jeff quadratic root 2

Percentage Accurate: 72.4% → 90.2%

Time: 27.3s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot -2}{2 \cdot a}\\ t_1 := \frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ t_2 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ \mathbf{if}\;b \leq -3 \cdot 10^{+149}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array}\\ \mathbf{elif}\;b \leq -3 \cdot 10^{-294}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+92}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot \left(-2\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* b -2.0) (* 2.0 a)))
        (t_1 (/ (* 2.0 c) (* 2.0 (- (* a (/ c b)) b))))
        (t_2 (sqrt (- (* b b) (* c (* a 4.0))))))
   (if (<= b -3e+149)
     (if (>= b 0.0) t_1 (/ b (- a)))
     (if (<= b -3e-294)
       (if (>= b 0.0) t_1 (/ (- t_2 b) (* 2.0 a)))
       (if (<= b 2e+92)
         (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_2)) t_0)
         (if (>= b 0.0) (/ (* c (- 2.0)) (+ b b)) t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = (b * -2.0) / (2.0 * a);
	double t_1 = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
	double t_2 = sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -3e+149) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = b / -a;
		}
		tmp_1 = tmp_2;
	} else if (b <= -3e-294) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = (t_2 - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 2e+92) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (2.0 * c) / (-b - t_2);
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c * -2.0) / (b + b);
	} 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    real(8) :: tmp_4
    t_0 = (b * (-2.0d0)) / (2.0d0 * a)
    t_1 = (2.0d0 * c) / (2.0d0 * ((a * (c / b)) - b))
    t_2 = sqrt(((b * b) - (c * (a * 4.0d0))))
    if (b <= (-3d+149)) then
        if (b >= 0.0d0) then
            tmp_2 = t_1
        else
            tmp_2 = b / -a
        end if
        tmp_1 = tmp_2
    else if (b <= (-3d-294)) then
        if (b >= 0.0d0) then
            tmp_3 = t_1
        else
            tmp_3 = (t_2 - b) / (2.0d0 * a)
        end if
        tmp_1 = tmp_3
    else if (b <= 2d+92) then
        if (b >= 0.0d0) then
            tmp_4 = (2.0d0 * c) / (-b - t_2)
        else
            tmp_4 = t_0
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = (c * -2.0d0) / (b + b)
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (b * -2.0) / (2.0 * a);
	double t_1 = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
	double t_2 = Math.sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -3e+149) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = b / -a;
		}
		tmp_1 = tmp_2;
	} else if (b <= -3e-294) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = (t_2 - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 2e+92) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (2.0 * c) / (-b - t_2);
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c * -2.0) / (b + b);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (b * -2.0) / (2.0 * a)
	t_1 = (2.0 * c) / (2.0 * ((a * (c / b)) - b))
	t_2 = math.sqrt(((b * b) - (c * (a * 4.0))))
	tmp_1 = 0
	if b <= -3e+149:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_1
		else:
			tmp_2 = b / -a
		tmp_1 = tmp_2
	elif b <= -3e-294:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = t_1
		else:
			tmp_3 = (t_2 - b) / (2.0 * a)
		tmp_1 = tmp_3
	elif b <= 2e+92:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = (2.0 * c) / (-b - t_2)
		else:
			tmp_4 = t_0
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = (c * -2.0) / (b + b)
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * -2.0) / Float64(2.0 * a))
	t_1 = Float64(Float64(2.0 * c) / Float64(2.0 * Float64(Float64(a * Float64(c / b)) - b)))
	t_2 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	tmp_1 = 0.0
	if (b <= -3e+149)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(b / Float64(-a));
		end
		tmp_1 = tmp_2;
	elseif (b <= -3e-294)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = Float64(Float64(t_2 - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (b <= 2e+92)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_2));
		else
			tmp_4 = t_0;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c * Float64(-2.0)) / Float64(b + b));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = (b * -2.0) / (2.0 * a);
	t_1 = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
	t_2 = sqrt(((b * b) - (c * (a * 4.0))));
	tmp_2 = 0.0;
	if (b <= -3e+149)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = b / -a;
		end
		tmp_2 = tmp_3;
	elseif (b <= -3e-294)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = t_1;
		else
			tmp_4 = (t_2 - b) / (2.0 * a);
		end
		tmp_2 = tmp_4;
	elseif (b <= 2e+92)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = (2.0 * c) / (-b - t_2);
		else
			tmp_5 = t_0;
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = (c * -2.0) / (b + b);
	else
		tmp_2 = t_0;
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * -2.0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * c), $MachinePrecision] / N[(2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -3e+149], If[GreaterEqual[b, 0.0], t$95$1, N[(b / (-a)), $MachinePrecision]], If[LessEqual[b, -3e-294], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(t$95$2 - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 2e+92], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$2), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(c * (-2.0)), $MachinePrecision] / N[(b + b), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -3.00000000000000003e149

