jeff quadratic root 1

Percentage Accurate: 71.2% → 89.6%

Time: 19.7s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 71.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\\ \mathbf{if}\;b \leq -2 \cdot 10^{+149}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{+57}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + t\_0}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t\_0 - b}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* b b) (* c (* a -4.0))))))
   (if (<= b -2e+149)
     (- 0.0 (/ c b))
     (if (<= b 5.5e+57)
       (if (>= b 0.0) (/ (/ (+ b t_0) -2.0) a) (/ (* c 2.0) (- t_0 b)))
       (- 0.0 (/ b a))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) + (c * (a * -4.0))));
	double tmp;
	if (b <= -2e+149) {
		tmp = 0.0 - (c / b);
	} else if (b <= 5.5e+57) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = ((b + t_0) / -2.0) / a;
		} else {
			tmp_1 = (c * 2.0) / (t_0 - b);
		}
		tmp = tmp_1;
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    t_0 = sqrt(((b * b) + (c * (a * (-4.0d0)))))
    if (b <= (-2d+149)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 5.5d+57) then
        if (b >= 0.0d0) then
            tmp_1 = ((b + t_0) / (-2.0d0)) / a
        else
            tmp_1 = (c * 2.0d0) / (t_0 - b)
        end if
        tmp = tmp_1
    else
        tmp = 0.0d0 - (b / 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) + (c * (a * -4.0))));
	double tmp;
	if (b <= -2e+149) {
		tmp = 0.0 - (c / b);
	} else if (b <= 5.5e+57) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = ((b + t_0) / -2.0) / a;
		} else {
			tmp_1 = (c * 2.0) / (t_0 - b);
		}
		tmp = tmp_1;
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) + (c * (a * -4.0))))
	tmp = 0
	if b <= -2e+149:
		tmp = 0.0 - (c / b)
	elif b <= 5.5e+57:
		tmp_1 = 0
		if b >= 0.0:
			tmp_1 = ((b + t_0) / -2.0) / a
		else:
			tmp_1 = (c * 2.0) / (t_0 - b)
		tmp = tmp_1
	else:
		tmp = 0.0 - (b / a)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0))))
	tmp = 0.0
	if (b <= -2e+149)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 5.5e+57)
		tmp_1 = 0.0
		if (b >= 0.0)
			tmp_1 = Float64(Float64(Float64(b + t_0) / -2.0) / a);
		else
			tmp_1 = Float64(Float64(c * 2.0) / Float64(t_0 - b));
		end
		tmp = tmp_1;
	else
		tmp = Float64(0.0 - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = sqrt(((b * b) + (c * (a * -4.0))));
	tmp = 0.0;
	if (b <= -2e+149)
		tmp = 0.0 - (c / b);
	elseif (b <= 5.5e+57)
		tmp_2 = 0.0;
		if (b >= 0.0)
			tmp_2 = ((b + t_0) / -2.0) / a;
		else
			tmp_2 = (c * 2.0) / (t_0 - b);
		end
		tmp = tmp_2;
	else
		tmp = 0.0 - (b / a);
	end
	tmp_3 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2e+149], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.5e+57], If[GreaterEqual[b, 0.0], N[(N[(N[(b + t$95$0), $MachinePrecision] / -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.0000000000000001e149

   1. Initial program 29.0%

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

   3. Simplified 29.0%

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

      1. flip-- N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 5.0%

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

   9. Simplified 5.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right)}{a}} \]

   10. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

          \[\leadsto \frac{c}{-1 \cdot \color{blue}{b}} \]

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

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

       5. mul-1-neg N/A

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

       6. neg-sub0 N/A

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

       7. --lowering--.f64 91.2%

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

   12. Simplified 91.2%

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

       1. sub0-neg N/A

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

       2. neg-lowering-neg.f64 91.2%

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

   14. Applied egg-rr 91.2%

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

   if -2.0000000000000001e149 < b < 5.5000000000000002e57

   1. Initial program 88.3%

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

   3. Simplified 88.3%

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

   if 5.5000000000000002e57 < b

   1. Initial program 70.9%

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

   3. Simplified 70.9%

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

      1. flip-- N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 70.9%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 70.9%

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

   9. Simplified 70.9%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right)}{a}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot b\right)}, a\right) \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

