jeff quadratic root 2

Percentage Accurate: 72.1% → 90.1%

Time: 31.0s

Alternatives: 6

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ \mathbf{if}\;b \leq -1.35 \cdot 10^{+154} \lor \neg \left(b \leq 7.5 \cdot 10^{+75}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0))))))
   (if (or (<= b -1.35e+154) (not (<= b 7.5e+75)))
     (if (>= b 0.0) (/ (* 2.0 c) (- (- b) b)) (/ (* b -2.0) (* 2.0 a)))
     (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_0)) (/ (- t_0 b) (* 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 <= -1.35e+154) || !(b <= 7.5e+75)) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (-b - b);
		} else {
			tmp_2 = (b * -2.0) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-b - t_0);
	} else {
		tmp_1 = (t_0 - b) / (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
    t_0 = sqrt(((b * b) - (c * (a * 4.0d0))))
    if ((b <= (-1.35d+154)) .or. (.not. (b <= 7.5d+75))) then
        if (b >= 0.0d0) then
            tmp_2 = (2.0d0 * c) / (-b - b)
        else
            tmp_2 = (b * (-2.0d0)) / (2.0d0 * a)
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (2.0d0 * c) / (-b - t_0)
    else
        tmp_1 = (t_0 - b) / (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 <= -1.35e+154) || !(b <= 7.5e+75)) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (-b - b);
		} else {
			tmp_2 = (b * -2.0) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-b - t_0);
	} else {
		tmp_1 = (t_0 - b) / (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 <= -1.35e+154) or not (b <= 7.5e+75):
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (2.0 * c) / (-b - b)
		else:
			tmp_2 = (b * -2.0) / (2.0 * a)
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = (2.0 * c) / (-b - t_0)
	else:
		tmp_1 = (t_0 - b) / (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 <= -1.35e+154) || !(b <= 7.5e+75))
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - b));
		else
			tmp_2 = Float64(Float64(b * -2.0) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0));
	else
		tmp_1 = Float64(Float64(t_0 - b) / Float64(2.0 * a));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	tmp_2 = 0.0;
	if ((b <= -1.35e+154) || ~((b <= 7.5e+75)))
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (2.0 * c) / (-b - b);
		else
			tmp_3 = (b * -2.0) / (2.0 * a);
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = (2.0 * c) / (-b - t_0);
	else
		tmp_2 = (t_0 - b) / (2.0 * a);
	end
	tmp_4 = 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[Or[LessEqual[b, -1.35e+154], N[Not[LessEqual[b, 7.5e+75]], $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - b), $MachinePrecision]), $MachinePrecision], N[(N[(b * -2.0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], 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]]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.35000000000000003e154 or 7.4999999999999995e75 < b

