jeff quadratic root 2

Percentage Accurate: 72.2% → 90.6%

Time: 16.5s

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: 72.2% accurate, 1.0× speedup

Math

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

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: 90.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}\\ \mathbf{if}\;b \leq -6 \cdot 10^{+126}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+105}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a \cdot 2}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* b b) (* (* a c) -4.0)))))
   (if (<= b -6e+126)
     (- 0.0 (/ b a))
     (if (<= b 3.1e+105)
       (if (>= b 0.0) (/ (* c -2.0) (+ b t_0)) (/ (- t_0 b) (* a 2.0)))
       (/ c (- 0.0 b))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) + ((a * c) * -4.0)));
	double tmp;
	if (b <= -6e+126) {
		tmp = 0.0 - (b / a);
	} else if (b <= 3.1e+105) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = (c * -2.0) / (b + t_0);
		} else {
			tmp_1 = (t_0 - b) / (a * 2.0);
		}
		tmp = tmp_1;
	} else {
		tmp = c / (0.0 - b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    t_0 = sqrt(((b * b) + ((a * c) * (-4.0d0))))
    if (b <= (-6d+126)) then
        tmp = 0.0d0 - (b / a)
    else if (b <= 3.1d+105) then
        if (b >= 0.0d0) then
            tmp_1 = (c * (-2.0d0)) / (b + t_0)
        else
            tmp_1 = (t_0 - b) / (a * 2.0d0)
        end if
        tmp = tmp_1
    else
        tmp = c / (0.0d0 - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) + ((a * c) * -4.0)));
	double tmp;
	if (b <= -6e+126) {
		tmp = 0.0 - (b / a);
	} else if (b <= 3.1e+105) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = (c * -2.0) / (b + t_0);
		} else {
			tmp_1 = (t_0 - b) / (a * 2.0);
		}
		tmp = tmp_1;
	} else {
		tmp = c / (0.0 - b);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) + ((a * c) * -4.0)))
	tmp = 0
	if b <= -6e+126:
		tmp = 0.0 - (b / a)
	elif b <= 3.1e+105:
		tmp_1 = 0
		if b >= 0.0:
			tmp_1 = (c * -2.0) / (b + t_0)
		else:
			tmp_1 = (t_0 - b) / (a * 2.0)
		tmp = tmp_1
	else:
		tmp = c / (0.0 - b)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) + Float64(Float64(a * c) * -4.0)))
	tmp = 0.0
	if (b <= -6e+126)
		tmp = Float64(0.0 - Float64(b / a));
	elseif (b <= 3.1e+105)
		tmp_1 = 0.0
		if (b >= 0.0)
			tmp_1 = Float64(Float64(c * -2.0) / Float64(b + t_0));
		else
			tmp_1 = Float64(Float64(t_0 - b) / Float64(a * 2.0));
		end
		tmp = tmp_1;
	else
		tmp = Float64(c / Float64(0.0 - b));
	end
	return tmp
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = sqrt(((b * b) + ((a * c) * -4.0)));
	tmp = 0.0;
	if (b <= -6e+126)
		tmp = 0.0 - (b / a);
	elseif (b <= 3.1e+105)
		tmp_2 = 0.0;
		if (b >= 0.0)
			tmp_2 = (c * -2.0) / (b + t_0);
		else
			tmp_2 = (t_0 - b) / (a * 2.0);
		end
		tmp = tmp_2;
	else
		tmp = c / (0.0 - b);
	end
	tmp_3 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -6e+126], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.1e+105], If[GreaterEqual[b, 0.0], N[(N[(c * -2.0), $MachinePrecision] / N[(b + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], N[(c / N[(0.0 - b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.0000000000000005e126

   1. Initial program 45.3%

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

   3. Simplified 45.3%

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

      1. div-inv N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. associate-*l/ N/A

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

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

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot c\right), \color{blue}{\left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \left(\color{blue}{b} + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(-4 \cdot \left(a \cdot c\right) + {b}^{2}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + -4 \cdot \left(a \cdot c\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot -4\right)\right)\right)\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({b}^{2}\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right)\right) \]

      17. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(-4 \cdot \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(c \cdot a\right)\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 2.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right)\right)\right) \]

   9. Simplified 2.0%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}} \]

   10. Taylor expanded in b around -inf

       \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\frac{b}{a}} \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right) \]

       4. /-lowering-/.f64 91.3%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   12. Simplified 91.3%

       \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   13. Step-by-step derivation

       1. sub0-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{b}{a}\right)\right) \]

       3. /-lowering-/.f64 91.3%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(b, a\right)\right) \]

   14. Applied egg-rr 91.3%

       \[\leadsto \color{blue}{-\frac{b}{a}} \]

   if -6.0000000000000005e126 < b < 3.10000000000000004e105

   1. Initial program 91.8%

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

   3. Simplified 91.8%

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

   if 3.10000000000000004e105 < b

   1. Initial program 43.9%

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

   3. Simplified 43.9%

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

      1. div-inv N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. associate-*l/ N/A

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

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

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

   6. Applied egg-rr 43.9%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot c\right), \color{blue}{\left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \left(\color{blue}{b} + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(-4 \cdot \left(a \cdot c\right) + {b}^{2}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + -4 \cdot \left(a \cdot c\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot -4\right)\right)\right)\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({b}^{2}\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right)\right) \]

      17. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(-4 \cdot \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(c \cdot a\right)\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 43.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right)\right)\right) \]

   9. Simplified 43.9%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}} \]

   10. Taylor expanded in c around 0

       \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

       4. /-lowering-/.f64 91.1%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   12. Simplified 91.1%