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

       3. --lowering--.f64 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+149}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{+57}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\\ \mathbf{if}\;b \leq -2 \cdot 10^{+149}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 5.9 \cdot 10^{+57}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(b + t\_0\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t\_0 - b}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* b b) (* c (* a -4.0))))))
   (if (<= b -2e+149)
     (- 0.0 (/ c b))
     (if (<= b 5.9e+57)
       (if (>= b 0.0) (* (+ b t_0) (/ -0.5 a)) (/ (* c 2.0) (- t_0 b)))
       (- 0.0 (/ b a))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) + (c * (a * -4.0))));
	double tmp;
	if (b <= -2e+149) {
		tmp = 0.0 - (c / b);
	} else if (b <= 5.9e+57) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = (b + t_0) * (-0.5 / a);
		} else {
			tmp_1 = (c * 2.0) / (t_0 - b);
		}
		tmp = tmp_1;
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    t_0 = sqrt(((b * b) + (c * (a * (-4.0d0)))))
    if (b <= (-2d+149)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 5.9d+57) then
        if (b >= 0.0d0) then
            tmp_1 = (b + t_0) * ((-0.5d0) / a)
        else
            tmp_1 = (c * 2.0d0) / (t_0 - b)
        end if
        tmp = tmp_1
    else
        tmp = 0.0d0 - (b / 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) + (c * (a * -4.0))));
	double tmp;
	if (b <= -2e+149) {
		tmp = 0.0 - (c / b);
	} else if (b <= 5.9e+57) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = (b + t_0) * (-0.5 / a);
		} else {
			tmp_1 = (c * 2.0) / (t_0 - b);
		}
		tmp = tmp_1;
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) + (c * (a * -4.0))))
	tmp = 0
	if b <= -2e+149:
		tmp = 0.0 - (c / b)
	elif b <= 5.9e+57:
		tmp_1 = 0
		if b >= 0.0:
			tmp_1 = (b + t_0) * (-0.5 / a)
		else:
			tmp_1 = (c * 2.0) / (t_0 - b)
		tmp = tmp_1
	else:
		tmp = 0.0 - (b / a)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0))))
	tmp = 0.0
	if (b <= -2e+149)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 5.9e+57)
		tmp_1 = 0.0
		if (b >= 0.0)
			tmp_1 = Float64(Float64(b + t_0) * Float64(-0.5 / a));
		else
			tmp_1 = Float64(Float64(c * 2.0) / Float64(t_0 - b));
		end
		tmp = tmp_1;
	else
		tmp = Float64(0.0 - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = sqrt(((b * b) + (c * (a * -4.0))));
	tmp = 0.0;
	if (b <= -2e+149)
		tmp = 0.0 - (c / b);
	elseif (b <= 5.9e+57)
		tmp_2 = 0.0;
		if (b >= 0.0)
			tmp_2 = (b + t_0) * (-0.5 / a);
		else
			tmp_2 = (c * 2.0) / (t_0 - b);
		end
		tmp = tmp_2;
	else
		tmp = 0.0 - (b / a);
	end
	tmp_3 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2e+149], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.9e+57], If[GreaterEqual[b, 0.0], N[(N[(b + t$95$0), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.0000000000000001e149

   1. Initial program 29.0%

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

   3. Simplified 29.0%

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

      1. flip-- N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 5.0%

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

   9. Simplified 5.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right)}{a}} \]

   10. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

          \[\leadsto \frac{c}{-1 \cdot \color{blue}{b}} \]

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

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

       5. mul-1-neg N/A

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

       6. neg-sub0 N/A

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

       7. --lowering--.f64 91.2%

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

   12. Simplified 91.2%

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

       1. sub0-neg N/A

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

       2. neg-lowering-neg.f64 91.2%

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

   14. Applied egg-rr 91.2%

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

   if -2.0000000000000001e149 < b < 5.90000000000000013e57

   1. Initial program 88.3%

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

   3. Simplified 88.3%

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

      1. div-inv N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. rem-square-sqrt N/A

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

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

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

      7. rem-square-sqrt N/A

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

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

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

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

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

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

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

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

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

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

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

      13. metadata-eval 88.2%

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

   6. Applied egg-rr 88.2%

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

   if 5.90000000000000013e57 < b

   1. Initial program 70.9%

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

   3. Simplified 70.9%

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

      1. flip-- N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 70.9%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 70.9%