   1. Initial program 36.4%

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

   3. Taylor expanded in a around 0 36.4%

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

      1. distribute-lft-out-- 36.4%

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

      2. associate-/l* 36.4%

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

   5. Simplified 36.4%

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

   6. Taylor expanded in b around -inf 95.4%

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

   if -3.00000000000000003e149 < b < -2.9999999999999999e-294

   1. Initial program 88.6%

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

   3. Taylor expanded in a around 0 88.6%

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

      1. distribute-lft-out-- 88.6%

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

      2. associate-/l* 88.6%

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

   5. Simplified 88.6%

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

   if -2.9999999999999999e-294 < b < 2.0000000000000001e92

   1. Initial program 84.9%

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

   3. Taylor expanded in b around -inf 84.9%

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

      1. *-commutative 84.9%

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

   5. Simplified 84.9%

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

   if 2.0000000000000001e92 < b

   1. Initial program 54.2%

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

   3. Taylor expanded in b around -inf 54.2%

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

      1. *-commutative 54.2%

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

   5. Simplified 54.2%

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

   6. Taylor expanded in b around inf 97.2%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{+149}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array}\\ \mathbf{elif}\;b \leq -3 \cdot 10^{-294}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+92}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot \left(-2\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ t_1 := \frac{b \cdot -2}{2 \cdot a}\\ t_2 := \sqrt{c \cdot \left(a \cdot -4\right)}\\ \mathbf{if}\;b \leq -4.8 \cdot 10^{-33}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq -3 \cdot 10^{-294}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-45}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + t\_2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + c \cdot \frac{a \cdot -2}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* 2.0 c) (* 2.0 (- (* a (/ c b)) b))))
        (t_1 (/ (* b -2.0) (* 2.0 a)))
        (t_2 (sqrt (* c (* a -4.0)))))
   (if (<= b -4.8e-33)
     (if (>= b 0.0) t_0 (* b (+ (/ c (pow b 2.0)) (/ -1.0 a))))
     (if (<= b -3e-294)
       (if (>= b 0.0) t_0 (/ (- t_2 b) (* 2.0 a)))
       (if (<= b 3.5e-45)
         (if (>= b 0.0) (/ (* c -2.0) (+ b t_2)) t_1)
         (if (>= b 0.0)
           (/ (* 2.0 c) (- (- b) (+ b (* c (/ (* a -2.0) b)))))
           t_1))))))