       \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+126}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+105}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{+109}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{+106}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{a}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.5e+109)
   (- 0.0 (/ b a))
   (if (<= b 2.9e+106)
     (if (>= b 0.0)
       (/ (* c -2.0) (+ b (sqrt (+ (* b b) (* (* a c) -4.0)))))
       (/ 0.5 (/ a (- (sqrt (+ (* b b) (* c (* a -4.0)))) b))))
     (/ c (- 0.0 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.5e+109) {
		tmp = 0.0 - (b / a);
	} else if (b <= 2.9e+106) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = (c * -2.0) / (b + sqrt(((b * b) + ((a * c) * -4.0))));
		} else {
			tmp_1 = 0.5 / (a / (sqrt(((b * b) + (c * (a * -4.0)))) - b));
		}
		tmp = tmp_1;
	} else {
		tmp = c / (0.0 - b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    real(8) :: tmp_1
    if (b <= (-2.5d+109)) then
        tmp = 0.0d0 - (b / a)
    else if (b <= 2.9d+106) then
        if (b >= 0.0d0) then
            tmp_1 = (c * (-2.0d0)) / (b + sqrt(((b * b) + ((a * c) * (-4.0d0)))))
        else
            tmp_1 = 0.5d0 / (a / (sqrt(((b * b) + (c * (a * (-4.0d0))))) - b))
        end if
        tmp = tmp_1
    else
        tmp = c / (0.0d0 - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.5e+109) {
		tmp = 0.0 - (b / a);
	} else if (b <= 2.9e+106) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = (c * -2.0) / (b + Math.sqrt(((b * b) + ((a * c) * -4.0))));
		} else {
			tmp_1 = 0.5 / (a / (Math.sqrt(((b * b) + (c * (a * -4.0)))) - b));
		}
		tmp = tmp_1;
	} else {
		tmp = c / (0.0 - b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.5e+109:
		tmp = 0.0 - (b / a)
	elif b <= 2.9e+106:
		tmp_1 = 0
		if b >= 0.0:
			tmp_1 = (c * -2.0) / (b + math.sqrt(((b * b) + ((a * c) * -4.0))))
		else:
			tmp_1 = 0.5 / (a / (math.sqrt(((b * b) + (c * (a * -4.0)))) - b))
		tmp = tmp_1
	else:
		tmp = c / (0.0 - b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.5e+109)
		tmp = Float64(0.0 - Float64(b / a));
	elseif (b <= 2.9e+106)
		tmp_1 = 0.0
		if (b >= 0.0)
			tmp_1 = Float64(Float64(c * -2.0) / Float64(b + sqrt(Float64(Float64(b * b) + Float64(Float64(a * c) * -4.0)))));
		else
			tmp_1 = Float64(0.5 / Float64(a / Float64(sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0)))) - b)));
		end
		tmp = tmp_1;
	else
		tmp = Float64(c / Float64(0.0 - b));
	end
	return tmp
end

MATLAB

function tmp_3 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.5e+109)
		tmp = 0.0 - (b / a);
	elseif (b <= 2.9e+106)
		tmp_2 = 0.0;
		if (b >= 0.0)
			tmp_2 = (c * -2.0) / (b + sqrt(((b * b) + ((a * c) * -4.0))));
		else
			tmp_2 = 0.5 / (a / (sqrt(((b * b) + (c * (a * -4.0)))) - b));
		end
		tmp = tmp_2;
	else
		tmp = c / (0.0 - b);
	end
	tmp_3 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.5e+109], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.9e+106], If[GreaterEqual[b, 0.0], N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(a / N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], N[(c / N[(0.0 - b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.5000000000000001e109

   1. Initial program 52.1%

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

   3. Simplified 52.1%

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

      1. div-inv N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. associate-*l/ N/A

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

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

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot c\right), \color{blue}{\left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \left(\color{blue}{b} + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(-4 \cdot \left(a \cdot c\right) + {b}^{2}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + -4 \cdot \left(a \cdot c\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot -4\right)\right)\right)\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({b}^{2}\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right)\right) \]

      17. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(-4 \cdot \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(c \cdot a\right)\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 4.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right)\right)\right) \]

   9. Simplified 4.0%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}} \]

   10. Taylor expanded in b around -inf

       \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\frac{b}{a}} \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right) \]

       4. /-lowering-/.f64 92.4%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   12. Simplified 92.4%

       \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   13. Step-by-step derivation

       1. sub0-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{b}{a}\right)\right) \]

       3. /-lowering-/.f64 92.4%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(b, a\right)\right) \]

   14. Applied egg-rr 92.4%

       \[\leadsto \color{blue}{-\frac{b}{a}} \]

   if -2.5000000000000001e109 < b < 2.9000000000000002e106

   1. Initial program 91.5%

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

   3. Simplified 91.5%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. associate-/r* N/A

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

      4. metadata-eval N/A

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

      5. flip3-- N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -4\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \frac{{\left(\sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4}\right)}^{3} - {b}^{3}}{\sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} \cdot \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} + \left(b \cdot b + \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} \cdot b\right)}\\ \end{array} \]

      6. clear-num N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -4\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \frac{1}{\frac{\sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} \cdot \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} + \left(b \cdot b + \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} \cdot b\right)}{{\left(\sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4}\right)}^{3} - {b}^{3}}}\\ \end{array} \]

      7. frac-times N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -4\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot 1}{a \cdot \frac{\sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} \cdot \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} + \left(b \cdot b + \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} \cdot b\right)}{{\left(\sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4}\right)}^{3} - {b}^{3}}}\\ \end{array} \]

      8. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -4\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{a \cdot \frac{\sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} \cdot \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} + \left(b \cdot b + \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} \cdot b\right)}{{\left(\sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4}\right)}^{3} - {b}^{3}}}\\ \end{array} \]

   6. Applied egg-rr 91.3%

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

   if 2.9000000000000002e106 < b

   1. Initial program 43.9%

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

   3. Simplified 43.9%

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

      1. div-inv N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. associate-*l/ N/A