   1. Initial program 47.8%

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

   3. Taylor expanded in b around inf 76.1%

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

   4. Taylor expanded in b around -inf 96.6%

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

      1. *-commutative 96.6%

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

   6. Simplified 96.6%

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

   if -1.35000000000000003e154 < b < 7.4999999999999995e75

   1. Initial program 90.9%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+154} \lor \neg \left(b \leq 7.5 \cdot 10^{+75}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ t_1 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array}\\ \mathbf{if}\;b \leq -5 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{1}{\frac{\left(b - b\right) - b \cdot \left(-2 \cdot \left(\frac{c}{b} \cdot \frac{a}{b}\right)\right)}{2 \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{+75}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\frac{-1}{a} - c \cdot \frac{\frac{-1}{b}}{b}\right)\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0)))))
        (t_1
         (if (>= b 0.0) (/ (* 2.0 c) (- (- b) b)) (/ (* b -2.0) (* 2.0 a)))))
   (if (<= b -5e+154)
     t_1
     (if (<= b -1e-310)
       (if (>= b 0.0)
         (/ 1.0 (/ (- (- b b) (* b (* -2.0 (* (/ c b) (/ a b))))) (* 2.0 c)))
         (/ (- t_0 b) (* 2.0 a)))
       (if (<= b 7.5e+75)
         (if (>= b 0.0)
           (/ (* 2.0 c) (- (- b) t_0))
           (* b (- (/ -1.0 a) (* c (/ (/ -1.0 b) b)))))
         t_1)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (-b - b);
	} else {
		tmp = (b * -2.0) / (2.0 * a);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b <= -5e+154) {
		tmp_1 = t_1;
	} else if (b <= -1e-310) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 1.0 / (((b - b) - (b * (-2.0 * ((c / b) * (a / b))))) / (2.0 * c));
		} else {
			tmp_2 = (t_0 - b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 7.5e+75) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_3 = b * ((-1.0 / a) - (c * ((-1.0 / b) / b)));
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = sqrt(((b * b) - (c * (a * 4.0d0))))
    if (b >= 0.0d0) then
        tmp = (2.0d0 * c) / (-b - b)
    else
        tmp = (b * (-2.0d0)) / (2.0d0 * a)
    end if
    t_1 = tmp
    if (b <= (-5d+154)) then
        tmp_1 = t_1
    else if (b <= (-1d-310)) then
        if (b >= 0.0d0) then
            tmp_2 = 1.0d0 / (((b - b) - (b * ((-2.0d0) * ((c / b) * (a / b))))) / (2.0d0 * c))
        else
            tmp_2 = (t_0 - b) / (2.0d0 * a)
        end if
        tmp_1 = tmp_2
    else if (b <= 7.5d+75) then
        if (b >= 0.0d0) then
            tmp_3 = (2.0d0 * c) / (-b - t_0)
        else
            tmp_3 = b * (((-1.0d0) / a) - (c * (((-1.0d0) / b) / b)))
        end if
        tmp_1 = tmp_3
    else
        tmp_1 = t_1
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (c * (a * 4.0))));
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (-b - b);
	} else {
		tmp = (b * -2.0) / (2.0 * a);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b <= -5e+154) {
		tmp_1 = t_1;
	} else if (b <= -1e-310) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 1.0 / (((b - b) - (b * (-2.0 * ((c / b) * (a / b))))) / (2.0 * c));
		} else {
			tmp_2 = (t_0 - b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 7.5e+75) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_3 = b * ((-1.0 / a) - (c * ((-1.0 / b) / b)));
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (c * (a * 4.0))))
	tmp = 0
	if b >= 0.0:
		tmp = (2.0 * c) / (-b - b)
	else:
		tmp = (b * -2.0) / (2.0 * a)
	t_1 = tmp
	tmp_1 = 0
	if b <= -5e+154:
		tmp_1 = t_1
	elif b <= -1e-310:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = 1.0 / (((b - b) - (b * (-2.0 * ((c / b) * (a / b))))) / (2.0 * c))
		else:
			tmp_2 = (t_0 - b) / (2.0 * a)
		tmp_1 = tmp_2
	elif b <= 7.5e+75:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (2.0 * c) / (-b - t_0)
		else:
			tmp_3 = b * ((-1.0 / a) - (c * ((-1.0 / b) / b)))
		tmp_1 = tmp_3
	else:
		tmp_1 = t_1
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) - b));
	else
		tmp = Float64(Float64(b * -2.0) / Float64(2.0 * a));
	end
	t_1 = tmp
	tmp_1 = 0.0
	if (b <= -5e+154)
		tmp_1 = t_1;
	elseif (b <= -1e-310)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(1.0 / Float64(Float64(Float64(b - b) - Float64(b * Float64(-2.0 * Float64(Float64(c / b) * Float64(a / b))))) / Float64(2.0 * c)));
		else
			tmp_2 = Float64(Float64(t_0 - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 7.5e+75)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0));
		else
			tmp_3 = Float64(b * Float64(Float64(-1.0 / a) - Float64(c * Float64(Float64(-1.0 / b) / b))));
		end
		tmp_1 = tmp_3;
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (2.0 * c) / (-b - b);
	else
		tmp = (b * -2.0) / (2.0 * a);
	end
	t_1 = tmp;
	tmp_2 = 0.0;
	if (b <= -5e+154)
		tmp_2 = t_1;
	elseif (b <= -1e-310)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = 1.0 / (((b - b) - (b * (-2.0 * ((c / b) * (a / b))))) / (2.0 * c));
		else
			tmp_3 = (t_0 - b) / (2.0 * a);
		end
		tmp_2 = tmp_3;
	elseif (b <= 7.5e+75)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (2.0 * c) / (-b - t_0);
		else
			tmp_4 = b * ((-1.0 / a) - (c * ((-1.0 / b) / b)));
		end
		tmp_2 = tmp_4;
	else
		tmp_2 = t_1;
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - b), $MachinePrecision]), $MachinePrecision], N[(N[(b * -2.0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]}, If[LessEqual[b, -5e+154], t$95$1, If[LessEqual[b, -1e-310], If[GreaterEqual[b, 0.0], N[(1.0 / N[(N[(N[(b - b), $MachinePrecision] - N[(b * N[(-2.0 * N[(N[(c / b), $MachinePrecision] * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 7.5e+75], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(b * N[(N[(-1.0 / a), $MachinePrecision] - N[(c * N[(N[(-1.0 / b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.00000000000000004e154 or 7.4999999999999995e75 < b