C

double code(double a, double b, double c) {
	double t_0 = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
	double t_1 = (b * -2.0) / (2.0 * a);
	double t_2 = sqrt((c * (a * -4.0)));
	double tmp_1;
	if (b <= -4.8e-33) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = b * ((c / pow(b, 2.0)) + (-1.0 / a));
		}
		tmp_1 = tmp_2;
	} else if (b <= -3e-294) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_0;
		} else {
			tmp_3 = (t_2 - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 3.5e-45) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c * -2.0) / (b + t_2);
		} else {
			tmp_4 = t_1;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-b - (b + (c * ((a * -2.0) / b))));
	} 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) :: t_2
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    real(8) :: tmp_4
    t_0 = (2.0d0 * c) / (2.0d0 * ((a * (c / b)) - b))
    t_1 = (b * (-2.0d0)) / (2.0d0 * a)
    t_2 = sqrt((c * (a * (-4.0d0))))
    if (b <= (-4.8d-33)) then
        if (b >= 0.0d0) then
            tmp_2 = t_0
        else
            tmp_2 = b * ((c / (b ** 2.0d0)) + ((-1.0d0) / a))
        end if
        tmp_1 = tmp_2
    else if (b <= (-3d-294)) then
        if (b >= 0.0d0) then
            tmp_3 = t_0
        else
            tmp_3 = (t_2 - b) / (2.0d0 * a)
        end if
        tmp_1 = tmp_3
    else if (b <= 3.5d-45) then
        if (b >= 0.0d0) then
            tmp_4 = (c * (-2.0d0)) / (b + t_2)
        else
            tmp_4 = t_1
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = (2.0d0 * c) / (-b - (b + (c * ((a * (-2.0d0)) / b))))
    else
        tmp_1 = t_1
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
	double t_1 = (b * -2.0) / (2.0 * a);
	double t_2 = Math.sqrt((c * (a * -4.0)));
	double tmp_1;
	if (b <= -4.8e-33) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = b * ((c / Math.pow(b, 2.0)) + (-1.0 / a));
		}
		tmp_1 = tmp_2;
	} else if (b <= -3e-294) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_0;
		} else {
			tmp_3 = (t_2 - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 3.5e-45) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c * -2.0) / (b + t_2);
		} else {
			tmp_4 = t_1;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-b - (b + (c * ((a * -2.0) / b))));
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (2.0 * c) / (2.0 * ((a * (c / b)) - b))
	t_1 = (b * -2.0) / (2.0 * a)
	t_2 = math.sqrt((c * (a * -4.0)))
	tmp_1 = 0
	if b <= -4.8e-33:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_0
		else:
			tmp_2 = b * ((c / math.pow(b, 2.0)) + (-1.0 / a))
		tmp_1 = tmp_2
	elif b <= -3e-294:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = t_0
		else:
			tmp_3 = (t_2 - b) / (2.0 * a)
		tmp_1 = tmp_3
	elif b <= 3.5e-45:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = (c * -2.0) / (b + t_2)
		else:
			tmp_4 = t_1
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = (2.0 * c) / (-b - (b + (c * ((a * -2.0) / b))))
	else:
		tmp_1 = t_1
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(2.0 * c) / Float64(2.0 * Float64(Float64(a * Float64(c / b)) - b)))
	t_1 = Float64(Float64(b * -2.0) / Float64(2.0 * a))
	t_2 = sqrt(Float64(c * Float64(a * -4.0)))
	tmp_1 = 0.0
	if (b <= -4.8e-33)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(b * Float64(Float64(c / (b ^ 2.0)) + Float64(-1.0 / a)));
		end
		tmp_1 = tmp_2;
	elseif (b <= -3e-294)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_0;
		else
			tmp_3 = Float64(Float64(t_2 - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (b <= 3.5e-45)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(c * -2.0) / Float64(b + t_2));
		else
			tmp_4 = t_1;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - Float64(b + Float64(c * Float64(Float64(a * -2.0) / b)))));
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
	t_1 = (b * -2.0) / (2.0 * a);
	t_2 = sqrt((c * (a * -4.0)));
	tmp_2 = 0.0;
	if (b <= -4.8e-33)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_0;
		else
			tmp_3 = b * ((c / (b ^ 2.0)) + (-1.0 / a));
		end
		tmp_2 = tmp_3;
	elseif (b <= -3e-294)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = t_0;
		else
			tmp_4 = (t_2 - b) / (2.0 * a);
		end
		tmp_2 = tmp_4;
	elseif (b <= 3.5e-45)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = (c * -2.0) / (b + t_2);
		else
			tmp_5 = t_1;
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = (2.0 * c) / (-b - (b + (c * ((a * -2.0) / b))));
	else
		tmp_2 = t_1;
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(2.0 * c), $MachinePrecision] / N[(2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * -2.0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -4.8e-33], If[GreaterEqual[b, 0.0], t$95$0, N[(b * N[(N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -3e-294], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(t$95$2 - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 3.5e-45], If[GreaterEqual[b, 0.0], N[(N[(c * -2.0), $MachinePrecision] / N[(b + t$95$2), $MachinePrecision]), $MachinePrecision], t$95$1], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[(b + N[(c * N[(N[(a * -2.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -4.8e-33

