98.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-1 \cdot b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

      2. mul-1-neg 98.3%

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

   8. Simplified 98.3%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{+85}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{+117}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-c}{b}\\ \mathbf{if}\;b \leq -1.95 \cdot 10^{+77}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-\left(b + b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-39}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- c) b)))
   (if (<= b -1.95e+77)
     (if (>= b 0.0) (/ c b) t_0)
     (if (<= b -2e-310)
       (if (>= b 0.0)
         (/ (- (+ b b)) (* a 2.0))
         (/ 2.0 (/ (- (sqrt (- (* b b) (* 4.0 (* c a)))) b) c)))
       (if (<= b 5.5e-39)
         (if (>= b 0.0)
           (/ (- (- b) (sqrt (* c (* a -4.0)))) (* a 2.0))
           (/ b a))
         (if (>= b 0.0) (- (/ c b) (/ b a)) t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = -c / b;
	double tmp_1;
	if (b <= -1.95e+77) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c / b;
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -(b + b) / (a * 2.0);
		} else {
			tmp_3 = 2.0 / ((sqrt(((b * b) - (4.0 * (c * a)))) - b) / c);
		}
		tmp_1 = tmp_3;
	} else if (b <= 5.5e-39) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (-b - sqrt((c * (a * -4.0)))) / (a * 2.0);
		} else {
			tmp_4 = b / a;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} 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
    real(8) :: tmp_4
    t_0 = -c / b
    if (b <= (-1.95d+77)) then
        if (b >= 0.0d0) then
            tmp_2 = c / b
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b <= (-2d-310)) then
        if (b >= 0.0d0) then
            tmp_3 = -(b + b) / (a * 2.0d0)
        else
            tmp_3 = 2.0d0 / ((sqrt(((b * b) - (4.0d0 * (c * a)))) - b) / c)
        end if
        tmp_1 = tmp_3
    else if (b <= 5.5d-39) then
        if (b >= 0.0d0) then
            tmp_4 = (-b - sqrt((c * (a * (-4.0d0))))) / (a * 2.0d0)
        else
            tmp_4 = b / a
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = (c / b) - (b / a)
    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 = -c / b;
	double tmp_1;
	if (b <= -1.95e+77) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c / b;
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -(b + b) / (a * 2.0);
		} else {
			tmp_3 = 2.0 / ((Math.sqrt(((b * b) - (4.0 * (c * a)))) - b) / c);
		}
		tmp_1 = tmp_3;
	} else if (b <= 5.5e-39) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (-b - Math.sqrt((c * (a * -4.0)))) / (a * 2.0);
		} else {
			tmp_4 = b / a;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = -c / b
	tmp_1 = 0
	if b <= -1.95e+77:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = c / b
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b <= -2e-310:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = -(b + b) / (a * 2.0)
		else:
			tmp_3 = 2.0 / ((math.sqrt(((b * b) - (4.0 * (c * a)))) - b) / c)
		tmp_1 = tmp_3
	elif b <= 5.5e-39:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = (-b - math.sqrt((c * (a * -4.0)))) / (a * 2.0)
		else:
			tmp_4 = b / a
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = (c / b) - (b / a)
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-c) / b)
	tmp_1 = 0.0
	if (b <= -1.95e+77)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c / b);
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b <= -2e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-Float64(b + b)) / Float64(a * 2.0));
		else
			tmp_3 = Float64(2.0 / Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a)))) - b) / c));
		end
		tmp_1 = tmp_3;
	elseif (b <= 5.5e-39)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(Float64(-b) - sqrt(Float64(c * Float64(a * -4.0)))) / Float64(a * 2.0));
		else
			tmp_4 = Float64(b / a);
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = -c / b;
	tmp_2 = 0.0;
	if (b <= -1.95e+77)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = c / b;
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b <= -2e-310)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = -(b + b) / (a * 2.0);
		else
			tmp_4 = 2.0 / ((sqrt(((b * b) - (4.0 * (c * a)))) - b) / c);
		end
		tmp_2 = tmp_4;
	elseif (b <= 5.5e-39)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = (-b - sqrt((c * (a * -4.0)))) / (a * 2.0);
		else
			tmp_5 = b / a;
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = (c / b) - (b / a);
	else
		tmp_2 = t_0;
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-c) / b), $MachinePrecision]}, If[LessEqual[b, -1.95e+77], If[GreaterEqual[b, 0.0], N[(c / b), $MachinePrecision], t$95$0], If[LessEqual[b, -2e-310], If[GreaterEqual[b, 0.0], N[((-N[(b + b), $MachinePrecision]) / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 5.5e-39], If[GreaterEqual[b, 0.0], N[(N[((-b) - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(b / a), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.9499999999999999e77