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

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

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

   6. Applied egg-rr 43.9%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot c\right), \color{blue}{\left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \left(\color{blue}{b} + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(-4 \cdot \left(a \cdot c\right) + {b}^{2}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + -4 \cdot \left(a \cdot c\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot -4\right)\right)\right)\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({b}^{2}\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right)\right) \]

      17. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(-4 \cdot \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(c \cdot a\right)\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 43.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right)\right)\right) \]

   9. Simplified 43.9%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}} \]

   10. Taylor expanded in c around 0

       \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

       4. /-lowering-/.f64 91.1%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   12. Simplified 91.1%

       \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{+109}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{+106}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{a}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+136}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{+106}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+136)
   (- 0.0 (/ b a))
   (if (<= b 1.55e+106)
     (if (>= b 0.0)
       (* c (/ -2.0 (+ b (sqrt (+ (* b b) (* c (* a -4.0)))))))
       (/ (- (sqrt (+ (* b b) (* (* a c) -4.0))) b) (* a 2.0)))
     (/ c (- 0.0 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+136) {
		tmp = 0.0 - (b / a);
	} else if (b <= 1.55e+106) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = c * (-2.0 / (b + sqrt(((b * b) + (c * (a * -4.0))))));
		} else {
			tmp_1 = (sqrt(((b * b) + ((a * c) * -4.0))) - b) / (a * 2.0);
		}
		tmp = tmp_1;
	} else {
		tmp = c / (0.0 - b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    real(8) :: tmp_1
    if (b <= (-5d+136)) then
        tmp = 0.0d0 - (b / a)
    else if (b <= 1.55d+106) then
        if (b >= 0.0d0) then
            tmp_1 = c * ((-2.0d0) / (b + sqrt(((b * b) + (c * (a * (-4.0d0)))))))
        else
            tmp_1 = (sqrt(((b * b) + ((a * c) * (-4.0d0)))) - b) / (a * 2.0d0)
        end if
        tmp = tmp_1
    else
        tmp = c / (0.0d0 - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+136) {
		tmp = 0.0 - (b / a);
	} else if (b <= 1.55e+106) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = c * (-2.0 / (b + Math.sqrt(((b * b) + (c * (a * -4.0))))));
		} else {
			tmp_1 = (Math.sqrt(((b * b) + ((a * c) * -4.0))) - b) / (a * 2.0);
		}
		tmp = tmp_1;
	} else {
		tmp = c / (0.0 - b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e+136:
		tmp = 0.0 - (b / a)
	elif b <= 1.55e+106:
		tmp_1 = 0
		if b >= 0.0:
			tmp_1 = c * (-2.0 / (b + math.sqrt(((b * b) + (c * (a * -4.0))))))
		else:
			tmp_1 = (math.sqrt(((b * b) + ((a * c) * -4.0))) - b) / (a * 2.0)
		tmp = tmp_1
	else:
		tmp = c / (0.0 - b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+136)
		tmp = Float64(0.0 - Float64(b / a));
	elseif (b <= 1.55e+106)
		tmp_1 = 0.0
		if (b >= 0.0)
			tmp_1 = Float64(c * Float64(-2.0 / Float64(b + sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0)))))));
		else
			tmp_1 = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(Float64(a * c) * -4.0))) - b) / Float64(a * 2.0));
		end
		tmp = tmp_1;
	else
		tmp = Float64(c / Float64(0.0 - b));
	end
	return tmp
end

MATLAB

function tmp_3 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e+136)
		tmp = 0.0 - (b / a);
	elseif (b <= 1.55e+106)
		tmp_2 = 0.0;
		if (b >= 0.0)
			tmp_2 = c * (-2.0 / (b + sqrt(((b * b) + (c * (a * -4.0))))));
		else
			tmp_2 = (sqrt(((b * b) + ((a * c) * -4.0))) - b) / (a * 2.0);
		end
		tmp = tmp_2;
	else
		tmp = c / (0.0 - b);
	end
	tmp_3 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+136], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.55e+106], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], N[(c / N[(0.0 - b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.0000000000000002e136

   1. Initial program 45.3%

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

   3. Simplified 45.3%

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

      1. div-inv N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. associate-*l/ N/A

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

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

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot c\right), \color{blue}{\left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \left(\color{blue}{b} + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(-4 \cdot \left(a \cdot c\right) + {b}^{2}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + -4 \cdot \left(a \cdot c\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot -4\right)\right)\right)\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({b}^{2}\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right)\right) \]

      17. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(-4 \cdot \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(c \cdot a\right)\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 2.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right)\right)\right) \]

   9. Simplified 2.0%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}} \]

   10. Taylor expanded in b around -inf

       \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\frac{b}{a}} \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right) \]

       4. /-lowering-/.f64 91.3%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   12. Simplified 91.3%

       \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   13. Step-by-step derivation

       1. sub0-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{b}{a}\right)\right) \]

       3. /-lowering-/.f64 91.3%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(b, a\right)\right) \]

   14. Applied egg-rr 91.3%

       \[\leadsto \color{blue}{-\frac{b}{a}} \]

   if -5.0000000000000002e136 < b < 1.55e106

   1. Initial program 91.8%

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

   3. Simplified 91.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. rem-square-sqrt N/A

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

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

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

      8. rem-square-sqrt N/A

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

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

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

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

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

      11. associate-*l* N/A

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

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

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

      13. *-lowering-*.f64 91.6%

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

   6. Applied egg-rr 91.6%

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

   if 1.55e106 < b

   1. Initial program 43.9%

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

   3. Simplified 43.9%

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

      1. div-inv N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. associate-*l/ N/A

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

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

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

   6. Applied egg-rr 43.9%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot c\right), \color{blue}{\left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \left(\color{blue}{b} + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(-4 \cdot \left(a \cdot c\right) + {b}^{2}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + -4 \cdot \left(a \cdot c\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot -4\right)\right)\right)\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({b}^{2}\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right)\right) \]