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

   9. Simplified 70.9%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right)}{a}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot b\right)}, a\right) \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

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

       3. --lowering--.f64 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+149}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 5.9 \cdot 10^{+57}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}\\ \mathbf{if}\;b \leq -5 \cdot 10^{+149}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-235}:\\ \;\;\;\;\frac{c \cdot 2}{t\_0 - b}\\ \mathbf{elif}\;b \leq 5.9 \cdot 10^{+57}:\\ \;\;\;\;\frac{-0.5 \cdot \left(b + t\_0\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* b b) (* -4.0 (* c a))))))
   (if (<= b -5e+149)
     (- 0.0 (/ c b))
     (if (<= b 1.8e-235)
       (/ (* c 2.0) (- t_0 b))
       (if (<= b 5.9e+57) (/ (* -0.5 (+ b t_0)) a) (- 0.0 (/ b a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) + (-4.0 * (c * a))));
	double tmp;
	if (b <= -5e+149) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1.8e-235) {
		tmp = (c * 2.0) / (t_0 - b);
	} else if (b <= 5.9e+57) {
		tmp = (-0.5 * (b + t_0)) / a;
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) + ((-4.0d0) * (c * a))))
    if (b <= (-5d+149)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 1.8d-235) then
        tmp = (c * 2.0d0) / (t_0 - b)
    else if (b <= 5.9d+57) then
        tmp = ((-0.5d0) * (b + t_0)) / a
    else
        tmp = 0.0d0 - (b / 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 * (c * a))));
	double tmp;
	if (b <= -5e+149) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1.8e-235) {
		tmp = (c * 2.0) / (t_0 - b);
	} else if (b <= 5.9e+57) {
		tmp = (-0.5 * (b + t_0)) / a;
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) + (-4.0 * (c * a))))
	tmp = 0
	if b <= -5e+149:
		tmp = 0.0 - (c / b)
	elif b <= 1.8e-235:
		tmp = (c * 2.0) / (t_0 - b)
	elif b <= 5.9e+57:
		tmp = (-0.5 * (b + t_0)) / a
	else:
		tmp = 0.0 - (b / a)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) + Float64(-4.0 * Float64(c * a))))
	tmp = 0.0
	if (b <= -5e+149)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 1.8e-235)
		tmp = Float64(Float64(c * 2.0) / Float64(t_0 - b));
	elseif (b <= 5.9e+57)
		tmp = Float64(Float64(-0.5 * Float64(b + t_0)) / a);
	else
		tmp = Float64(0.0 - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) + (-4.0 * (c * a))));
	tmp = 0.0;
	if (b <= -5e+149)
		tmp = 0.0 - (c / b);
	elseif (b <= 1.8e-235)
		tmp = (c * 2.0) / (t_0 - b);
	elseif (b <= 5.9e+57)
		tmp = (-0.5 * (b + t_0)) / a;
	else
		tmp = 0.0 - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -5e+149], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.8e-235], N[(N[(c * 2.0), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.9e+57], N[(N[(-0.5 * N[(b + t$95$0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -4.9999999999999999e149

   1. Initial program 29.0%

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

   3. Simplified 29.0%

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

      1. flip-- N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 5.0%

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

   9. Simplified 5.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right)}{a}} \]

   10. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

          \[\leadsto \frac{c}{-1 \cdot \color{blue}{b}} \]

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

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

       5. mul-1-neg N/A

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

       6. neg-sub0 N/A

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

       7. --lowering--.f64 91.2%

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

   12. Simplified 91.2%

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

       1. sub0-neg N/A

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

       2. neg-lowering-neg.f64 91.2%

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

   14. Applied egg-rr 91.2%

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

   if -4.9999999999999999e149 < b < 1.79999999999999999e-235

   1. Initial program 90.0%

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

   3. Simplified 90.0%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

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

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

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

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

      5. rem-square-sqrt N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 90.0%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

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

      2. associate-*r/ N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

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

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

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

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

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

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

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

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

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

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 90.0%

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

   9. Simplified 90.0%

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

   if 1.79999999999999999e-235 < b < 5.90000000000000013e57

   1. Initial program 85.3%

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

   3. Simplified 85.3%

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

      1. flip-- N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 85.3%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 85.3%