   1. Initial program 51.6%

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

   2. Taylor expanded in b around -inf 96.2%

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

      1. associate-*r/ 96.2%

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

      2. neg-mul-1 96.2%

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

   4. Simplified 96.2%

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

   5. Taylor expanded in b around inf 96.2%

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

   6. Taylor expanded in b around 0 96.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

   if -1.9499999999999999e77 < b < -1.999999999999994e-310

   1. Initial program 81.9%

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

   3. Simplified 81.7%

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

   4. Taylor expanded in b around inf 81.7%

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

   if -1.999999999999994e-310 < b < 5.50000000000000018e-39

   1. Initial program 82.3%

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

   3. Simplified 82.3%

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

   4. Taylor expanded in b around -inf 82.3%

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

      1. +-commutative 82.3%

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

      2. fma-def 82.3%

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

   6. Simplified 82.3%

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

   7. Taylor expanded in b around 0 73.1%

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

      1. associate-*r* 73.1%

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

      2. metadata-eval 73.1%

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

      3. distribute-lft-neg-in 73.1%

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

      4. *-commutative 73.1%

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

      5. distribute-lft-neg-in 73.1%

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

      6. metadata-eval 73.1%

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

   9. Simplified 73.1%

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

   10. Taylor expanded in a around inf 73.1%

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

   if 5.50000000000000018e-39 < b

   1. Initial program 62.1%

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

   2. Taylor expanded in b around -inf 62.1%

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

      1. associate-*r/ 62.1%

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

      2. neg-mul-1 62.1%

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

   4. Simplified 62.1%

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

   5. Taylor expanded in b around inf 86.2%

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

   6. Taylor expanded in b around 0 88.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
   7. Step-by-step derivation

      1. +-commutative 88.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

      2. mul-1-neg 88.4%

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

      3. unsub-neg 88.4%

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

   8. Simplified 88.4%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.95 \cdot 10^{+77}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-\left(b + b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-39}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{c \cdot \left(a \cdot -4\right)}\\ t_1 := \frac{-c}{b}\\ \mathbf{if}\;b \leq -2.15 \cdot 10^{-44}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-\left(b + b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t_0 - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-33}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \frac{a}{b}, -2 \cdot \frac{b}{c}\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* c (* a -4.0)))) (t_1 (/ (- c) b)))
   (if (<= b -2.15e-44)
     (if (>= b 0.0) (/ c b) t_1)
     (if (<= b -2e-310)
       (if (>= b 0.0) (/ (- (+ b b)) (* a 2.0)) (/ 2.0 (/ (- t_0 b) c)))
       (if (<= b 1.1e-33)
         (if (>= b 0.0)
           (/ (- (- b) t_0) (* a 2.0))
           (/ 2.0 (fma 2.0 (/ a b) (* -2.0 (/ b c)))))
         (if (>= b 0.0) (- (/ c b) (/ b a)) t_1))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((c * (a * -4.0)));
	double t_1 = -c / b;
	double tmp_1;
	if (b <= -2.15e-44) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c / b;
		} else {
			tmp_2 = t_1;
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -(b + b) / (a * 2.0);
		} else {
			tmp_3 = 2.0 / ((t_0 - b) / c);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.1e-33) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (-b - t_0) / (a * 2.0);
		} else {
			tmp_4 = 2.0 / fma(2.0, (a / b), (-2.0 * (b / c)));
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(c * Float64(a * -4.0)))
	t_1 = Float64(Float64(-c) / b)
	tmp_1 = 0.0
	if (b <= -2.15e-44)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c / b);
		else
			tmp_2 = t_1;
		end
		tmp_1 = tmp_2;
	elseif (b <= -2e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-Float64(b + b)) / Float64(a * 2.0));
		else
			tmp_3 = Float64(2.0 / Float64(Float64(t_0 - b) / c));
		end
		tmp_1 = tmp_3;
	elseif (b <= 1.1e-33)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(Float64(-b) - t_0) / Float64(a * 2.0));
		else
			tmp_4 = Float64(2.0 / fma(2.0, Float64(a / b), Float64(-2.0 * Float64(b / c))));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-c) / b), $MachinePrecision]}, If[LessEqual[b, -2.15e-44], If[GreaterEqual[b, 0.0], N[(c / b), $MachinePrecision], t$95$1], If[LessEqual[b, -2e-310], If[GreaterEqual[b, 0.0], N[((-N[(b + b), $MachinePrecision]) / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$0 - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.1e-33], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(a / b), $MachinePrecision] + N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.15000000000000007e-44

   1. Initial program 62.7%

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

   2. Taylor expanded in b around -inf 87.2%

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

      1. associate-*r/ 87.2%

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

      2. neg-mul-1 87.2%

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

   4. Simplified 87.2%

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

   5. Taylor expanded in b around inf 87.2%

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

   6. Taylor expanded in b around 0 87.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

   if -2.15000000000000007e-44 < b < -1.999999999999994e-310

   1. Initial program 79.3%

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

   3. Simplified 79.2%

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

   4. Taylor expanded in b around inf 79.2%

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

   5. Taylor expanded in b around 0 68.5%

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

      1. associate-*r* 23.6%

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

      2. metadata-eval 23.6%

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

      3. distribute-lft-neg-in 23.6%

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

      4. *-commutative 23.6%

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

      5. distribute-lft-neg-in 23.6%

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

      6. metadata-eval 23.6%

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

   7. Simplified 68.5%

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

   if -1.999999999999994e-310 < b < 1.10000000000000003e-33

   1. Initial program 82.3%

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

   3. Simplified 82.3%

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

   4. Taylor expanded in b around -inf 82.3%

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

      1. +-commutative 82.3%

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

      2. fma-def 82.3%

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

   6. Simplified 82.3%

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

   7. Taylor expanded in b around 0 73.1%

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

      1. associate-*r* 73.1%

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

      2. metadata-eval 73.1%

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

      3. distribute-lft-neg-in 73.1%

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

      4. *-commutative 73.1%

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

      5. distribute-lft-neg-in 73.1%

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

      6. metadata-eval 73.1%

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

   9. Simplified 73.1%

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

   if 1.10000000000000003e-33 < b

   1. Initial program 62.1%

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

   2. Taylor expanded in b around -inf 62.1%

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

      1. associate-*r/ 62.1%

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

      2. neg-mul-1 62.1%

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

   4. Simplified 62.1%

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

   5. Taylor expanded in b around inf 86.2%

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

   6. Taylor expanded in b around 0 88.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
   7. Step-by-step derivation

      1. +-commutative 88.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

      2. mul-1-neg 88.4%

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

      3. unsub-neg 88.4%

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

   8. Simplified 88.4%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.15 \cdot 10^{-44}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-\left(b + b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-33}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \frac{a}{b}, -2 \cdot \frac{b}{c}\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{c \cdot \left(a \cdot -4\right)}\\ t_1 := \frac{-c}{b}\\ \mathbf{if}\;b \leq -6.5 \cdot 10^{-43}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-\left(b + b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t_0 - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-37}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* c (* a -4.0)))) (t_1 (/ (- c) b)))
   (if (<= b -6.5e-43)
     (if (>= b 0.0) (/ c b) t_1)
     (if (<= b -2e-310)
       (if (>= b 0.0) (/ (- (+ b b)) (* a 2.0)) (/ 2.0 (/ (- t_0 b) c)))
       (if (<= b 1.15e-37)
         (if (>= b 0.0) (/ (- (- b) t_0) (* a 2.0)) (/ b a))
         (if (>= b 0.0) (- (/ c b) (/ b a)) t_1))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((c * (a * -4.0)));
	double t_1 = -c / b;
	double tmp_1;
	if (b <= -6.5e-43) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c / b;
		} else {
			tmp_2 = t_1;
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -(b + b) / (a * 2.0);
		} else {
			tmp_3 = 2.0 / ((t_0 - b) / c);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.15e-37) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (-b - t_0) / (a * 2.0);
		} else {
			tmp_4 = b / a;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} 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
    real(8) :: tmp_4
    t_0 = sqrt((c * (a * (-4.0d0))))
    t_1 = -c / b
    if (b <= (-6.5d-43)) then
        if (b >= 0.0d0) then
            tmp_2 = c / b
        else
            tmp_2 = t_1
        end if
        tmp_1 = tmp_2
    else if (b <= (-2d-310)) then
        if (b >= 0.0d0) then
            tmp_3 = -(b + b) / (a * 2.0d0)
        else
            tmp_3 = 2.0d0 / ((t_0 - b) / c)
        end if
        tmp_1 = tmp_3
    else if (b <= 1.15d-37) then
        if (b >= 0.0d0) then
            tmp_4 = (-b - t_0) / (a * 2.0d0)
        else
            tmp_4 = b / a
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = (c / b) - (b / a)
    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((c * (a * -4.0)));
	double t_1 = -c / b;
	double tmp_1;
	if (b <= -6.5e-43) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c / b;
		} else {
			tmp_2 = t_1;
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -(b + b) / (a * 2.0);
		} else {
			tmp_3 = 2.0 / ((t_0 - b) / c);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.15e-37) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (-b - t_0) / (a * 2.0);
		} else {
			tmp_4 = b / a;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt((c * (a * -4.0)))
	t_1 = -c / b
	tmp_1 = 0
	if b <= -6.5e-43:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = c / b
		else:
			tmp_2 = t_1
		tmp_1 = tmp_2
	elif b <= -2e-310:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = -(b + b) / (a * 2.0)
		else:
			tmp_3 = 2.0 / ((t_0 - b) / c)
		tmp_1 = tmp_3
	elif b <= 1.15e-37:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = (-b - t_0) / (a * 2.0)
		else:
			tmp_4 = b / a
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = (c / b) - (b / a)
	else:
		tmp_1 = t_1
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(c * Float64(a * -4.0)))
	t_1 = Float64(Float64(-c) / b)
	tmp_1 = 0.0
	if (b <= -6.5e-43)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c / b);
		else
			tmp_2 = t_1;
		end
		tmp_1 = tmp_2;
	elseif (b <= -2e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-Float64(b + b)) / Float64(a * 2.0));
		else
			tmp_3 = Float64(2.0 / Float64(Float64(t_0 - b) / c));
		end
		tmp_1 = tmp_3;
	elseif (b <= 1.15e-37)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(Float64(-b) - t_0) / Float64(a * 2.0));
		else
			tmp_4 = Float64(b / a);
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = sqrt((c * (a * -4.0)));
	t_1 = -c / b;
	tmp_2 = 0.0;
	if (b <= -6.5e-43)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = c / b;
		else
			tmp_3 = t_1;
		end
		tmp_2 = tmp_3;
	elseif (b <= -2e-310)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = -(b + b) / (a * 2.0);
		else
			tmp_4 = 2.0 / ((t_0 - b) / c);
		end
		tmp_2 = tmp_4;
	elseif (b <= 1.15e-37)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = (-b - t_0) / (a * 2.0);
		else
			tmp_5 = b / a;
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = (c / b) - (b / a);
	else
		tmp_2 = t_1;
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-c) / b), $MachinePrecision]}, If[LessEqual[b, -6.5e-43], If[GreaterEqual[b, 0.0], N[(c / b), $MachinePrecision], t$95$1], If[LessEqual[b, -2e-310], If[GreaterEqual[b, 0.0], N[((-N[(b + b), $MachinePrecision]) / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$0 - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.15e-37], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(b / a), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -6.50000000000000001e-43