      17. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(-4 \cdot \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(c \cdot a\right)\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 43.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right)\right)\right) \]

   9. Simplified 43.9%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}} \]

   10. Taylor expanded in c around 0

       \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

       4. /-lowering-/.f64 91.1%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   12. Simplified 91.1%

       \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+136}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{+106}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.8 \cdot 10^{-35}:\\ \;\;\;\;b \cdot \left(\frac{c}{b \cdot b} + \frac{-1}{a}\right)\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+105}:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.8e-35)
   (* b (+ (/ c (* b b)) (/ -1.0 a)))
   (if (<= b 4e+105)
     (/ (* c -2.0) (+ b (sqrt (+ (* b b) (* (* a c) -4.0)))))
     (/ c (- 0.0 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.8e-35) {
		tmp = b * ((c / (b * b)) + (-1.0 / a));
	} else if (b <= 4e+105) {
		tmp = (c * -2.0) / (b + sqrt(((b * b) + ((a * c) * -4.0))));
	} else {
		tmp = c / (0.0 - b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-7.8d-35)) then
        tmp = b * ((c / (b * b)) + ((-1.0d0) / a))
    else if (b <= 4d+105) then
        tmp = (c * (-2.0d0)) / (b + sqrt(((b * b) + ((a * c) * (-4.0d0)))))
    else
        tmp = c / (0.0d0 - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.8e-35) {
		tmp = b * ((c / (b * b)) + (-1.0 / a));
	} else if (b <= 4e+105) {
		tmp = (c * -2.0) / (b + Math.sqrt(((b * b) + ((a * c) * -4.0))));
	} else {
		tmp = c / (0.0 - b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.8e-35:
		tmp = b * ((c / (b * b)) + (-1.0 / a))
	elif b <= 4e+105:
		tmp = (c * -2.0) / (b + math.sqrt(((b * b) + ((a * c) * -4.0))))
	else:
		tmp = c / (0.0 - b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.8e-35)
		tmp = Float64(b * Float64(Float64(c / Float64(b * b)) + Float64(-1.0 / a)));
	elseif (b <= 4e+105)
		tmp = Float64(Float64(c * -2.0) / Float64(b + sqrt(Float64(Float64(b * b) + Float64(Float64(a * c) * -4.0)))));
	else
		tmp = Float64(c / Float64(0.0 - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.8e-35)
		tmp = b * ((c / (b * b)) + (-1.0 / a));
	elseif (b <= 4e+105)
		tmp = (c * -2.0) / (b + sqrt(((b * b) + ((a * c) * -4.0))));
	else
		tmp = c / (0.0 - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.8e-35], N[(b * N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4e+105], N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c / N[(0.0 - b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.79999999999999961e-35

   1. Initial program 71.4%

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

   3. Simplified 71.4%

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

      1. div-inv N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. associate-*l/ N/A

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

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

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

   6. Applied egg-rr 8.6%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot c\right), \color{blue}{\left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \left(\color{blue}{b} + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(-4 \cdot \left(a \cdot c\right) + {b}^{2}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + -4 \cdot \left(a \cdot c\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot -4\right)\right)\right)\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({b}^{2}\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right)\right) \]

      17. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(-4 \cdot \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(c \cdot a\right)\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 12.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right)\right)\right) \]

   9. Simplified 12.3%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}} \]

   10. Taylor expanded in b around -inf

       \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(-1 \cdot b\right) \cdot \color{blue}{\left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot b\right), \color{blue}{\left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)}\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(b\right)\right), \left(\color{blue}{-1 \cdot \frac{c}{{b}^{2}}} + \frac{1}{a}\right)\right) \]

       4. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(0 - b\right), \left(\color{blue}{-1 \cdot \frac{c}{{b}^{2}}} + \frac{1}{a}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \left(\color{blue}{-1 \cdot \frac{c}{{b}^{2}}} + \frac{1}{a}\right)\right) \]

       6. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \left(\frac{1}{a} + \color{blue}{-1 \cdot \frac{c}{{b}^{2}}}\right)\right) \]

       7. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \left(\frac{1}{a} + \left(\mathsf{neg}\left(\frac{c}{{b}^{2}}\right)\right)\right)\right) \]

       8. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \left(\frac{1}{a} - \color{blue}{\frac{c}{{b}^{2}}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{\_.f64}\left(\left(\frac{1}{a}\right), \color{blue}{\left(\frac{c}{{b}^{2}}\right)}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \left(\frac{\color{blue}{c}}{{b}^{2}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(c, \color{blue}{\left({b}^{2}\right)}\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(c, \left(b \cdot \color{blue}{b}\right)\right)\right)\right) \]

       13. *-lowering-*.f64 87.1%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right) \]

   12. Simplified 87.1%

       \[\leadsto \color{blue}{\left(0 - b\right) \cdot \left(\frac{1}{a} - \frac{c}{b \cdot b}\right)} \]

   if -7.79999999999999961e-35 < b < 3.9999999999999998e105

   1. Initial program 89.8%

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

   3. Simplified 89.8%

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

      1. div-inv N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. associate-*l/ N/A

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

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

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

   6. Applied egg-rr 83.9%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot c\right), \color{blue}{\left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \left(\color{blue}{b} + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(-4 \cdot \left(a \cdot c\right) + {b}^{2}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + -4 \cdot \left(a \cdot c\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot -4\right)\right)\right)\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({b}^{2}\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right)\right) \]

      17. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(-4 \cdot \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(c \cdot a\right)\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 84.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right)\right)\right) \]

   9. Simplified 84.3%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}} \]

   if 3.9999999999999998e105 < b

   1. Initial program 43.9%

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

   3. Simplified 43.9%

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

      1. div-inv N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. associate-*l/ N/A