   1. Initial program 63.5%

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

   3. Taylor expanded in a around 0 63.5%

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

      1. distribute-lft-out-- 63.5%

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

      2. associate-/l* 63.5%

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

   5. Simplified 63.5%

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

   6. Taylor expanded in b around -inf 87.3%

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

      1. associate-*r* 87.3%

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

      2. mul-1-neg 87.3%

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

      3. +-commutative 87.3%

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

      4. mul-1-neg 87.3%

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

      5. unsub-neg 87.3%

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

   8. Simplified 87.3%

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

   if -4.8e-33 < b < -2.9999999999999999e-294

   1. Initial program 85.7%

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

   3. Taylor expanded in a around 0 85.7%

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

      1. distribute-lft-out-- 85.7%

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

      2. associate-/l* 85.7%

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

   5. Simplified 85.7%

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

   6. Taylor expanded in b around 0 73.0%

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

      1. associate-*r* 21.3%

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

      2. *-commutative 21.3%

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

   8. Simplified 72.6%

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

   if -2.9999999999999999e-294 < b < 3.5e-45

   1. Initial program 73.9%

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

   3. Taylor expanded in b around -inf 73.9%

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

      1. *-commutative 73.9%

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

   5. Simplified 73.9%

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

   6. Taylor expanded in b around 0 66.5%

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

      1. associate-*r* 66.5%

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

      2. *-commutative 66.5%

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

   8. Simplified 66.5%

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

      1. div-inv 66.4%

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

      2. associate-*l* 66.4%

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

   10. Applied egg-rr 66.4%

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

       1. un-div-inv 66.5%

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

       2. frac-2neg 66.5%

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

       3. *-commutative 66.5%

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

       4. distribute-rgt-neg-in 66.5%

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

       5. metadata-eval 66.5%

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

       6. neg-sub0 66.5%

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

       7. add-sqr-sqrt 0.1%

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

       8. sqrt-unprod 64.3%

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

       9. sqr-neg 64.3%

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

       10. sqrt-prod 64.2%

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

       11. add-sqr-sqrt 64.2%

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

       12. associate-+l- 64.2%

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

       13. neg-sub0 64.2%

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

       14. add-sqr-sqrt 0.1%

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

       15. sqrt-unprod 65.8%

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

       16. sqr-neg 65.8%

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

       17. sqrt-prod 66.5%

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

       18. add-sqr-sqrt 66.5%

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

       19. *-commutative 66.5%

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

       20. *-commutative 66.5%

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

       21. associate-*r* 66.5%

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

       22. *-commutative 66.5%

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

   12. Applied egg-rr 66.5%

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

   if 3.5e-45 < b

   1. Initial program 68.9%

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

   3. Taylor expanded in b around -inf 68.9%

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

      1. *-commutative 68.9%

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

   5. Simplified 68.9%

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

   6. Taylor expanded in a around 0 84.0%

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

      1. associate-*r/ 84.0%

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

      2. associate-*r* 84.0%

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

      3. *-commutative 84.0%

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

      4. *-rgt-identity 84.0%

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

      5. times-frac 90.6%

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

      6. /-rgt-identity 90.6%

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