   1. Initial program 62.7%

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

   2. Taylor expanded in b around -inf 87.2%

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

      1. associate-*r/ 87.2%

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

      2. neg-mul-1 87.2%

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

   4. Simplified 87.2%

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

   5. Taylor expanded in b around inf 87.2%

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

   6. Taylor expanded in b around 0 87.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

   if -6.50000000000000001e-43 < b < -1.999999999999994e-310

   1. Initial program 79.3%

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

   3. Simplified 79.2%

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

   4. Taylor expanded in b around inf 79.2%

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

   5. Taylor expanded in b around 0 68.5%

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

      1. associate-*r* 23.6%

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

      2. metadata-eval 23.6%

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

      3. distribute-lft-neg-in 23.6%

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

      4. *-commutative 23.6%

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

      5. distribute-lft-neg-in 23.6%

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

      6. metadata-eval 23.6%

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

   7. Simplified 68.5%

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

   if -1.999999999999994e-310 < b < 1.15e-37

   1. Initial program 82.3%

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

   3. Simplified 82.3%

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

   4. Taylor expanded in b around -inf 82.3%

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

      1. +-commutative 82.3%

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

      2. fma-def 82.3%

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

   6. Simplified 82.3%

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

   7. Taylor expanded in b around 0 73.1%

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

      1. associate-*r* 73.1%

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

      2. metadata-eval 73.1%

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

      3. distribute-lft-neg-in 73.1%

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

      4. *-commutative 73.1%

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

      5. distribute-lft-neg-in 73.1%

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

      6. metadata-eval 73.1%

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

   9. Simplified 73.1%

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

   10. Taylor expanded in a around inf 73.1%

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

   if 1.15e-37 < b

   1. Initial program 62.1%

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

   2. Taylor expanded in b around -inf 62.1%

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

      1. associate-*r/ 62.1%

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

      2. neg-mul-1 62.1%

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

   4. Simplified 62.1%

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

   5. Taylor expanded in b around inf 86.2%

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

   6. Taylor expanded in b around 0 88.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
   7. Step-by-step derivation

      1. +-commutative 88.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

      2. mul-1-neg 88.4%

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

      3. unsub-neg 88.4%

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

   8. Simplified 88.4%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-43}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-\left(b + b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-37}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot \left(c \cdot -4\right)}\\ t_1 := \frac{-c}{b}\\ \mathbf{if}\;b \leq -2.35 \cdot 10^{-92}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-\left(b + b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{b + t_0}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-37}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0.5 \cdot \left(b - t_0\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \frac{a}{b}, -2 \cdot \frac{b}{c}\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a (* c -4.0)))) (t_1 (/ (- c) b)))
   (if (<= b -2.35e-92)
     (if (>= b 0.0) (/ c b) t_1)
     (if (<= b -2e-310)
       (if (>= b 0.0) (/ (- (+ b b)) (* a 2.0)) (* c (/ 2.0 (+ b t_0))))
       (if (<= b 4.2e-37)
         (if (>= b 0.0)
           (/ (* 0.5 (- b t_0)) a)
           (/ 2.0 (fma 2.0 (/ a b) (* -2.0 (/ b c)))))
         (if (>= b 0.0) (- (/ c b) (/ b a)) t_1))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((a * (c * -4.0)));
	double t_1 = -c / b;
	double tmp_1;
	if (b <= -2.35e-92) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c / b;
		} else {
			tmp_2 = t_1;
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -(b + b) / (a * 2.0);
		} else {
			tmp_3 = c * (2.0 / (b + t_0));
		}
		tmp_1 = tmp_3;
	} else if (b <= 4.2e-37) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (0.5 * (b - t_0)) / a;
		} else {
			tmp_4 = 2.0 / fma(2.0, (a / b), (-2.0 * (b / c)));
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(a * Float64(c * -4.0)))
	t_1 = Float64(Float64(-c) / b)
	tmp_1 = 0.0
	if (b <= -2.35e-92)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c / b);
		else
			tmp_2 = t_1;
		end
		tmp_1 = tmp_2;
	elseif (b <= -2e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-Float64(b + b)) / Float64(a * 2.0));
		else
			tmp_3 = Float64(c * Float64(2.0 / Float64(b + t_0)));
		end
		tmp_1 = tmp_3;
	elseif (b <= 4.2e-37)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(0.5 * Float64(b - t_0)) / a);
		else
			tmp_4 = Float64(2.0 / fma(2.0, Float64(a / b), Float64(-2.0 * Float64(b / c))));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-c) / b), $MachinePrecision]}, If[LessEqual[b, -2.35e-92], If[GreaterEqual[b, 0.0], N[(c / b), $MachinePrecision], t$95$1], If[LessEqual[b, -2e-310], If[GreaterEqual[b, 0.0], N[((-N[(b + b), $MachinePrecision]) / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c * N[(2.0 / N[(b + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 4.2e-37], If[GreaterEqual[b, 0.0], N[(N[(0.5 * N[(b - t$95$0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(2.0 / N[(2.0 * N[(a / b), $MachinePrecision] + N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.34999999999999996e-92