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

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

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

   6. Applied egg-rr 43.9%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot c\right), \color{blue}{\left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \left(\color{blue}{b} + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(-4 \cdot \left(a \cdot c\right) + {b}^{2}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + -4 \cdot \left(a \cdot c\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot -4\right)\right)\right)\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({b}^{2}\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right)\right) \]

      17. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(-4 \cdot \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(c \cdot a\right)\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 43.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right)\right)\right) \]

   9. Simplified 43.9%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}} \]

   10. Taylor expanded in c around 0

       \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

       4. /-lowering-/.f64 91.1%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   12. Simplified 91.1%

       \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.8 \cdot 10^{-35}:\\ \;\;\;\;b \cdot \left(\frac{c}{b \cdot b} + \frac{-1}{a}\right)\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+105}:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-39}:\\ \;\;\;\;b \cdot \left(\frac{c}{b \cdot b} + \frac{-1}{a}\right)\\ \mathbf{elif}\;b \leq 8 \cdot 10^{+105}:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8e-39)
   (* b (+ (/ c (* b b)) (/ -1.0 a)))
   (if (<= b 8e+105)
     (* c (/ -2.0 (+ b (sqrt (+ (* b b) (* c (* a -4.0)))))))
     (/ c (- 0.0 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8e-39) {
		tmp = b * ((c / (b * b)) + (-1.0 / a));
	} else if (b <= 8e+105) {
		tmp = c * (-2.0 / (b + sqrt(((b * b) + (c * (a * -4.0))))));
	} else {
		tmp = c / (0.0 - b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-8d-39)) then
        tmp = b * ((c / (b * b)) + ((-1.0d0) / a))
    else if (b <= 8d+105) then
        tmp = c * ((-2.0d0) / (b + sqrt(((b * b) + (c * (a * (-4.0d0)))))))
    else
        tmp = c / (0.0d0 - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8e-39) {
		tmp = b * ((c / (b * b)) + (-1.0 / a));
	} else if (b <= 8e+105) {
		tmp = c * (-2.0 / (b + Math.sqrt(((b * b) + (c * (a * -4.0))))));
	} else {
		tmp = c / (0.0 - b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8e-39:
		tmp = b * ((c / (b * b)) + (-1.0 / a))
	elif b <= 8e+105:
		tmp = c * (-2.0 / (b + math.sqrt(((b * b) + (c * (a * -4.0))))))
	else:
		tmp = c / (0.0 - b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8e-39)
		tmp = Float64(b * Float64(Float64(c / Float64(b * b)) + Float64(-1.0 / a)));
	elseif (b <= 8e+105)
		tmp = Float64(c * Float64(-2.0 / Float64(b + sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0)))))));
	else
		tmp = Float64(c / Float64(0.0 - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8e-39)
		tmp = b * ((c / (b * b)) + (-1.0 / a));
	elseif (b <= 8e+105)
		tmp = c * (-2.0 / (b + sqrt(((b * b) + (c * (a * -4.0))))));
	else
		tmp = c / (0.0 - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8e-39], N[(b * N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8e+105], N[(c * N[(-2.0 / N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c / N[(0.0 - b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.99999999999999943e-39

   1. Initial program 71.4%

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

   3. Simplified 71.4%

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

      1. div-inv N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. associate-*l/ N/A

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

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

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

   6. Applied egg-rr 8.6%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot c\right), \color{blue}{\left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \left(\color{blue}{b} + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(-4 \cdot \left(a \cdot c\right) + {b}^{2}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + -4 \cdot \left(a \cdot c\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot -4\right)\right)\right)\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({b}^{2}\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right)\right) \]

      17. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(-4 \cdot \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(c \cdot a\right)\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 12.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right)\right)\right) \]

   9. Simplified 12.3%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}} \]

   10. Taylor expanded in b around -inf

       \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(-1 \cdot b\right) \cdot \color{blue}{\left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot b\right), \color{blue}{\left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)}\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(b\right)\right), \left(\color{blue}{-1 \cdot \frac{c}{{b}^{2}}} + \frac{1}{a}\right)\right) \]

       4. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(0 - b\right), \left(\color{blue}{-1 \cdot \frac{c}{{b}^{2}}} + \frac{1}{a}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \left(\color{blue}{-1 \cdot \frac{c}{{b}^{2}}} + \frac{1}{a}\right)\right) \]

       6. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \left(\frac{1}{a} + \color{blue}{-1 \cdot \frac{c}{{b}^{2}}}\right)\right) \]

       7. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \left(\frac{1}{a} + \left(\mathsf{neg}\left(\frac{c}{{b}^{2}}\right)\right)\right)\right) \]

       8. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \left(\frac{1}{a} - \color{blue}{\frac{c}{{b}^{2}}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{\_.f64}\left(\left(\frac{1}{a}\right), \color{blue}{\left(\frac{c}{{b}^{2}}\right)}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \left(\frac{\color{blue}{c}}{{b}^{2}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(c, \color{blue}{\left({b}^{2}\right)}\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(c, \left(b \cdot \color{blue}{b}\right)\right)\right)\right) \]

       13. *-lowering-*.f64 87.1%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right) \]

   12. Simplified 87.1%

       \[\leadsto \color{blue}{\left(0 - b\right) \cdot \left(\frac{1}{a} - \frac{c}{b \cdot b}\right)} \]

   if -7.99999999999999943e-39 < b < 7.9999999999999995e105

   1. Initial program 89.8%

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

   3. Simplified 89.8%

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

      1. div-inv N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. associate-*l/ N/A

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

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

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

   6. Applied egg-rr 83.9%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot c\right), \color{blue}{\left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \left(\color{blue}{b} + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(-4 \cdot \left(a \cdot c\right) + {b}^{2}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + -4 \cdot \left(a \cdot c\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot -4\right)\right)\right)\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({b}^{2}\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right)\right) \]