   8. Simplified 90.6%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{-33}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq -3 \cdot 10^{-294}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-45}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + c \cdot \frac{a \cdot -2}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ t_1 := \frac{b \cdot -2}{2 \cdot a}\\ \mathbf{if}\;b \leq -4.4 \cdot 10^{+147}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array}\\ \mathbf{elif}\;b \leq -3 \cdot 10^{-294}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-47}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + c \cdot \frac{a \cdot -2}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* 2.0 c) (* 2.0 (- (* a (/ c b)) b))))
        (t_1 (/ (* b -2.0) (* 2.0 a))))
   (if (<= b -4.4e+147)
     (if (>= b 0.0) t_0 (/ b (- a)))
     (if (<= b -3e-294)
       (if (>= b 0.0)
         t_0
         (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a)))
       (if (<= b 4e-47)
         (if (>= b 0.0) (/ (* c -2.0) (+ b (sqrt (* c (* a -4.0))))) t_1)
         (if (>= b 0.0)
           (/ (* 2.0 c) (- (- b) (+ b (* c (/ (* a -2.0) b)))))
           t_1))))))

C

double code(double a, double b, double c) {
	double t_0 = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
	double t_1 = (b * -2.0) / (2.0 * a);
	double tmp_1;
	if (b <= -4.4e+147) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = b / -a;
		}
		tmp_1 = tmp_2;
	} else if (b <= -3e-294) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_0;
		} else {
			tmp_3 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 4e-47) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c * -2.0) / (b + sqrt((c * (a * -4.0))));
		} else {
			tmp_4 = t_1;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-b - (b + (c * ((a * -2.0) / b))));
	} 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 = (2.0d0 * c) / (2.0d0 * ((a * (c / b)) - b))
    t_1 = (b * (-2.0d0)) / (2.0d0 * a)
    if (b <= (-4.4d+147)) then
        if (b >= 0.0d0) then
            tmp_2 = t_0
        else
            tmp_2 = b / -a
        end if
        tmp_1 = tmp_2
    else if (b <= (-3d-294)) then
        if (b >= 0.0d0) then
            tmp_3 = t_0
        else
            tmp_3 = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (2.0d0 * a)
        end if
        tmp_1 = tmp_3
    else if (b <= 4d-47) then
        if (b >= 0.0d0) then
            tmp_4 = (c * (-2.0d0)) / (b + sqrt((c * (a * (-4.0d0)))))
        else
            tmp_4 = t_1
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = (2.0d0 * c) / (-b - (b + (c * ((a * (-2.0d0)) / b))))
    else
        tmp_1 = t_1
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
	double t_1 = (b * -2.0) / (2.0 * a);
	double tmp_1;
	if (b <= -4.4e+147) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = b / -a;
		}
		tmp_1 = tmp_2;
	} else if (b <= -3e-294) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_0;
		} else {
			tmp_3 = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 4e-47) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c * -2.0) / (b + Math.sqrt((c * (a * -4.0))));
		} else {
			tmp_4 = t_1;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-b - (b + (c * ((a * -2.0) / b))));
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (2.0 * c) / (2.0 * ((a * (c / b)) - b))
	t_1 = (b * -2.0) / (2.0 * a)
	tmp_1 = 0
	if b <= -4.4e+147:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_0
		else:
			tmp_2 = b / -a
		tmp_1 = tmp_2
	elif b <= -3e-294:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = t_0
		else:
			tmp_3 = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)
		tmp_1 = tmp_3
	elif b <= 4e-47:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = (c * -2.0) / (b + math.sqrt((c * (a * -4.0))))
		else:
			tmp_4 = t_1
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = (2.0 * c) / (-b - (b + (c * ((a * -2.0) / b))))
	else:
		tmp_1 = t_1
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(2.0 * c) / Float64(2.0 * Float64(Float64(a * Float64(c / b)) - b)))
	t_1 = Float64(Float64(b * -2.0) / Float64(2.0 * a))
	tmp_1 = 0.0
	if (b <= -4.4e+147)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(b / Float64(-a));
		end
		tmp_1 = tmp_2;
	elseif (b <= -3e-294)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_0;
		else
			tmp_3 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (b <= 4e-47)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(c * -2.0) / Float64(b + sqrt(Float64(c * Float64(a * -4.0)))));
		else
			tmp_4 = t_1;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - Float64(b + Float64(c * Float64(Float64(a * -2.0) / b)))));
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
	t_1 = (b * -2.0) / (2.0 * a);
	tmp_2 = 0.0;
	if (b <= -4.4e+147)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_0;
		else
			tmp_3 = b / -a;
		end
		tmp_2 = tmp_3;
	elseif (b <= -3e-294)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = t_0;
		else
			tmp_4 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
		end
		tmp_2 = tmp_4;
	elseif (b <= 4e-47)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = (c * -2.0) / (b + sqrt((c * (a * -4.0))));
		else
			tmp_5 = t_1;
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = (2.0 * c) / (-b - (b + (c * ((a * -2.0) / b))));
	else
		tmp_2 = t_1;
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(2.0 * c), $MachinePrecision] / N[(2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * -2.0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4.4e+147], If[GreaterEqual[b, 0.0], t$95$0, N[(b / (-a)), $MachinePrecision]], If[LessEqual[b, -3e-294], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 4e-47], If[GreaterEqual[b, 0.0], N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[(b + N[(c * N[(N[(a * -2.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -4.4000000000000003e147