   1. Initial program 67.4%

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

   2. Taylor expanded in b around -inf 82.6%

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

      1. associate-*r/ 82.6%

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

      2. neg-mul-1 82.6%

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

   4. Simplified 82.6%

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

   5. Taylor expanded in b around inf 82.6%

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

   6. Taylor expanded in b around 0 82.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

   if -2.34999999999999996e-92 < b < -1.999999999999994e-310

   1. Initial program 73.5%

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

   3. Simplified 73.3%

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

   4. Taylor expanded in b around inf 73.3%

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

   5. Taylor expanded in b around 0 73.3%

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

      1. associate-*r* 14.8%

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

      2. metadata-eval 14.8%

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

      3. distribute-lft-neg-in 14.8%

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

      4. *-commutative 14.8%

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

      5. distribute-lft-neg-in 14.8%

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

      6. metadata-eval 14.8%

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

   7. Simplified 73.3%

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

      1. associate-/r/ 73.2%

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

      2. add-sqr-sqrt 73.2%

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

      3. sqrt-unprod 71.5%

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

      4. sqr-neg 71.5%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 71.3%

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

      7. associate-*r* 71.3%

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

   9. Applied egg-rr 71.3%

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

   if -1.999999999999994e-310 < b < 4.2000000000000002e-37

   1. Initial program 82.3%

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

   3. Simplified 82.3%

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

   4. Taylor expanded in b around -inf 82.3%

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

      1. +-commutative 82.3%

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

      2. fma-def 82.3%

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

   6. Simplified 82.3%

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

   7. Taylor expanded in b around 0 73.1%

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

      1. associate-*r* 73.1%

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

      2. metadata-eval 73.1%

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

      3. distribute-lft-neg-in 73.1%

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

      4. *-commutative 73.1%

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

      5. distribute-lft-neg-in 73.1%

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

      6. metadata-eval 73.1%

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

   9. Simplified 73.1%

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

       1. div-sub 73.1%

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

       2. sub-neg 73.1%

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

       3. *-un-lft-identity 73.1%

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

       4. *-commutative 73.1%

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

       5. times-frac 73.1%

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

       6. metadata-eval 73.1%

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

       7. add-sqr-sqrt 0.0%

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

       8. sqrt-unprod 69.8%

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

       9. sqr-neg 69.8%

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

       10. sqrt-unprod 69.6%

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

       11. add-sqr-sqrt 69.6%

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

       12. *-un-lft-identity 69.6%

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

       13. *-commutative 69.6%

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

       14. times-frac 69.6%

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

       15. metadata-eval 69.6%

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

       16. associate-*r* 69.6%

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

       17. *-commutative 69.6%

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

   11. Applied egg-rr 69.6%

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

       1. sub-neg 69.6%

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

       2. distribute-lft-out-- 69.6%

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

       3. div-sub 69.6%

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

       4. associate-*r/ 69.6%

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

   13. Simplified 69.6%

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

   if 4.2000000000000002e-37 < b

   1. Initial program 62.1%

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

   2. Taylor expanded in b around -inf 62.1%

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

      1. associate-*r/ 62.1%

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

      2. neg-mul-1 62.1%

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

   4. Simplified 62.1%

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

   5. Taylor expanded in b around inf 86.2%

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

   6. Taylor expanded in b around 0 88.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
   7. Step-by-step derivation

      1. +-commutative 88.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

      2. mul-1-neg 88.4%

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

      3. unsub-neg 88.4%

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

   8. Simplified 88.4%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.35 \cdot 10^{-92}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-\left(b + b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{b + \sqrt{a \cdot \left(c \cdot -4\right)}}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-37}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0.5 \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \frac{a}{b}, -2 \cdot \frac{b}{c}\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 7: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-c}{b}\\ \mathbf{if}\;b \leq -3.9 \cdot 10^{-94}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-\left(b + b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{c}}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-33}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0.5 \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \frac{a}{b}, -2 \cdot \frac{b}{c}\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- c) b)))
   (if (<= b -3.9e-94)
     (if (>= b 0.0) (/ c b) t_0)
     (if (<= b -2e-310)
       (if (>= b 0.0)
         (/ (- (+ b b)) (* a 2.0))
         (/ 2.0 (/ (+ b (sqrt (* c (* a -4.0)))) c)))
       (if (<= b 4.4e-33)
         (if (>= b 0.0)
           (/ (* 0.5 (- b (sqrt (* a (* c -4.0))))) a)
           (/ 2.0 (fma 2.0 (/ a b) (* -2.0 (/ b c)))))
         (if (>= b 0.0) (- (/ c b) (/ b a)) t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = -c / b;
	double tmp_1;
	if (b <= -3.9e-94) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c / b;
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -(b + b) / (a * 2.0);
		} else {
			tmp_3 = 2.0 / ((b + sqrt((c * (a * -4.0)))) / c);
		}
		tmp_1 = tmp_3;
	} else if (b <= 4.4e-33) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (0.5 * (b - sqrt((a * (c * -4.0))))) / a;
		} else {
			tmp_4 = 2.0 / fma(2.0, (a / b), (-2.0 * (b / c)));
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-c) / b)
	tmp_1 = 0.0
	if (b <= -3.9e-94)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c / b);
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b <= -2e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-Float64(b + b)) / Float64(a * 2.0));
		else
			tmp_3 = Float64(2.0 / Float64(Float64(b + sqrt(Float64(c * Float64(a * -4.0)))) / c));
		end
		tmp_1 = tmp_3;
	elseif (b <= 4.4e-33)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(0.5 * Float64(b - sqrt(Float64(a * Float64(c * -4.0))))) / a);
		else
			tmp_4 = Float64(2.0 / fma(2.0, Float64(a / b), Float64(-2.0 * Float64(b / c))));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-c) / b), $MachinePrecision]}, If[LessEqual[b, -3.9e-94], If[GreaterEqual[b, 0.0], N[(c / b), $MachinePrecision], t$95$0], If[LessEqual[b, -2e-310], If[GreaterEqual[b, 0.0], N[((-N[(b + b), $MachinePrecision]) / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 4.4e-33], If[GreaterEqual[b, 0.0], N[(N[(0.5 * N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(2.0 / N[(2.0 * N[(a / b), $MachinePrecision] + N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -3.9000000000000002e-94

   1. Initial program 67.4%

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

   2. Taylor expanded in b around -inf 82.6%

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

      1. associate-*r/ 82.6%

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

      2. neg-mul-1 82.6%

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

   4. Simplified 82.6%

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

   5. Taylor expanded in b around inf 82.6%

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

   6. Taylor expanded in b around 0 82.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

   if -3.9000000000000002e-94 < b < -1.999999999999994e-310

   1. Initial program 73.5%

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

   3. Simplified 73.3%

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

   4. Taylor expanded in b around inf 73.3%

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

   5. Taylor expanded in b around 0 73.3%

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

      1. associate-*r* 14.8%

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

      2. metadata-eval 14.8%

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

      3. distribute-lft-neg-in 14.8%

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

      4. *-commutative 14.8%

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

      5. distribute-lft-neg-in 14.8%

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

      6. metadata-eval 14.8%

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

   7. Simplified 73.3%

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

      1. expm1-log1p-u 50.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{c \cdot \left(-4 \cdot a\right)}}{c}\right)\right)}}\\ \end{array} \]