      17. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(-4 \cdot \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(c \cdot a\right)\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 84.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right)\right)\right) \]

   9. Simplified 84.3%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}} \]
   10. Step-by-step derivation

       1. pow1/2 N/A

          \[\leadsto \frac{-2 \cdot c}{b + {\left(b \cdot b + -4 \cdot \left(c \cdot a\right)\right)}^{\color{blue}{\frac{1}{2}}}} \]

       2. *-commutative N/A

          \[\leadsto \frac{-2 \cdot c}{b + {\left(b \cdot b + \left(c \cdot a\right) \cdot -4\right)}^{\frac{1}{2}}} \]

       3. associate-*r* N/A

          \[\leadsto \frac{-2 \cdot c}{b + {\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)}^{\frac{1}{2}}} \]

       4. metadata-eval N/A

          \[\leadsto \frac{-2 \cdot c}{b + {\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)}^{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}} \]

       5. pow-flip N/A

          \[\leadsto \frac{-2 \cdot c}{b + \frac{1}{\color{blue}{{\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)}^{\frac{-1}{2}}}}} \]

       6. associate-*l/ N/A

          \[\leadsto \frac{-2}{b + \frac{1}{{\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)}^{\frac{-1}{2}}}} \cdot \color{blue}{c} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{-2}{b + \frac{1}{{\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right)}^{\frac{-1}{2}}}}\right), \color{blue}{c}\right) \]

   11. Applied egg-rr 84.0%

       \[\leadsto \color{blue}{\frac{-2}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}} \cdot c} \]

   if 7.9999999999999995e105 < b

   1. Initial program 43.9%

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

   3. Simplified 43.9%

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

      1. div-inv N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. associate-*l/ N/A

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

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

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

   6. Applied egg-rr 43.9%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot c\right), \color{blue}{\left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \left(\color{blue}{b} + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(-4 \cdot \left(a \cdot c\right) + {b}^{2}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + -4 \cdot \left(a \cdot c\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot -4\right)\right)\right)\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({b}^{2}\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right)\right) \]

      17. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(-4 \cdot \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(c \cdot a\right)\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 43.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right)\right)\right) \]

   9. Simplified 43.9%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}} \]

   10. Taylor expanded in c around 0

       \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

       4. /-lowering-/.f64 91.1%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   12. Simplified 91.1%

       \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-39}:\\ \;\;\;\;b \cdot \left(\frac{c}{b \cdot b} + \frac{-1}{a}\right)\\ \mathbf{elif}\;b \leq 8 \cdot 10^{+105}:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{-103}:\\ \;\;\;\;b \cdot \left(\frac{c}{b \cdot b} + \frac{-1}{a}\right)\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-61}:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{\left(a \cdot c\right) \cdot -4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.5e-103)
   (* b (+ (/ c (* b b)) (/ -1.0 a)))
   (if (<= b 1.65e-61)
     (/ (* c -2.0) (+ b (sqrt (* (* a c) -4.0))))
     (/ c (- 0.0 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.5e-103) {
		tmp = b * ((c / (b * b)) + (-1.0 / a));
	} else if (b <= 1.65e-61) {
		tmp = (c * -2.0) / (b + sqrt(((a * c) * -4.0)));
	} else {
		tmp = c / (0.0 - b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.5d-103)) then
        tmp = b * ((c / (b * b)) + ((-1.0d0) / a))
    else if (b <= 1.65d-61) then
        tmp = (c * (-2.0d0)) / (b + sqrt(((a * c) * (-4.0d0))))
    else
        tmp = c / (0.0d0 - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.5e-103) {
		tmp = b * ((c / (b * b)) + (-1.0 / a));
	} else if (b <= 1.65e-61) {
		tmp = (c * -2.0) / (b + Math.sqrt(((a * c) * -4.0)));
	} else {
		tmp = c / (0.0 - b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.5e-103:
		tmp = b * ((c / (b * b)) + (-1.0 / a))
	elif b <= 1.65e-61:
		tmp = (c * -2.0) / (b + math.sqrt(((a * c) * -4.0)))
	else:
		tmp = c / (0.0 - b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.5e-103)
		tmp = Float64(b * Float64(Float64(c / Float64(b * b)) + Float64(-1.0 / a)));
	elseif (b <= 1.65e-61)
		tmp = Float64(Float64(c * -2.0) / Float64(b + sqrt(Float64(Float64(a * c) * -4.0))));
	else
		tmp = Float64(c / Float64(0.0 - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.5e-103)
		tmp = b * ((c / (b * b)) + (-1.0 / a));
	elseif (b <= 1.65e-61)
		tmp = (c * -2.0) / (b + sqrt(((a * c) * -4.0)));
	else
		tmp = c / (0.0 - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.5e-103], N[(b * N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.65e-61], N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c / N[(0.0 - b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.49999999999999983e-103

   1. Initial program 74.5%

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

   3. Simplified 74.5%

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

      1. div-inv N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. associate-*l/ N/A

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

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

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

   6. Applied egg-rr 13.4%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot c\right), \color{blue}{\left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \left(\color{blue}{b} + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(-4 \cdot \left(a \cdot c\right) + {b}^{2}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + -4 \cdot \left(a \cdot c\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot -4\right)\right)\right)\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({b}^{2}\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right)\right) \]

      17. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(-4 \cdot \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(c \cdot a\right)\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 16.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right)\right)\right) \]

   9. Simplified 16.8%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}} \]

   10. Taylor expanded in b around -inf

       \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(-1 \cdot b\right) \cdot \color{blue}{\left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot b\right), \color{blue}{\left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)}\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(b\right)\right), \left(\color{blue}{-1 \cdot \frac{c}{{b}^{2}}} + \frac{1}{a}\right)\right) \]

       4. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(0 - b\right), \left(\color{blue}{-1 \cdot \frac{c}{{b}^{2}}} + \frac{1}{a}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \left(\color{blue}{-1 \cdot \frac{c}{{b}^{2}}} + \frac{1}{a}\right)\right) \]