   1. Initial program 47.8%

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

   3. Taylor expanded in b around inf 76.1%

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

   4. Taylor expanded in b around -inf 96.6%

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

      1. *-commutative 96.6%

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

   6. Simplified 96.6%

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

   if -5.00000000000000004e154 < b < -9.999999999999969e-311

   1. Initial program 88.7%

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

   3. Taylor expanded in b around inf 88.7%

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

      1. distribute-lft-in 88.7%

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

      2. *-rgt-identity 88.7%

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

      3. *-commutative 88.7%

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

      4. unpow2 88.7%

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

      5. times-frac 88.7%

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

   5. Simplified 88.7%

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

      1. clear-num 88.7%

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

      2. clear-num 88.7%

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

      3. associate--r+ 88.7%

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

      4. add-sqr-sqrt 88.7%

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

      5. sqrt-unprod 88.7%

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

      6. sqr-neg 88.7%

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

      7. sqrt-prod 88.7%

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

      8. add-sqr-sqrt 88.7%

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

      9. *-commutative 88.7%

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

   7. Applied egg-rr 88.7%

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

   if -9.999999999999969e-311 < b < 7.4999999999999995e75

   1. Initial program 93.6%

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

   3. Taylor expanded in b around -inf 93.6%

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

      1. mul-1-neg 93.6%

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

      2. *-commutative 93.6%

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

      3. distribute-rgt-neg-in 93.6%

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

      4. +-commutative 93.6%

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

      5. mul-1-neg 93.6%

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

      6. unsub-neg 93.6%

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

      7. unpow2 93.6%

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

   5. Simplified 93.6%

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

      1. associate-/r* 93.6%

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

      2. div-inv 93.6%

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

      3. *-un-lft-identity 93.6%

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

      4. times-frac 93.6%

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

   7. Applied egg-rr 93.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{a} - \frac{c}{1} \cdot \frac{\frac{1}{b}}{b}\right) \cdot \left(-b\right)\\ \end{array} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{1}{\frac{\left(b - b\right) - b \cdot \left(-2 \cdot \left(\frac{c}{b} \cdot \frac{a}{b}\right)\right)}{2 \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{+75}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\frac{-1}{a} - c \cdot \frac{\frac{-1}{b}}{b}\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-b\right) - b\\ t_1 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array}\\ \mathbf{if}\;b \leq -1 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -9.5 \cdot 10^{-304}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{1}{\frac{\left(b - b\right) - b \cdot \left(-2 \cdot \left(\frac{c}{b} \cdot \frac{a}{b}\right)\right)}{2 \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-38}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(c \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2 \cdot a}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (- b) b))
        (t_1 (if (>= b 0.0) (/ (* 2.0 c) t_0) (/ (* b -2.0) (* 2.0 a)))))
   (if (<= b -1e+154)
     t_1
     (if (<= b -9.5e-304)
       (if (>= b 0.0)
         (/ 1.0 (/ (- (- b b) (* b (* -2.0 (* (/ c b) (/ a b))))) (* 2.0 c)))
         (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a)))
       (if (<= b 3.4e-38)
         (if (>= b 0.0)
           (/ (* 2.0 c) (- (- b) (sqrt (* -4.0 (* c a)))))
           (/ t_0 (* 2.0 a)))
         t_1)))))