      2. expm1-udef 26.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{c \cdot \left(-4 \cdot a\right)}}{c}\right)} - 1}}\\ \end{array} \]

      3. add-sqr-sqrt 26.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{\mathsf{log1p}\left(\frac{\sqrt{-b} \cdot \sqrt{-b} + \sqrt{c \cdot \left(-4 \cdot a\right)}}{c}\right)} - 1}\\ \end{array} \]

      4. sqrt-unprod 24.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{\mathsf{log1p}\left(\frac{\sqrt{\left(-b\right) \cdot \left(-b\right)} + \sqrt{c \cdot \left(-4 \cdot a\right)}}{c}\right)} - 1}\\ \end{array} \]

      5. sqr-neg 24.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{\mathsf{log1p}\left(\frac{\sqrt{b \cdot b} + \sqrt{c \cdot \left(-4 \cdot a\right)}}{c}\right)} - 1}\\ \end{array} \]

      6. sqrt-unprod 0.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{\mathsf{log1p}\left(\frac{\sqrt{b} \cdot \sqrt{b} + \sqrt{c \cdot \left(-4 \cdot a\right)}}{c}\right)} - 1}\\ \end{array} \]

      7. add-sqr-sqrt 24.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{\mathsf{log1p}\left(\frac{b + \sqrt{c \cdot \left(-4 \cdot a\right)}}{c}\right)} - 1}\\ \end{array} \]

      8. associate-*r* 24.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{\mathsf{log1p}\left(\frac{b + \sqrt{\left(c \cdot -4\right) \cdot a}}{c}\right)} - 1}\\ \end{array} \]

   9. Applied egg-rr 24.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{e^{\mathsf{log1p}\left(\frac{b + \sqrt{\left(c \cdot -4\right) \cdot a}}{c}\right)} - 1}}\\ \end{array} \]
   10. Step-by-step derivation

       1. expm1-def 48.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b + \sqrt{\left(c \cdot -4\right) \cdot a}}{c}\right)\right)}}\\ \end{array} \]

       2. expm1-log1p 71.3%

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

       3. associate-*l* 71.3%

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

       4. *-commutative 71.3%

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

   11. Simplified 71.3%

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

   if -1.999999999999994e-310 < b < 4.40000000000000011e-33

   1. Initial program 82.3%

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

   3. Simplified 82.3%

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

   4. Taylor expanded in b around -inf 82.3%

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

      1. +-commutative 82.3%

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

      2. fma-def 82.3%

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

   6. Simplified 82.3%

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

   7. Taylor expanded in b around 0 73.1%

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

      1. associate-*r* 73.1%

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

      2. metadata-eval 73.1%

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

      3. distribute-lft-neg-in 73.1%

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

      4. *-commutative 73.1%

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

      5. distribute-lft-neg-in 73.1%

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

      6. metadata-eval 73.1%

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

   9. Simplified 73.1%

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

       1. div-sub 73.1%

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

       2. sub-neg 73.1%

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

       3. *-un-lft-identity 73.1%

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

       4. *-commutative 73.1%

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

       5. times-frac 73.1%

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

       6. metadata-eval 73.1%

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

       7. add-sqr-sqrt 0.0%

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

       8. sqrt-unprod 69.8%

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

       9. sqr-neg 69.8%

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

       10. sqrt-unprod 69.6%

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

       11. add-sqr-sqrt 69.6%

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

       12. *-un-lft-identity 69.6%

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

       13. *-commutative 69.6%

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

       14. times-frac 69.6%

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

       15. metadata-eval 69.6%

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

       16. associate-*r* 69.6%

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

       17. *-commutative 69.6%

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

   11. Applied egg-rr 69.6%

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

       1. sub-neg 69.6%

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

       2. distribute-lft-out-- 69.6%

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

       3. div-sub 69.6%

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

       4. associate-*r/ 69.6%

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

   13. Simplified 69.6%

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

   if 4.40000000000000011e-33 < b

   1. Initial program 62.1%

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

   2. Taylor expanded in b around -inf 62.1%

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

      1. associate-*r/ 62.1%

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

      2. neg-mul-1 62.1%

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

   4. Simplified 62.1%

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

   5. Taylor expanded in b around inf 86.2%

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

   6. Taylor expanded in b around 0 88.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
   7. Step-by-step derivation

      1. +-commutative 88.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

      2. mul-1-neg 88.4%

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

      3. unsub-neg 88.4%

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

   8. Simplified 88.4%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.9 \cdot 10^{-94}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-\left(b + b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{c}}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-33}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0.5 \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \frac{a}{b}, -2 \cdot \frac{b}{c}\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 8: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-c}{b}\\ \mathbf{if}\;b \leq -1.2 \cdot 10^{-94}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-\left(b + b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{b + \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-39}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0.5 \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \frac{a}{b}, -2 \cdot \frac{b}{c}\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- c) b)))
   (if (<= b -1.2e-94)
     (if (>= b 0.0) (/ c b) t_0)
     (if (<= b -2e-310)
       (if (>= b 0.0)
         (/ (- (+ b b)) (* a 2.0))
         (/ (* c 2.0) (+ b (sqrt (* c (* a -4.0))))))
       (if (<= b 4.2e-39)
         (if (>= b 0.0)
           (/ (* 0.5 (- b (sqrt (* a (* c -4.0))))) a)
           (/ 2.0 (fma 2.0 (/ a b) (* -2.0 (/ b c)))))
         (if (>= b 0.0) (- (/ c b) (/ b a)) t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = -c / b;
	double tmp_1;
	if (b <= -1.2e-94) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c / b;
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -(b + b) / (a * 2.0);
		} else {
			tmp_3 = (c * 2.0) / (b + sqrt((c * (a * -4.0))));
		}
		tmp_1 = tmp_3;
	} else if (b <= 4.2e-39) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (0.5 * (b - sqrt((a * (c * -4.0))))) / a;
		} else {
			tmp_4 = 2.0 / fma(2.0, (a / b), (-2.0 * (b / c)));
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-c) / b)
	tmp_1 = 0.0
	if (b <= -1.2e-94)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c / b);
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b <= -2e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-Float64(b + b)) / Float64(a * 2.0));
		else
			tmp_3 = Float64(Float64(c * 2.0) / Float64(b + sqrt(Float64(c * Float64(a * -4.0)))));
		end
		tmp_1 = tmp_3;
	elseif (b <= 4.2e-39)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(0.5 * Float64(b - sqrt(Float64(a * Float64(c * -4.0))))) / a);
		else
			tmp_4 = Float64(2.0 / fma(2.0, Float64(a / b), Float64(-2.0 * Float64(b / c))));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-c) / b), $MachinePrecision]}, If[LessEqual[b, -1.2e-94], If[GreaterEqual[b, 0.0], N[(c / b), $MachinePrecision], t$95$0], If[LessEqual[b, -2e-310], If[GreaterEqual[b, 0.0], N[((-N[(b + b), $MachinePrecision]) / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 4.2e-39], If[GreaterEqual[b, 0.0], N[(N[(0.5 * N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(2.0 / N[(2.0 * N[(a / b), $MachinePrecision] + N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.2e-94