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

   9. Simplified 85.3%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right)}{a}} \]

   if 5.90000000000000013e57 < b

   1. Initial program 70.9%

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

   3. Simplified 70.9%

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

      1. flip-- N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 70.9%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 70.9%

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

   9. Simplified 70.9%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right)}{a}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot b\right)}, a\right) \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

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

       3. --lowering--.f64 100.0%

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

   12. Simplified 100.0%

       \[\leadsto \frac{\color{blue}{0 - b}}{a} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+149}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-235}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)} - b}\\ \mathbf{elif}\;b \leq 5.9 \cdot 10^{+57}:\\ \;\;\;\;\frac{-0.5 \cdot \left(b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-7}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 5.7 \cdot 10^{+57}:\\ \;\;\;\;\frac{-0.5 \cdot \left(b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.15e-7)
   (- 0.0 (/ c b))
   (if (<= b 5.7e+57)
     (/ (* -0.5 (+ b (sqrt (+ (* b b) (* -4.0 (* c a)))))) a)
     (- 0.0 (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.15e-7) {
		tmp = 0.0 - (c / b);
	} else if (b <= 5.7e+57) {
		tmp = (-0.5 * (b + sqrt(((b * b) + (-4.0 * (c * a)))))) / a;
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.15d-7)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 5.7d+57) then
        tmp = ((-0.5d0) * (b + sqrt(((b * b) + ((-4.0d0) * (c * a)))))) / a
    else
        tmp = 0.0d0 - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.15e-7) {
		tmp = 0.0 - (c / b);
	} else if (b <= 5.7e+57) {
		tmp = (-0.5 * (b + Math.sqrt(((b * b) + (-4.0 * (c * a)))))) / a;
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.15e-7:
		tmp = 0.0 - (c / b)
	elif b <= 5.7e+57:
		tmp = (-0.5 * (b + math.sqrt(((b * b) + (-4.0 * (c * a)))))) / a
	else:
		tmp = 0.0 - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.15e-7)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 5.7e+57)
		tmp = Float64(Float64(-0.5 * Float64(b + sqrt(Float64(Float64(b * b) + Float64(-4.0 * Float64(c * a)))))) / a);
	else
		tmp = Float64(0.0 - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.15e-7)
		tmp = 0.0 - (c / b);
	elseif (b <= 5.7e+57)
		tmp = (-0.5 * (b + sqrt(((b * b) + (-4.0 * (c * a)))))) / a;
	else
		tmp = 0.0 - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.15e-7], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.7e+57], N[(N[(-0.5 * N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.14999999999999997e-7

   1. Initial program 65.7%

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

   3. Simplified 65.7%

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

      1. flip-- N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 7.9%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 21.9%

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

   9. Simplified 21.9%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right)}{a}} \]

   10. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

          \[\leadsto \frac{c}{-1 \cdot \color{blue}{b}} \]

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

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

       5. mul-1-neg N/A

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

       6. neg-sub0 N/A

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

       7. --lowering--.f64 86.7%

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

   12. Simplified 86.7%

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

       1. sub0-neg N/A

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

       2. neg-lowering-neg.f64 86.7%

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

   14. Applied egg-rr 86.7%

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

   if -1.14999999999999997e-7 < b < 5.6999999999999998e57

   1. Initial program 86.1%

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

   3. Simplified 86.1%

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

      1. flip-- N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 81.2%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 81.5%

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

   9. Simplified 81.5%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right)}{a}} \]

   if 5.6999999999999998e57 < b

   1. Initial program 70.9%

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

   3. Simplified 70.9%

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

      1. flip-- N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 70.9%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 70.9%

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

   9. Simplified 70.9%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right)}{a}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot b\right)}, a\right) \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