C

double code(double a, double b, double c) {
	double t_0 = -b - b;
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / t_0;
	} else {
		tmp = (b * -2.0) / (2.0 * a);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b <= -1e+154) {
		tmp_1 = t_1;
	} else if (b <= -9.5e-304) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 1.0 / (((b - b) - (b * (-2.0 * ((c / b) * (a / b))))) / (2.0 * c));
		} else {
			tmp_2 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 3.4e-38) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * c) / (-b - sqrt((-4.0 * (c * a))));
		} else {
			tmp_3 = t_0 / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = -b - b
    if (b >= 0.0d0) then
        tmp = (2.0d0 * c) / t_0
    else
        tmp = (b * (-2.0d0)) / (2.0d0 * a)
    end if
    t_1 = tmp
    if (b <= (-1d+154)) then
        tmp_1 = t_1
    else if (b <= (-9.5d-304)) then
        if (b >= 0.0d0) then
            tmp_2 = 1.0d0 / (((b - b) - (b * ((-2.0d0) * ((c / b) * (a / b))))) / (2.0d0 * c))
        else
            tmp_2 = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (2.0d0 * a)
        end if
        tmp_1 = tmp_2
    else if (b <= 3.4d-38) then
        if (b >= 0.0d0) then
            tmp_3 = (2.0d0 * c) / (-b - sqrt(((-4.0d0) * (c * a))))
        else
            tmp_3 = t_0 / (2.0d0 * a)
        end if
        tmp_1 = tmp_3
    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 = -b - b;
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / t_0;
	} else {
		tmp = (b * -2.0) / (2.0 * a);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b <= -1e+154) {
		tmp_1 = t_1;
	} else if (b <= -9.5e-304) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 1.0 / (((b - b) - (b * (-2.0 * ((c / b) * (a / b))))) / (2.0 * c));
		} else {
			tmp_2 = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 3.4e-38) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * c) / (-b - Math.sqrt((-4.0 * (c * a))));
		} else {
			tmp_3 = t_0 / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = -b - b
	tmp = 0
	if b >= 0.0:
		tmp = (2.0 * c) / t_0
	else:
		tmp = (b * -2.0) / (2.0 * a)
	t_1 = tmp
	tmp_1 = 0
	if b <= -1e+154:
		tmp_1 = t_1
	elif b <= -9.5e-304:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = 1.0 / (((b - b) - (b * (-2.0 * ((c / b) * (a / b))))) / (2.0 * c))
		else:
			tmp_2 = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)
		tmp_1 = tmp_2
	elif b <= 3.4e-38:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (2.0 * c) / (-b - math.sqrt((-4.0 * (c * a))))
		else:
			tmp_3 = t_0 / (2.0 * a)
		tmp_1 = tmp_3
	else:
		tmp_1 = t_1
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-b) - b)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(2.0 * c) / t_0);
	else
		tmp = Float64(Float64(b * -2.0) / Float64(2.0 * a));
	end
	t_1 = tmp
	tmp_1 = 0.0
	if (b <= -1e+154)
		tmp_1 = t_1;
	elseif (b <= -9.5e-304)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(1.0 / Float64(Float64(Float64(b - b) - Float64(b * Float64(-2.0 * Float64(Float64(c / b) * Float64(a / b))))) / Float64(2.0 * c)));
		else
			tmp_2 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 3.4e-38)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - sqrt(Float64(-4.0 * Float64(c * a)))));
		else
			tmp_3 = Float64(t_0 / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = -b - b;
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (2.0 * c) / t_0;
	else
		tmp = (b * -2.0) / (2.0 * a);
	end
	t_1 = tmp;
	tmp_2 = 0.0;
	if (b <= -1e+154)
		tmp_2 = t_1;
	elseif (b <= -9.5e-304)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = 1.0 / (((b - b) - (b * (-2.0 * ((c / b) * (a / b))))) / (2.0 * c));
		else
			tmp_3 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
		end
		tmp_2 = tmp_3;
	elseif (b <= 3.4e-38)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (2.0 * c) / (-b - sqrt((-4.0 * (c * a))));
		else
			tmp_4 = t_0 / (2.0 * a);
		end
		tmp_2 = tmp_4;
	else
		tmp_2 = t_1;
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) - b), $MachinePrecision]}, Block[{t$95$1 = If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(b * -2.0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]}, If[LessEqual[b, -1e+154], t$95$1, If[LessEqual[b, -9.5e-304], If[GreaterEqual[b, 0.0], N[(1.0 / N[(N[(N[(b - b), $MachinePrecision] - N[(b * N[(-2.0 * N[(N[(c / b), $MachinePrecision] * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 3.4e-38], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.00000000000000004e154 or 3.4000000000000002e-38 < b