   1. Initial program 67.4%

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

   2. Taylor expanded in b around -inf 82.6%

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

      1. associate-*r/ 82.6%

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

      2. neg-mul-1 82.6%

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

   4. Simplified 82.6%

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

   5. Taylor expanded in b around inf 82.6%

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

   6. Taylor expanded in b around 0 82.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

   if -1.2e-94 < b < -1.999999999999994e-310

   1. Initial program 73.5%

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

   3. Simplified 73.3%

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

   4. Taylor expanded in b around inf 73.3%

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

   5. Taylor expanded in b around 0 73.3%

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

      1. associate-*r* 14.8%

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

      2. metadata-eval 14.8%

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

      3. distribute-lft-neg-in 14.8%

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

      4. *-commutative 14.8%

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

      5. distribute-lft-neg-in 14.8%

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

      6. metadata-eval 14.8%

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

   7. Simplified 73.3%

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

      1. expm1-log1p-u 51.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\frac{\left(-b\right) + \sqrt{c \cdot \left(-4 \cdot a\right)}}{c}}\right)\right)\\ \end{array} \]

      2. expm1-udef 10.1%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{2}{\frac{\left(-b\right) + \sqrt{c \cdot \left(-4 \cdot a\right)}}{c}}\right)} - 1\\ \end{array} \]

      3. div-inv 10.1%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(2 \cdot \frac{1}{\frac{\left(-b\right) + \sqrt{c \cdot \left(-4 \cdot a\right)}}{c}}\right)} - 1\\ \end{array} \]

      4. clear-num 10.1%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(2 \cdot \frac{c}{\left(-b\right) + \sqrt{c \cdot \left(-4 \cdot a\right)}}\right)} - 1\\ \end{array} \]

      5. add-sqr-sqrt 10.1%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(2 \cdot \frac{c}{\sqrt{-b} \cdot \sqrt{-b} + \sqrt{c \cdot \left(-4 \cdot a\right)}}\right)} - 1\\ \end{array} \]

      6. sqrt-unprod 9.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(2 \cdot \frac{c}{\sqrt{\left(-b\right) \cdot \left(-b\right)} + \sqrt{c \cdot \left(-4 \cdot a\right)}}\right)} - 1\\ \end{array} \]

      7. sqr-neg 9.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(2 \cdot \frac{c}{\sqrt{b \cdot b} + \sqrt{c \cdot \left(-4 \cdot a\right)}}\right)} - 1\\ \end{array} \]

      8. sqrt-unprod 0.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(2 \cdot \frac{c}{\sqrt{b} \cdot \sqrt{b} + \sqrt{c \cdot \left(-4 \cdot a\right)}}\right)} - 1\\ \end{array} \]

      9. add-sqr-sqrt 9.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(2 \cdot \frac{c}{b + \sqrt{c \cdot \left(-4 \cdot a\right)}}\right)} - 1\\ \end{array} \]

      10. associate-*r* 9.0%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(2 \cdot \frac{c}{b + \sqrt{\left(c \cdot -4\right) \cdot a}}\right)} - 1\\ \end{array} \]

   9. Applied egg-rr 9.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(2 \cdot \frac{c}{b + \sqrt{\left(c \cdot -4\right) \cdot a}}\right)} - 1\\ \end{array} \]
   10. Step-by-step derivation

       1. expm1-def 50.1%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \frac{c}{b + \sqrt{\left(c \cdot -4\right) \cdot a}}\right)\right)\\ \end{array} \]