   1. Initial program 36.4%

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

   3. Taylor expanded in a around 0 36.4%

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

      1. distribute-lft-out-- 36.4%

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

      2. associate-/l* 36.4%

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

   5. Simplified 36.4%

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

   6. Taylor expanded in b around -inf 95.4%

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

   if -4.4000000000000003e147 < b < -2.9999999999999999e-294

   1. Initial program 88.6%

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

   3. Taylor expanded in a around 0 88.6%

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

      1. distribute-lft-out-- 88.6%

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

      2. associate-/l* 88.6%

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

   5. Simplified 88.6%

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

   if -2.9999999999999999e-294 < b < 3.9999999999999999e-47

   1. Initial program 73.9%

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

   3. Taylor expanded in b around -inf 73.9%

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

      1. *-commutative 73.9%

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

   5. Simplified 73.9%

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

   6. Taylor expanded in b around 0 66.5%

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

      1. associate-*r* 66.5%

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

      2. *-commutative 66.5%

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

   8. Simplified 66.5%

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

      1. div-inv 66.4%

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

      2. associate-*l* 66.4%

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

   10. Applied egg-rr 66.4%

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

       1. un-div-inv 66.5%

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

       2. frac-2neg 66.5%

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

       3. *-commutative 66.5%

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

       4. distribute-rgt-neg-in 66.5%

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

       5. metadata-eval 66.5%

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

       6. neg-sub0 66.5%

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

       7. add-sqr-sqrt 0.1%

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

       8. sqrt-unprod 64.3%

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

       9. sqr-neg 64.3%

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

       10. sqrt-prod 64.2%

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

       11. add-sqr-sqrt 64.2%

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

       12. associate-+l- 64.2%

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

       13. neg-sub0 64.2%

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

       14. add-sqr-sqrt 0.1%

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

       15. sqrt-unprod 65.8%

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

       16. sqr-neg 65.8%

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

       17. sqrt-prod 66.5%

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

       18. add-sqr-sqrt 66.5%

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

       19. *-commutative 66.5%

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

       20. *-commutative 66.5%

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

       21. associate-*r* 66.5%

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

       22. *-commutative 66.5%

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

   12. Applied egg-rr 66.5%

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

   if 3.9999999999999999e-47 < b

   1. Initial program 68.9%

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

   3. Taylor expanded in b around -inf 68.9%

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

      1. *-commutative 68.9%

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

   5. Simplified 68.9%

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

   6. Taylor expanded in a around 0 84.0%

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

      1. associate-*r/ 84.0%

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

      2. associate-*r* 84.0%

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

      3. *-commutative 84.0%

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

      4. *-rgt-identity 84.0%

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

      5. times-frac 90.6%

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

      6. /-rgt-identity 90.6%

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

   8. Simplified 90.6%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{+147}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array}\\ \mathbf{elif}\;b \leq -3 \cdot 10^{-294}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-47}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + c \cdot \frac{a \cdot -2}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ \mathbf{if}\;b \leq -5 \cdot 10^{+147}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{+98}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot \left(-2\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0))))))
   (if (<= b -5e+147)
     (if (>= b 0.0) (/ (* 2.0 c) (* 2.0 (- (* a (/ c b)) b))) (/ b (- a)))
     (if (<= b 1e+98)
       (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_0)) (/ (- t_0 b) (* 2.0 a)))
       (if (>= b 0.0) (/ (* c (- 2.0)) (+ b b)) (/ (* b -2.0) (* 2.0 a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -5e+147) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
		} else {
			tmp_2 = b / -a;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1e+98) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_3 = (t_0 - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c * -2.0) / (b + b);
	} else {
		tmp_1 = (b * -2.0) / (2.0 * a);
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = sqrt(((b * b) - (c * (a * 4.0d0))))
    if (b <= (-5d+147)) then
        if (b >= 0.0d0) then
            tmp_2 = (2.0d0 * c) / (2.0d0 * ((a * (c / b)) - b))
        else
            tmp_2 = b / -a
        end if
        tmp_1 = tmp_2
    else if (b <= 1d+98) then
        if (b >= 0.0d0) then
            tmp_3 = (2.0d0 * c) / (-b - t_0)
        else
            tmp_3 = (t_0 - b) / (2.0d0 * a)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = (c * -2.0d0) / (b + b)
    else
        tmp_1 = (b * (-2.0d0)) / (2.0d0 * a)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -5e+147) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
		} else {
			tmp_2 = b / -a;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1e+98) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_3 = (t_0 - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c * -2.0) / (b + b);
	} else {
		tmp_1 = (b * -2.0) / (2.0 * a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (c * (a * 4.0))))
	tmp_1 = 0
	if b <= -5e+147:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (2.0 * c) / (2.0 * ((a * (c / b)) - b))
		else:
			tmp_2 = b / -a
		tmp_1 = tmp_2
	elif b <= 1e+98:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (2.0 * c) / (-b - t_0)
		else:
			tmp_3 = (t_0 - b) / (2.0 * a)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = (c * -2.0) / (b + b)
	else:
		tmp_1 = (b * -2.0) / (2.0 * a)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	tmp_1 = 0.0
	if (b <= -5e+147)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(2.0 * c) / Float64(2.0 * Float64(Float64(a * Float64(c / b)) - b)));
		else
			tmp_2 = Float64(b / Float64(-a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1e+98)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0));
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c * Float64(-2.0)) / Float64(b + b));
	else
		tmp_1 = Float64(Float64(b * -2.0) / Float64(2.0 * a));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	tmp_2 = 0.0;
	if (b <= -5e+147)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
		else
			tmp_3 = b / -a;
		end
		tmp_2 = tmp_3;
	elseif (b <= 1e+98)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (2.0 * c) / (-b - t_0);
		else
			tmp_4 = (t_0 - b) / (2.0 * a);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = (c * -2.0) / (b + b);
	else
		tmp_2 = (b * -2.0) / (2.0 * a);
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -5e+147], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / (-a)), $MachinePrecision]], If[LessEqual[b, 1e+98], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c * (-2.0)), $MachinePrecision] / N[(b + b), $MachinePrecision]), $MachinePrecision], N[(N[(b * -2.0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.0000000000000002e147

   1. Initial program 36.4%

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

   3. Taylor expanded in a around 0 36.4%

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

      1. distribute-lft-out-- 36.4%

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

      2. associate-/l* 36.4%

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

   5. Simplified 36.4%

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

   6. Taylor expanded in b around -inf 95.4%

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

   if -5.0000000000000002e147 < b < 9.99999999999999998e97

   1. Initial program 86.9%

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

   if 9.99999999999999998e97 < b

   1. Initial program 54.2%

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

   3. Taylor expanded in b around -inf 54.2%

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

      1. *-commutative 54.2%

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

   5. Simplified 54.2%

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

   6. Taylor expanded in b around inf 97.2%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+147}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{+98}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot \left(-2\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot -2}{2 \cdot a}\\ \mathbf{if}\;b \leq 2.9 \cdot 10^{-45}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + c \cdot \frac{a \cdot -2}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* b -2.0) (* 2.0 a))))
   (if (<= b 2.9e-45)
     (if (>= b 0.0) (* c (/ -2.0 (+ b (sqrt (* c (* a -4.0)))))) t_0)
     (if (>= b 0.0)
       (/ (* 2.0 c) (- (- b) (+ b (* c (/ (* a -2.0) b)))))
       t_0))))