   1. Initial program 56.2%

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

   3. Taylor expanded in b around inf 75.4%

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

   4. Taylor expanded in b around -inf 92.2%

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

      1. *-commutative 92.2%

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

   6. Simplified 92.2%

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

   if -1.00000000000000004e154 < b < -9.50000000000000023e-304

   1. Initial program 89.7%

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

   3. Taylor expanded in b around inf 89.7%

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

      1. distribute-lft-in 89.7%

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

      2. *-rgt-identity 89.7%

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

      3. *-commutative 89.7%

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

      4. unpow2 89.7%

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

      5. times-frac 89.7%

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

   5. Simplified 89.7%

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

      1. clear-num 89.7%

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

      2. clear-num 89.7%

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

      3. associate--r+ 89.7%

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

      4. add-sqr-sqrt 89.7%

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

      5. sqrt-unprod 89.7%

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

      6. sqr-neg 89.7%

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

      7. sqrt-prod 89.7%

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

      8. add-sqr-sqrt 89.7%

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

      9. *-commutative 89.7%

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

   7. Applied egg-rr 89.7%

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

   if -9.50000000000000023e-304 < b < 3.4000000000000002e-38

   1. Initial program 90.7%

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

   3. Taylor expanded in b around -inf 90.7%

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

   4. Taylor expanded in b around 0 80.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + -1 \cdot b}{2 \cdot a}\\ \end{array} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq -9.5 \cdot 10^{-304}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{1}{\frac{\left(b - b\right) - b \cdot \left(-2 \cdot \left(\frac{c}{b} \cdot \frac{a}{b}\right)\right)}{2 \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-38}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(c \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-4 \cdot \left(c \cdot a\right)}\\ t_1 := \left(-b\right) - b\\ t_2 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array}\\ \mathbf{if}\;b \leq -9 \cdot 10^{-61}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;b \leq -9.5 \cdot 10^{-304}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + b \cdot \left(-2 \cdot \left(\frac{c}{b} \cdot \frac{a}{b}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{-41}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{2 \cdot a}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* -4.0 (* c a))))
        (t_1 (- (- b) b))
        (t_2 (if (>= b 0.0) (/ (* 2.0 c) t_1) (/ (* b -2.0) (* 2.0 a)))))
   (if (<= b -9e-61)
     t_2
     (if (<= b -9.5e-304)
       (if (>= b 0.0)
         (/ (* 2.0 c) (- (- b) (+ b (* b (* -2.0 (* (/ c b) (/ a b)))))))
         (/ (- t_0 b) (* 2.0 a)))
       (if (<= b 1.02e-41)
         (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_0)) (/ t_1 (* 2.0 a)))
         t_2)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((-4.0 * (c * a)));
	double t_1 = -b - b;
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / t_1;
	} else {
		tmp = (b * -2.0) / (2.0 * a);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b <= -9e-61) {
		tmp_1 = t_2;
	} else if (b <= -9.5e-304) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (-b - (b + (b * (-2.0 * ((c / b) * (a / b))))));