       2. expm1-log1p 71.5%

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

       3. associate-*r/ 71.5%

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

       4. associate-*l* 71.5%

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

       5. *-commutative 71.5%

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

   11. Simplified 71.5%

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

   if -1.999999999999994e-310 < b < 4.19999999999999987e-39

   1. Initial program 82.3%

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

   3. Simplified 82.3%

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

   4. Taylor expanded in b around -inf 82.3%

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

      1. +-commutative 82.3%

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

      2. fma-def 82.3%

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

   6. Simplified 82.3%

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

   7. Taylor expanded in b around 0 73.1%

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

      1. associate-*r* 73.1%

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

      2. metadata-eval 73.1%

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

      3. distribute-lft-neg-in 73.1%

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

      4. *-commutative 73.1%

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

      5. distribute-lft-neg-in 73.1%

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

      6. metadata-eval 73.1%

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

   9. Simplified 73.1%

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

       1. div-sub 73.1%

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

       2. sub-neg 73.1%

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

       3. *-un-lft-identity 73.1%

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

       4. *-commutative 73.1%

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

       5. times-frac 73.1%

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

       6. metadata-eval 73.1%

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

       7. add-sqr-sqrt 0.0%

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

       8. sqrt-unprod 69.8%

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

       9. sqr-neg 69.8%

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

       10. sqrt-unprod 69.6%

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

       11. add-sqr-sqrt 69.6%

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

       12. *-un-lft-identity 69.6%

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

       13. *-commutative 69.6%

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

       14. times-frac 69.6%

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

       15. metadata-eval 69.6%

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

       16. associate-*r* 69.6%

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

       17. *-commutative 69.6%

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

   11. Applied egg-rr 69.6%

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

       1. sub-neg 69.6%

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

       2. distribute-lft-out-- 69.6%

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

       3. div-sub 69.6%

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

       4. associate-*r/ 69.6%

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

   13. Simplified 69.6%

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

   if 4.19999999999999987e-39 < b

   1. Initial program 62.1%

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

   2. Taylor expanded in b around -inf 62.1%

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

      1. associate-*r/ 62.1%

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

      2. neg-mul-1 62.1%

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

   4. Simplified 62.1%

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

   5. Taylor expanded in b around inf 86.2%

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

   6. Taylor expanded in b around 0 88.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
   7. Step-by-step derivation

      1. +-commutative 88.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

      2. mul-1-neg 88.4%

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

      3. unsub-neg 88.4%

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

   8. Simplified 88.4%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{-94}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-\left(b + b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{b + \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-39}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0.5 \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \frac{a}{b}, -2 \cdot \frac{b}{c}\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 9: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-c}{b}\\ \mathbf{if}\;b \leq -5.4 \cdot 10^{-48}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-\left(b + b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-39}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0.5 \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \frac{a}{b}, -2 \cdot \frac{b}{c}\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- c) b)))
   (if (<= b -5.4e-48)
     (if (>= b 0.0) (/ c b) t_0)
     (if (<= b -2e-310)
       (if (>= b 0.0)
         (/ (- (+ b b)) (* a 2.0))
         (/ 2.0 (/ (- (sqrt (* c (* a -4.0))) b) c)))
       (if (<= b 8.5e-39)
         (if (>= b 0.0)
           (/ (* 0.5 (- b (sqrt (* a (* c -4.0))))) a)
           (/ 2.0 (fma 2.0 (/ a b) (* -2.0 (/ b c)))))
         (if (>= b 0.0) (- (/ c b) (/ b a)) t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = -c / b;
	double tmp_1;
	if (b <= -5.4e-48) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c / b;
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -(b + b) / (a * 2.0);
		} else {
			tmp_3 = 2.0 / ((sqrt((c * (a * -4.0))) - b) / c);
		}
		tmp_1 = tmp_3;
	} else if (b <= 8.5e-39) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (0.5 * (b - sqrt((a * (c * -4.0))))) / a;
		} else {
			tmp_4 = 2.0 / fma(2.0, (a / b), (-2.0 * (b / c)));
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-c) / b)
	tmp_1 = 0.0
	if (b <= -5.4e-48)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c / b);
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b <= -2e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-Float64(b + b)) / Float64(a * 2.0));
		else
			tmp_3 = Float64(2.0 / Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / c));
		end
		tmp_1 = tmp_3;
	elseif (b <= 8.5e-39)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(0.5 * Float64(b - sqrt(Float64(a * Float64(c * -4.0))))) / a);
		else
			tmp_4 = Float64(2.0 / fma(2.0, Float64(a / b), Float64(-2.0 * Float64(b / c))));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-c) / b), $MachinePrecision]}, If[LessEqual[b, -5.4e-48], If[GreaterEqual[b, 0.0], N[(c / b), $MachinePrecision], t$95$0], If[LessEqual[b, -2e-310], If[GreaterEqual[b, 0.0], N[((-N[(b + b), $MachinePrecision]) / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 8.5e-39], If[GreaterEqual[b, 0.0], N[(N[(0.5 * N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(2.0 / N[(2.0 * N[(a / b), $MachinePrecision] + N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -5.40000000000000023e-48

   1. Initial program 62.7%

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

   2. Taylor expanded in b around -inf 87.2%

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

      1. associate-*r/ 87.2%

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

      2. neg-mul-1 87.2%

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

   4. Simplified 87.2%

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

   5. Taylor expanded in b around inf 87.2%

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

   6. Taylor expanded in b around 0 87.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

   if -5.40000000000000023e-48 < b < -1.999999999999994e-310

   1. Initial program 79.3%

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

   3. Simplified 79.2%

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

   4. Taylor expanded in b around inf 79.2%

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

   5. Taylor expanded in b around 0 68.5%

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

      1. associate-*r* 23.6%

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

      2. metadata-eval 23.6%

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

      3. distribute-lft-neg-in 23.6%

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

      4. *-commutative 23.6%

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

      5. distribute-lft-neg-in 23.6%

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

      6. metadata-eval 23.6%

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

   7. Simplified 68.5%

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

   if -1.999999999999994e-310 < b < 8.5000000000000005e-39

   1. Initial program 82.3%

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

   3. Simplified 82.3%

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

   4. Taylor expanded in b around -inf 82.3%

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

      1. +-commutative 82.3%

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

      2. fma-def 82.3%

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

   6. Simplified 82.3%

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

   7. Taylor expanded in b around 0 73.1%

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

      1. associate-*r* 73.1%

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

      2. metadata-eval 73.1%

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

      3. distribute-lft-neg-in 73.1%

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

      4. *-commutative 73.1%

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

      5. distribute-lft-neg-in 73.1%

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

      6. metadata-eval 73.1%

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

   9. Simplified 73.1%

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

       1. div-sub 73.1%

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

       2. sub-neg 73.1%

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

       3. *-un-lft-identity 73.1%

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

       4. *-commutative 73.1%

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

       5. times-frac 73.1%

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

       6. metadata-eval 73.1%

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

       7. add-sqr-sqrt 0.0%

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

       8. sqrt-unprod 69.8%

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

       9. sqr-neg 69.8%

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

       10. sqrt-unprod 69.6%

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

       11. add-sqr-sqrt 69.6%

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

       12. *-un-lft-identity 69.6%

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

       13. *-commutative 69.6%

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

       14. times-frac 69.6%

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

       15. metadata-eval 69.6%

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

       16. associate-*r* 69.6%

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

       17. *-commutative 69.6%

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

   11. Applied egg-rr 69.6%

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

       1. sub-neg 69.6%

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

       2. distribute-lft-out-- 69.6%

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

       3. div-sub 69.6%

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

       4. associate-*r/ 69.6%

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

   13. Simplified 69.6%

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

   if 8.5000000000000005e-39 < b

   1. Initial program 62.1%

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

   2. Taylor expanded in b around -inf 62.1%

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

      1. associate-*r/ 62.1%

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

      2. neg-mul-1 62.1%

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

   4. Simplified 62.1%

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

   5. Taylor expanded in b around inf 86.2%

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

   6. Taylor expanded in b around 0 88.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
   7. Step-by-step derivation

      1. +-commutative 88.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

      2. mul-1-neg 88.4%

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

      3. unsub-neg 88.4%

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

   8. Simplified 88.4%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{-48}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-\left(b + b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-39}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0.5 \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \frac{a}{b}, -2 \cdot \frac{b}{c}\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 10: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.32 \cdot 10^{-33}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \frac{a}{b}, -2 \cdot \frac{b}{c}\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.32e-33)
   (if (>= b 0.0)
     (* (- b (sqrt (* a (* c -4.0)))) (/ 0.5 a))
     (/ 2.0 (fma 2.0 (/ a b) (* -2.0 (/ b c)))))
   (if (>= b 0.0) (- (/ c b) (/ b a)) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= 1.32e-33) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (b - sqrt((a * (c * -4.0)))) * (0.5 / a);
		} else {
			tmp_2 = 2.0 / fma(2.0, (a / b), (-2.0 * (b / c)));
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = -c / b;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= 1.32e-33)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(b - sqrt(Float64(a * Float64(c * -4.0)))) * Float64(0.5 / a));
		else
			tmp_2 = Float64(2.0 / fma(2.0, Float64(a / b), Float64(-2.0 * Float64(b / c))));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp_1 = Float64(Float64(-c) / b);
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.32e-33], If[GreaterEqual[b, 0.0], N[(N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(a / b), $MachinePrecision] + N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.31999999999999993e-33