C

double code(double a, double b, double c) {
	double t_0 = (b * -2.0) / (2.0 * a);
	double tmp_1;
	if (b <= 2.9e-45) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / (b + sqrt((c * (a * -4.0)))));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-b - (b + (c * ((a * -2.0) / b))));
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = (b * (-2.0d0)) / (2.0d0 * a)
    if (b <= 2.9d-45) then
        if (b >= 0.0d0) then
            tmp_2 = c * ((-2.0d0) / (b + sqrt((c * (a * (-4.0d0))))))
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (2.0d0 * c) / (-b - (b + (c * ((a * (-2.0d0)) / b))))
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (b * -2.0) / (2.0 * a);
	double tmp_1;
	if (b <= 2.9e-45) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / (b + Math.sqrt((c * (a * -4.0)))));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-b - (b + (c * ((a * -2.0) / b))));
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (b * -2.0) / (2.0 * a)
	tmp_1 = 0
	if b <= 2.9e-45:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = c * (-2.0 / (b + math.sqrt((c * (a * -4.0)))))
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = (2.0 * c) / (-b - (b + (c * ((a * -2.0) / b))))
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * -2.0) / Float64(2.0 * a))
	tmp_1 = 0.0
	if (b <= 2.9e-45)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * Float64(-2.0 / Float64(b + sqrt(Float64(c * Float64(a * -4.0))))));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - Float64(b + Float64(c * Float64(Float64(a * -2.0) / b)))));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = (b * -2.0) / (2.0 * a);
	tmp_2 = 0.0;
	if (b <= 2.9e-45)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = c * (-2.0 / (b + sqrt((c * (a * -4.0)))));
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = (2.0 * c) / (-b - (b + (c * ((a * -2.0) / b))));
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * -2.0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 2.9e-45], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[(b + N[(c * N[(N[(a * -2.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.9e-45

   1. Initial program 72.2%

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

   3. Taylor expanded in b around -inf 64.2%

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

      1. *-commutative 64.2%

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

   5. Simplified 64.2%

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

   6. Taylor expanded in b around 0 62.6%

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

      1. associate-*r* 62.6%

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

      2. *-commutative 62.6%

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

   8. Simplified 62.6%

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

      1. div-inv 62.6%

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

      2. associate-*l* 62.6%

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

   10. Applied egg-rr 62.6%

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

       1. un-div-inv 62.6%

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

       2. frac-2neg 62.6%

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

       3. *-commutative 62.6%

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

       4. distribute-rgt-neg-in 62.6%

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

       5. metadata-eval 62.6%

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

       6. neg-sub0 62.6%

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

       7. add-sqr-sqrt 48.4%

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

       8. sqrt-unprod 62.2%

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

       9. sqr-neg 62.2%

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

       10. sqrt-prod 62.1%

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

       11. add-sqr-sqrt 62.1%

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

       12. associate-+l- 62.1%

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

       13. neg-sub0 62.1%

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

       14. add-sqr-sqrt 48.4%

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

       15. sqrt-unprod 62.5%

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

       16. sqr-neg 62.5%

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

       17. sqrt-prod 62.6%

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

       18. add-sqr-sqrt 62.6%

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

       19. *-commutative 62.6%

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

       20. *-commutative 62.6%

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

       21. associate-*r* 62.6%

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

       22. *-commutative 62.6%

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

   12. Applied egg-rr 62.6%

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

       1. associate-/l* 62.6%

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

       2. *-commutative 62.6%

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

   14. Simplified 62.6%

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

   if 2.9e-45 < b

   1. Initial program 68.9%

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

   3. Taylor expanded in b around -inf 68.9%

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

      1. *-commutative 68.9%

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

   5. Simplified 68.9%

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

   6. Taylor expanded in a around 0 84.0%

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

      1. associate-*r/ 84.0%

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

      2. associate-*r* 84.0%

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

      3. *-commutative 84.0%

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

      4. *-rgt-identity 84.0%

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

      5. times-frac 90.6%

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

      6. /-rgt-identity 90.6%

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

   8. Simplified 90.6%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-45}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + c \cdot \frac{a \cdot -2}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot -2}{2 \cdot a}\\ \mathbf{if}\;b \leq 1.8 \cdot 10^{-48}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + c \cdot \frac{a \cdot -2}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* b -2.0) (* 2.0 a))))
   (if (<= b 1.8e-48)
     (if (>= b 0.0) (/ (* c -2.0) (+ b (sqrt (* c (* a -4.0))))) t_0)
     (if (>= b 0.0)
       (/ (* 2.0 c) (- (- b) (+ b (* c (/ (* a -2.0) b)))))
       t_0))))