		} else {
			tmp_2 = (t_0 - b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.02e-41) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_3 = t_1 / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = t_2;
	}
	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
    t_0 = sqrt(((-4.0d0) * (c * a)))
    t_1 = -b - b
    if (b >= 0.0d0) then
        tmp = (2.0d0 * c) / t_1
    else
        tmp = (b * (-2.0d0)) / (2.0d0 * a)
    end if
    t_2 = tmp
    if (b <= (-9d-61)) then
        tmp_1 = t_2
    else if (b <= (-9.5d-304)) then
        if (b >= 0.0d0) then
            tmp_2 = (2.0d0 * c) / (-b - (b + (b * ((-2.0d0) * ((c / b) * (a / b))))))
        else
            tmp_2 = (t_0 - b) / (2.0d0 * a)
        end if
        tmp_1 = tmp_2
    else if (b <= 1.02d-41) then
        if (b >= 0.0d0) then
            tmp_3 = (2.0d0 * c) / (-b - t_0)
        else
            tmp_3 = t_1 / (2.0d0 * a)
        end if
        tmp_1 = tmp_3
    else
        tmp_1 = t_2
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt((-4.0 * (c * a)));
	double t_1 = -b - b;
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / t_1;
	} else {
		tmp = (b * -2.0) / (2.0 * a);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b <= -9e-61) {
		tmp_1 = t_2;
	} else if (b <= -9.5e-304) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (-b - (b + (b * (-2.0 * ((c / b) * (a / b))))));
		} else {
			tmp_2 = (t_0 - b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.02e-41) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_3 = t_1 / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = t_2;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt((-4.0 * (c * a)))
	t_1 = -b - b
	tmp = 0
	if b >= 0.0:
		tmp = (2.0 * c) / t_1
	else:
		tmp = (b * -2.0) / (2.0 * a)
	t_2 = tmp
	tmp_1 = 0
	if b <= -9e-61:
		tmp_1 = t_2
	elif b <= -9.5e-304:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (2.0 * c) / (-b - (b + (b * (-2.0 * ((c / b) * (a / b))))))
		else:
			tmp_2 = (t_0 - b) / (2.0 * a)
		tmp_1 = tmp_2
	elif b <= 1.02e-41:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (2.0 * c) / (-b - t_0)
		else:
			tmp_3 = t_1 / (2.0 * a)
		tmp_1 = tmp_3
	else:
		tmp_1 = t_2
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(-4.0 * Float64(c * a)))
	t_1 = Float64(Float64(-b) - b)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(2.0 * c) / t_1);
	else
		tmp = Float64(Float64(b * -2.0) / Float64(2.0 * a));
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b <= -9e-61)
		tmp_1 = t_2;
	elseif (b <= -9.5e-304)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - Float64(b + Float64(b * Float64(-2.0 * Float64(Float64(c / b) * Float64(a / b)))))));
		else
			tmp_2 = Float64(Float64(t_0 - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.02e-41)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0));
		else
			tmp_3 = Float64(t_1 / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	else
		tmp_1 = t_2;
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt((-4.0 * (c * a)));
	t_1 = -b - b;
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (2.0 * c) / t_1;
	else
		tmp = (b * -2.0) / (2.0 * a);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b <= -9e-61)
		tmp_2 = t_2;
	elseif (b <= -9.5e-304)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (2.0 * c) / (-b - (b + (b * (-2.0 * ((c / b) * (a / b))))));
		else
			tmp_3 = (t_0 - b) / (2.0 * a);
		end
		tmp_2 = tmp_3;
	elseif (b <= 1.02e-41)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (2.0 * c) / (-b - t_0);
		else
			tmp_4 = t_1 / (2.0 * a);
		end
		tmp_2 = tmp_4;
	else
		tmp_2 = t_2;
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-b) - b), $MachinePrecision]}, Block[{t$95$2 = If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(b * -2.0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]}, If[LessEqual[b, -9e-61], t$95$2, If[LessEqual[b, -9.5e-304], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[(b + N[(b * N[(-2.0 * N[(N[(c / b), $MachinePrecision] * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.02e-41], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -9e-61 or 1.02e-41 < b