       6. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \left(\frac{1}{a} + \color{blue}{-1 \cdot \frac{c}{{b}^{2}}}\right)\right) \]

       7. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \left(\frac{1}{a} + \left(\mathsf{neg}\left(\frac{c}{{b}^{2}}\right)\right)\right)\right) \]

       8. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \left(\frac{1}{a} - \color{blue}{\frac{c}{{b}^{2}}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{\_.f64}\left(\left(\frac{1}{a}\right), \color{blue}{\left(\frac{c}{{b}^{2}}\right)}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \left(\frac{\color{blue}{c}}{{b}^{2}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(c, \color{blue}{\left({b}^{2}\right)}\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(c, \left(b \cdot \color{blue}{b}\right)\right)\right)\right) \]

       13. *-lowering-*.f64 83.4%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right) \]

   12. Simplified 83.4%

       \[\leadsto \color{blue}{\left(0 - b\right) \cdot \left(\frac{1}{a} - \frac{c}{b \cdot b}\right)} \]

   if -2.49999999999999983e-103 < b < 1.64999999999999998e-61

   1. Initial program 84.5%

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

   3. Simplified 84.5%

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

      1. div-inv N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. associate-*l/ N/A

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

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

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

   6. Applied egg-rr 79.8%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot c\right), \color{blue}{\left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \left(\color{blue}{b} + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(-4 \cdot \left(a \cdot c\right) + {b}^{2}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + -4 \cdot \left(a \cdot c\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot -4\right)\right)\right)\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({b}^{2}\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right)\right) \]

      17. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(-4 \cdot \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(c \cdot a\right)\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 80.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right)\right)\right) \]

   9. Simplified 80.4%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}} \]

   10. Taylor expanded in b around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right)}\right)\right)\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \left(a \cdot c\right)\right)\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \left(c \cdot a\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 77.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right)\right) \]

   12. Simplified 77.1%

       \[\leadsto \frac{-2 \cdot c}{b + \sqrt{\color{blue}{-4 \cdot \left(c \cdot a\right)}}} \]

   if 1.64999999999999998e-61 < b

   1. Initial program 71.6%

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

   3. Simplified 71.6%

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

      1. div-inv N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. associate-*l/ N/A

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

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

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

   6. Applied egg-rr 71.6%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot c\right), \color{blue}{\left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \left(\color{blue}{b} + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(-4 \cdot \left(a \cdot c\right) + {b}^{2}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + -4 \cdot \left(a \cdot c\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot -4\right)\right)\right)\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({b}^{2}\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right)\right) \]

      17. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(-4 \cdot \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(c \cdot a\right)\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 71.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right)\right)\right) \]

   9. Simplified 71.6%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}} \]

   10. Taylor expanded in c around 0

       \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

       4. /-lowering-/.f64 82.0%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   12. Simplified 82.0%

       \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{-103}:\\ \;\;\;\;b \cdot \left(\frac{c}{b \cdot b} + \frac{-1}{a}\right)\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-61}:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{\left(a \cdot c\right) \cdot -4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.9% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (- 0.0 (/ b a)) (/ c (- 0.0 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = 0.0 - (b / a);
	} else {
		tmp = c / (0.0 - b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = 0.0d0 - (b / a)
    else
        tmp = c / (0.0d0 - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = 0.0 - (b / a);
	} else {
		tmp = c / (0.0 - b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = 0.0 - (b / a)
	else:
		tmp = c / (0.0 - b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(0.0 - Float64(b / a));
	else
		tmp = Float64(c / Float64(0.0 - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-310)
		tmp = 0.0 - (b / a);
	else
		tmp = c / (0.0 - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(c / N[(0.0 - b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 78.0%

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

   3. Simplified 78.0%

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

      1. div-inv N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. associate-*l/ N/A

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

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

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

   6. Applied egg-rr 31.6%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot c\right), \color{blue}{\left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \left(\color{blue}{b} + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(-4 \cdot \left(a \cdot c\right) + {b}^{2}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + -4 \cdot \left(a \cdot c\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot -4\right)\right)\right)\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({b}^{2}\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right)\right) \]

      17. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(-4 \cdot \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(c \cdot a\right)\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 34.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right)\right)\right) \]

   9. Simplified 34.3%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}} \]

   10. Taylor expanded in b around -inf

       \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\frac{b}{a}} \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right) \]

       4. /-lowering-/.f64 65.2%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   12. Simplified 65.2%

       \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   13. Step-by-step derivation

       1. sub0-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{b}{a}\right)\right) \]

       3. /-lowering-/.f64 65.2%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(b, a\right)\right) \]

   14. Applied egg-rr 65.2%

       \[\leadsto \color{blue}{-\frac{b}{a}} \]

   if -1.999999999999994e-310 < b

   1. Initial program 73.7%

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

   3. Simplified 73.7%

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

      1. div-inv N/A

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

      2. flip-- N/A

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

      3. +-commutative N/A

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

      4. associate-*l/ N/A

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

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

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

   6. Applied egg-rr 73.7%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot c\right), \color{blue}{\left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \left(\color{blue}{b} + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(-4 \cdot \left(a \cdot c\right) + {b}^{2}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + -4 \cdot \left(a \cdot c\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot -4\right)\right)\right)\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({b}^{2}\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right)\right) \]

      17. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(-4 \cdot \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(c \cdot a\right)\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 73.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right)\right)\right) \]

   9. Simplified 73.7%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}} \]

   10. Taylor expanded in c around 0

       \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

       4. /-lowering-/.f64 67.2%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   12. Simplified 67.2%

       \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.2%

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

Alternative 8: 34.6% accurate, 24.2× speedup

Math

\[\begin{array}{l} \\ 0 - \frac{b}{a} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return 0.0 - (b / a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0 - (b / a)
end function