   1. Initial program 72.3%

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

   3. Simplified 72.2%

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

   4. Taylor expanded in b around -inf 66.1%

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

      1. +-commutative 66.1%

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

      2. fma-def 66.1%

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

   6. Simplified 66.1%

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

   7. Taylor expanded in b around 0 64.0%

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

      1. associate-*r* 64.0%

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

      2. metadata-eval 64.0%

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

      3. distribute-lft-neg-in 64.0%

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

      4. *-commutative 64.0%

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

      5. distribute-lft-neg-in 64.0%

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

      6. metadata-eval 64.0%

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

   9. Simplified 64.0%

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

       1. div-sub 64.0%

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

       2. sub-neg 64.0%

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

       3. *-un-lft-identity 64.0%

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

       4. *-commutative 64.0%

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

       5. times-frac 64.0%

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

       6. metadata-eval 64.0%

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

       7. add-sqr-sqrt 47.1%

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

       8. sqrt-unprod 63.2%

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

       9. sqr-neg 63.2%

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

       10. sqrt-unprod 63.2%

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

       11. add-sqr-sqrt 63.2%

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

       12. *-un-lft-identity 63.2%

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

       13. *-commutative 63.2%

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

       14. times-frac 63.2%

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

       15. metadata-eval 63.2%

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

       16. associate-*r* 63.2%

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

       17. *-commutative 63.2%

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

   11. Applied egg-rr 63.2%

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

       1. sub-neg 63.2%

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

       2. distribute-lft-out-- 63.2%

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

       3. div-sub 63.2%

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

       4. associate-*r/ 63.2%

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

       5. associate-*l/ 63.1%

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

   13. Simplified 63.1%

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

   if 1.31999999999999993e-33 < b

   1. Initial program 62.1%

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

   2. Taylor expanded in b around -inf 62.1%

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

      1. associate-*r/ 62.1%

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

      2. neg-mul-1 62.1%

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

   4. Simplified 62.1%

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

   5. Taylor expanded in b around inf 86.2%

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

   6. Taylor expanded in b around 0 88.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
   7. Step-by-step derivation

      1. +-commutative 88.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

      2. mul-1-neg 88.4%

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

      3. unsub-neg 88.4%

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

   8. Simplified 88.4%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.32 \cdot 10^{-33}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \frac{a}{b}, -2 \cdot \frac{b}{c}\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 11: 74.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.5 \cdot 10^{-39}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0.5 \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \frac{a}{b}, -2 \cdot \frac{b}{c}\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8.5e-39)
   (if (>= b 0.0)
     (/ (* 0.5 (- b (sqrt (* a (* c -4.0))))) a)
     (/ 2.0 (fma 2.0 (/ a b) (* -2.0 (/ b c)))))
   (if (>= b 0.0) (- (/ c b) (/ b a)) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= 8.5e-39) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (0.5 * (b - sqrt((a * (c * -4.0))))) / a;
		} else {
			tmp_2 = 2.0 / fma(2.0, (a / b), (-2.0 * (b / c)));
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = -c / b;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= 8.5e-39)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(0.5 * Float64(b - sqrt(Float64(a * Float64(c * -4.0))))) / a);
		else
			tmp_2 = Float64(2.0 / fma(2.0, Float64(a / b), Float64(-2.0 * Float64(b / c))));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp_1 = Float64(Float64(-c) / b);
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8.5e-39], If[GreaterEqual[b, 0.0], N[(N[(0.5 * N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(2.0 / N[(2.0 * N[(a / b), $MachinePrecision] + N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 8.5000000000000005e-39

   1. Initial program 72.3%

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

   3. Simplified 72.2%

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

   4. Taylor expanded in b around -inf 66.1%

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

      1. +-commutative 66.1%

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

      2. fma-def 66.1%

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

   6. Simplified 66.1%

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

   7. Taylor expanded in b around 0 64.0%

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

      1. associate-*r* 64.0%

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

      2. metadata-eval 64.0%

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

      3. distribute-lft-neg-in 64.0%

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

      4. *-commutative 64.0%

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

      5. distribute-lft-neg-in 64.0%

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

      6. metadata-eval 64.0%

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

   9. Simplified 64.0%

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

       1. div-sub 64.0%

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

       2. sub-neg 64.0%

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

       3. *-un-lft-identity 64.0%

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

       4. *-commutative 64.0%

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

       5. times-frac 64.0%

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

       6. metadata-eval 64.0%

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

       7. add-sqr-sqrt 47.1%

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

       8. sqrt-unprod 63.2%

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

       9. sqr-neg 63.2%

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

       10. sqrt-unprod 63.2%

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

       11. add-sqr-sqrt 63.2%

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

       12. *-un-lft-identity 63.2%

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

       13. *-commutative 63.2%

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

       14. times-frac 63.2%

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

       15. metadata-eval 63.2%

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

       16. associate-*r* 63.2%

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

       17. *-commutative 63.2%

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

   11. Applied egg-rr 63.2%

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

       1. sub-neg 63.2%

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

       2. distribute-lft-out-- 63.2%

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

       3. div-sub 63.2%

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

       4. associate-*r/ 63.2%

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

   13. Simplified 63.2%

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

   if 8.5000000000000005e-39 < b

   1. Initial program 62.1%

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

   2. Taylor expanded in b around -inf 62.1%

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

      1. associate-*r/ 62.1%

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

      2. neg-mul-1 62.1%

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

   4. Simplified 62.1%

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

   5. Taylor expanded in b around inf 86.2%

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

   6. Taylor expanded in b around 0 88.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
   7. Step-by-step derivation

      1. +-commutative 88.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

      2. mul-1-neg 88.4%

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

      3. unsub-neg 88.4%

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

   8. Simplified 88.4%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.5 \cdot 10^{-39}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0.5 \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \frac{a}{b}, -2 \cdot \frac{b}{c}\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 12: 68.1% accurate, 13.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.5%

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

2. Taylor expanded in b around -inf 64.8%

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

   1. associate-*r/ 64.8%

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

   2. neg-mul-1 64.8%

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

4. Simplified 64.8%

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

5. Taylor expanded in b around inf 64.9%

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

6. Taylor expanded in b around 0 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
7. Step-by-step derivation

   1. +-commutative 65.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

   2. mul-1-neg 65.8%

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

   3. unsub-neg 65.8%

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

8. Simplified 65.8%

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

9. Final simplification 65.8%

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

Alternative 13: 36.2% accurate, 19.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.5%

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

2. Taylor expanded in b around -inf 64.8%

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

   1. associate-*r/ 64.8%

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

   2. neg-mul-1 64.8%

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

4. Simplified 64.8%

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

5. Taylor expanded in b around inf 64.9%

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

6. Taylor expanded in b around 0 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

7. Final simplification 31.4%

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

Alternative 14: 68.0% accurate, 19.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -(b / a);
	} else {
		tmp = -c / b;
	}
	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 = -(b / a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.5%

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

2. Taylor expanded in b around -inf 64.8%

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

   1. associate-*r/ 64.8%

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

   2. neg-mul-1 64.8%

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

4. Simplified 64.8%

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

5. Taylor expanded in b around inf 64.9%

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

6. Taylor expanded in b around inf 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
7. Step-by-step derivation

   1. associate-*r/ 65.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-1 \cdot b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

   2. mul-1-neg 65.5%

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

8. Simplified 65.5%

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

9. Final simplification 65.5%

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

Reproduce

herbie shell --seed 2023322 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  :precision binary64
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))