C

double code(double a, double b, double c) {
	double t_0 = (b * -2.0) / (2.0 * a);
	double tmp_1;
	if (b <= 1.8e-48) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c * -2.0) / (b + sqrt((c * (a * -4.0))));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-b - (b + (c * ((a * -2.0) / b))));
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = (b * (-2.0d0)) / (2.0d0 * a)
    if (b <= 1.8d-48) then
        if (b >= 0.0d0) then
            tmp_2 = (c * (-2.0d0)) / (b + sqrt((c * (a * (-4.0d0)))))
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (2.0d0 * c) / (-b - (b + (c * ((a * (-2.0d0)) / b))))
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (b * -2.0) / (2.0 * a);
	double tmp_1;
	if (b <= 1.8e-48) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c * -2.0) / (b + Math.sqrt((c * (a * -4.0))));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-b - (b + (c * ((a * -2.0) / b))));
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (b * -2.0) / (2.0 * a)
	tmp_1 = 0
	if b <= 1.8e-48:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (c * -2.0) / (b + math.sqrt((c * (a * -4.0))))
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = (2.0 * c) / (-b - (b + (c * ((a * -2.0) / b))))
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * -2.0) / Float64(2.0 * a))
	tmp_1 = 0.0
	if (b <= 1.8e-48)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(c * -2.0) / Float64(b + sqrt(Float64(c * Float64(a * -4.0)))));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - Float64(b + Float64(c * Float64(Float64(a * -2.0) / b)))));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = (b * -2.0) / (2.0 * a);
	tmp_2 = 0.0;
	if (b <= 1.8e-48)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (c * -2.0) / (b + sqrt((c * (a * -4.0))));
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = (2.0 * c) / (-b - (b + (c * ((a * -2.0) / b))));
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * -2.0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.8e-48], If[GreaterEqual[b, 0.0], N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[(b + N[(c * N[(N[(a * -2.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.8000000000000001e-48

   1. Initial program 72.2%

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

   3. Taylor expanded in b around -inf 64.2%

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

      1. *-commutative 64.2%

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

   5. Simplified 64.2%

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

   6. Taylor expanded in b around 0 62.6%

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

      1. associate-*r* 62.6%

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

      2. *-commutative 62.6%

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

   8. Simplified 62.6%

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

      1. div-inv 62.6%

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

      2. associate-*l* 62.6%

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

   10. Applied egg-rr 62.6%

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

       1. un-div-inv 62.6%

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

       2. frac-2neg 62.6%

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

       3. *-commutative 62.6%

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

       4. distribute-rgt-neg-in 62.6%

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

       5. metadata-eval 62.6%

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

       6. neg-sub0 62.6%

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

       7. add-sqr-sqrt 48.4%

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

       8. sqrt-unprod 62.2%

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

       9. sqr-neg 62.2%

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

       10. sqrt-prod 62.1%

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

       11. add-sqr-sqrt 62.1%

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

       12. associate-+l- 62.1%

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

       13. neg-sub0 62.1%

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

       14. add-sqr-sqrt 48.4%

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

       15. sqrt-unprod 62.5%

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

       16. sqr-neg 62.5%

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

       17. sqrt-prod 62.6%

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

       18. add-sqr-sqrt 62.6%

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

       19. *-commutative 62.6%

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

       20. *-commutative 62.6%

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

       21. associate-*r* 62.6%

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

       22. *-commutative 62.6%

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

   12. Applied egg-rr 62.6%

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

   if 1.8000000000000001e-48 < b

   1. Initial program 68.9%

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

   3. Taylor expanded in b around -inf 68.9%

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

      1. *-commutative 68.9%

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

   5. Simplified 68.9%

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

   6. Taylor expanded in a around 0 84.0%

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

      1. associate-*r/ 84.0%

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

      2. associate-*r* 84.0%

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

      3. *-commutative 84.0%

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

      4. *-rgt-identity 84.0%

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

      5. times-frac 90.6%

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

      6. /-rgt-identity 90.6%

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

   8. Simplified 90.6%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-48}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + c \cdot \frac{a \cdot -2}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.8% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.9%

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

3. Taylor expanded in a around 0 69.4%

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

   1. distribute-lft-out-- 69.4%

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

   2. associate-/l* 72.1%

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

5. Simplified 72.1%

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

6. Taylor expanded in b around -inf 67.6%

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

7. Final simplification 67.6%

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

Alternative 8: 67.6% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.9%

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

3. Taylor expanded in b around -inf 66.1%

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

   1. *-commutative 66.1%

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

5. Simplified 66.1%

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

6. Taylor expanded in b around inf 67.2%

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

7. Final simplification 67.2%

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

Alternative 9: 47.5% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.9%

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

3. Taylor expanded in b around -inf 66.1%

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

   1. *-commutative 66.1%

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

5. Simplified 66.1%

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

6. Taylor expanded in b around 0 48.3%

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

   1. associate-*r* 48.3%

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

   2. *-commutative 48.3%

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

8. Simplified 48.3%

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

9. Taylor expanded in b around inf 43.9%

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

10. Final simplification 43.9%

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

Reproduce

herbie shell --seed 2024089 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))