Java

public static double code(double a, double b, double c) {
	return 0.0 - (b / a);
}

Python

def code(a, b, c):
	return 0.0 - (b / a)

Julia

function code(a, b, c)
	return Float64(0.0 - Float64(b / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 - (b / a);
end

Wolfram

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

Derivation

1. Initial program 75.9%

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

3. Simplified 75.9%

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

   1. div-inv N/A

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

   2. flip-- N/A

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

   3. +-commutative N/A

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

   4. associate-*l/ N/A

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

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

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

6. Applied egg-rr 52.3%

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

7. Taylor expanded in b around 0

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

   1. if-same N/A

      \[\leadsto -2 \cdot \color{blue}{\frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{-2 \cdot c}{\color{blue}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot c\right), \color{blue}{\left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \left(\color{blue}{b} + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(-4 \cdot \left(a \cdot c\right) + {b}^{2}\right)\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + -4 \cdot \left(a \cdot c\right)\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot -4\right)\right)\right)\right) \]

   9. rem-square-sqrt N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right) \]

   11. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({b}^{2}\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

   15. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right)\right) \]

   17. rem-square-sqrt N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(-4 \cdot \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

   20. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(c \cdot a\right)\right)\right)\right)\right)\right) \]

   21. *-lowering-*.f64 53.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right)\right)\right) \]

9. Simplified 53.7%

   \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}} \]

10. Taylor expanded in b around -inf

    \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
11. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

    2. neg-sub0 N/A

       \[\leadsto 0 - \color{blue}{\frac{b}{a}} \]

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

       \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right) \]

    4. /-lowering-/.f64 34.4%

       \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

12. Simplified 34.4%

    \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
13. Step-by-step derivation

    1. sub0-neg N/A

       \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

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

       \[\leadsto \mathsf{neg.f64}\left(\left(\frac{b}{a}\right)\right) \]

    3. /-lowering-/.f64 34.4%

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(b, a\right)\right) \]

14. Applied egg-rr 34.4%

    \[\leadsto \color{blue}{-\frac{b}{a}} \]

15. Final simplification 34.4%

    \[\leadsto 0 - \frac{b}{a} \]
16. Add Preprocessing

Alternative 9: 2.5% accurate, 40.3× speedup

Math

\[\begin{array}{l} \\ \frac{b}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ b a))

C

double code(double a, double b, double c) {
	return b / a;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = b / a
end function

Java

public static double code(double a, double b, double c) {
	return b / a;
}

Python

def code(a, b, c):
	return b / a

Julia

function code(a, b, c)
	return Float64(b / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = b / a;
end

Wolfram

code[a_, b_, c_] := N[(b / a), $MachinePrecision]

Derivation

1. Initial program 75.9%

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

3. Simplified 75.9%

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

   1. div-inv N/A

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

   2. flip-- N/A

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

   3. +-commutative N/A

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

   4. associate-*l/ N/A

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

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

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

6. Applied egg-rr 52.3%

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

7. Taylor expanded in b around 0

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

   1. if-same N/A

      \[\leadsto -2 \cdot \color{blue}{\frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{-2 \cdot c}{\color{blue}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot c\right), \color{blue}{\left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \left(\color{blue}{b} + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(-4 \cdot \left(a \cdot c\right) + {b}^{2}\right)\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + -4 \cdot \left(a \cdot c\right)\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot -4\right)\right)\right)\right) \]

   9. rem-square-sqrt N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + \left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right) \]

   11. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({b}^{2}\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right)\right) \]

   15. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot {\left(\sqrt{-4}\right)}^{2}\right)\right)\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot \left(\sqrt{-4} \cdot \sqrt{-4}\right)\right)\right)\right)\right)\right) \]

   17. rem-square-sqrt N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(-4 \cdot \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

   20. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \left(c \cdot a\right)\right)\right)\right)\right)\right) \]

   21. *-lowering-*.f64 53.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right)\right)\right)\right) \]

9. Simplified 53.7%

   \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}} \]

10. Taylor expanded in c around 0

    \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \color{blue}{\left(-2 \cdot \frac{a \cdot c}{b} + 2 \cdot b\right)}\right) \]
11. Step-by-step derivation

    1. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \left(-2 \cdot \frac{a \cdot c}{b} + \left(\mathsf{neg}\left(-2\right)\right) \cdot b\right)\right) \]

    2. cancel-sign-sub-inv N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \left(-2 \cdot \frac{a \cdot c}{b} - \color{blue}{-2 \cdot b}\right)\right) \]

    3. distribute-lft-out-- N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \left(-2 \cdot \color{blue}{\left(\frac{a \cdot c}{b} - b\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{*.f64}\left(-2, \color{blue}{\left(\frac{a \cdot c}{b} - b\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{*.f64}\left(-2, \mathsf{\_.f64}\left(\left(\frac{a \cdot c}{b}\right), \color{blue}{b}\right)\right)\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{*.f64}\left(-2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(a \cdot c\right), b\right), b\right)\right)\right) \]

    7. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{*.f64}\left(-2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(c \cdot a\right), b\right), b\right)\right)\right) \]

    8. *-lowering-*.f64 33.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, c\right), \mathsf{*.f64}\left(-2, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, a\right), b\right), b\right)\right)\right) \]

12. Simplified 33.9%

    \[\leadsto \frac{-2 \cdot c}{\color{blue}{-2 \cdot \left(\frac{c \cdot a}{b} - b\right)}} \]

13. Taylor expanded in c around inf

    \[\leadsto \color{blue}{\frac{b}{a}} \]
14. Step-by-step derivation

    1. /-lowering-/.f64 2.7%

       \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{a}\right) \]

15. Simplified 2.7%

    \[\leadsto \color{blue}{\frac{b}{a}} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(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))))