   1. Initial program 66.5%

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

   3. Taylor expanded in b around inf 80.2%

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

   4. Taylor expanded in b around -inf 87.3%

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

      1. *-commutative 87.3%

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

   6. Simplified 87.3%

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

   if -9e-61 < b < -9.50000000000000023e-304

   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%

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

      1. distribute-lft-in 84.9%

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

      2. *-rgt-identity 84.9%

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

      3. *-commutative 84.9%

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

      4. unpow2 84.9%

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

      5. times-frac 84.9%

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

   5. Simplified 84.9%

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

   6. Taylor expanded in b around 0 82.2%

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

   if -9.50000000000000023e-304 < b < 1.02e-41

   1. Initial program 90.7%

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

   3. Taylor expanded in b around -inf 90.7%

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

   4. Taylor expanded in b around 0 80.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + -1 \cdot b}{2 \cdot a}\\ \end{array} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-61}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq -9.5 \cdot 10^{-304}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + b \cdot \left(-2 \cdot \left(\frac{c}{b} \cdot \frac{a}{b}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(c \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{-41}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(c \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-b\right) - b\\ \mathbf{if}\;b \leq 5.3 \cdot 10^{-40}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(c \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (- b) b)))
   (if (<= b 5.3e-40)
     (if (>= b 0.0)
       (/ (* 2.0 c) (- (- b) (sqrt (* -4.0 (* c a)))))
       (/ t_0 (* 2.0 a)))
     (if (>= b 0.0) (/ (* 2.0 c) t_0) (/ (* b -2.0) (* 2.0 a))))))

C

double code(double a, double b, double c) {
	double t_0 = -b - b;
	double tmp_1;
	if (b <= 5.3e-40) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (-b - sqrt((-4.0 * (c * a))));
		} else {
			tmp_2 = t_0 / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / t_0;
	} else {
		tmp_1 = (b * -2.0) / (2.0 * a);
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = -b - b
    if (b <= 5.3d-40) then
        if (b >= 0.0d0) then
            tmp_2 = (2.0d0 * c) / (-b - sqrt(((-4.0d0) * (c * a))))
        else
            tmp_2 = t_0 / (2.0d0 * a)
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (2.0d0 * c) / t_0
    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 = -b - b;
	double tmp_1;
	if (b <= 5.3e-40) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (-b - Math.sqrt((-4.0 * (c * a))));
		} else {
			tmp_2 = t_0 / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / t_0;
	} else {
		tmp_1 = (b * -2.0) / (2.0 * a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = -b - b
	tmp_1 = 0
	if b <= 5.3e-40:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (2.0 * c) / (-b - math.sqrt((-4.0 * (c * a))))
		else:
			tmp_2 = t_0 / (2.0 * a)
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = (2.0 * c) / t_0
	else:
		tmp_1 = (b * -2.0) / (2.0 * a)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-b) - b)
	tmp_1 = 0.0
	if (b <= 5.3e-40)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - sqrt(Float64(-4.0 * Float64(c * a)))));
		else
			tmp_2 = Float64(t_0 / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / t_0);
	else
		tmp_1 = Float64(Float64(b * -2.0) / Float64(2.0 * a));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = -b - b;
	tmp_2 = 0.0;
	if (b <= 5.3e-40)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (2.0 * c) / (-b - sqrt((-4.0 * (c * a))));
		else
			tmp_3 = t_0 / (2.0 * a);
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = (2.0 * c) / t_0;
	else
		tmp_2 = (b * -2.0) / (2.0 * a);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) - b), $MachinePrecision]}, If[LessEqual[b, 5.3e-40], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(b * -2.0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 5.3000000000000002e-40

   1. Initial program 79.2%

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

   3. Taylor expanded in b around -inf 74.2%

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

   4. Taylor expanded in b around 0 71.6%

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

   if 5.3000000000000002e-40 < b

   1. Initial program 60.8%

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

   3. Taylor expanded in b around inf 88.7%

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

   4. Taylor expanded in b around -inf 88.7%

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

      1. *-commutative 88.7%

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

   6. Simplified 88.7%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.3 \cdot 10^{-40}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(c \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.2% accurate, 9.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (-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 = (2.0d0 * c) / (-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 = (2.0 * c) / (-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 = (2.0 * c) / (-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(2.0 * c) / Float64(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 = (2.0 * c) / (-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[(2.0 * c), $MachinePrecision] / N[((-b) - b), $MachinePrecision]), $MachinePrecision], N[(N[(b * -2.0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 72.7%

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

3. Taylor expanded in b around inf 69.9%

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

4. Taylor expanded in b around -inf 66.7%

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

   1. *-commutative 66.7%

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

6. Simplified 66.7%

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

Reproduce

herbie shell --seed 2024097 
(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))))