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

       3. --lowering--.f64 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-7}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 5.7 \cdot 10^{+57}:\\ \;\;\;\;\frac{-0.5 \cdot \left(b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-8}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 5.9 \cdot 10^{+57}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.5e-8)
   (- 0.0 (/ c b))
   (if (<= b 5.9e+57)
     (* (/ -0.5 a) (+ b (sqrt (+ (* b b) (* -4.0 (* c a))))))
     (- 0.0 (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.5e-8) {
		tmp = 0.0 - (c / b);
	} else if (b <= 5.9e+57) {
		tmp = (-0.5 / a) * (b + sqrt(((b * b) + (-4.0 * (c * a)))));
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-9.5d-8)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 5.9d+57) then
        tmp = ((-0.5d0) / a) * (b + sqrt(((b * b) + ((-4.0d0) * (c * a)))))
    else
        tmp = 0.0d0 - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.5e-8) {
		tmp = 0.0 - (c / b);
	} else if (b <= 5.9e+57) {
		tmp = (-0.5 / a) * (b + Math.sqrt(((b * b) + (-4.0 * (c * a)))));
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9.5e-8:
		tmp = 0.0 - (c / b)
	elif b <= 5.9e+57:
		tmp = (-0.5 / a) * (b + math.sqrt(((b * b) + (-4.0 * (c * a)))))
	else:
		tmp = 0.0 - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9.5e-8)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 5.9e+57)
		tmp = Float64(Float64(-0.5 / a) * Float64(b + sqrt(Float64(Float64(b * b) + Float64(-4.0 * Float64(c * a))))));
	else
		tmp = Float64(0.0 - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -9.5e-8)
		tmp = 0.0 - (c / b);
	elseif (b <= 5.9e+57)
		tmp = (-0.5 / a) * (b + sqrt(((b * b) + (-4.0 * (c * a)))));
	else
		tmp = 0.0 - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9.5e-8], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.9e+57], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.50000000000000036e-8

   1. Initial program 65.7%

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

   3. Simplified 65.7%

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

      1. flip-- N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 7.9%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 21.9%

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

   9. Simplified 21.9%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right)}{a}} \]

   10. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

          \[\leadsto \frac{c}{-1 \cdot \color{blue}{b}} \]

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

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

       5. mul-1-neg N/A

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

       6. neg-sub0 N/A

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

       7. --lowering--.f64 86.7%

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

   12. Simplified 86.7%

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

       1. sub0-neg N/A

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

       2. neg-lowering-neg.f64 86.7%

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

   14. Applied egg-rr 86.7%

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

   if -9.50000000000000036e-8 < b < 5.90000000000000013e57

   1. Initial program 86.1%

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

   3. Simplified 86.1%

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

      1. flip-- N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 81.2%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 81.5%

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

   9. Simplified 81.5%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right)}{a}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. associate-/l* N/A

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

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

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

   11. Applied egg-rr 81.3%

       \[\leadsto \color{blue}{\left(b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}\right) \cdot \frac{-0.5}{a}} \]

   if 5.90000000000000013e57 < b

   1. Initial program 70.9%

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

   3. Simplified 70.9%

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

      1. flip-- N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 70.9%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 70.9%

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

   9. Simplified 70.9%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right)}{a}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot b\right)}, a\right) \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

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

       3. --lowering--.f64 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-8}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 5.9 \cdot 10^{+57}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-8}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-133}:\\ \;\;\;\;\frac{-0.5 \cdot \left(b + \sqrt{-4 \cdot \left(c \cdot a\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.5e-8)
   (- 0.0 (/ c b))
   (if (<= b 1.65e-133)
     (/ (* -0.5 (+ b (sqrt (* -4.0 (* c a))))) a)
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.5e-8) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1.65e-133) {
		tmp = (-0.5 * (b + sqrt((-4.0 * (c * a))))) / a;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-9.5d-8)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 1.65d-133) then
        tmp = ((-0.5d0) * (b + sqrt(((-4.0d0) * (c * a))))) / a
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.5e-8) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1.65e-133) {
		tmp = (-0.5 * (b + Math.sqrt((-4.0 * (c * a))))) / a;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9.5e-8:
		tmp = 0.0 - (c / b)
	elif b <= 1.65e-133:
		tmp = (-0.5 * (b + math.sqrt((-4.0 * (c * a))))) / a
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9.5e-8)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 1.65e-133)
		tmp = Float64(Float64(-0.5 * Float64(b + sqrt(Float64(-4.0 * Float64(c * a))))) / a);
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -9.5e-8)
		tmp = 0.0 - (c / b);
	elseif (b <= 1.65e-133)
		tmp = (-0.5 * (b + sqrt((-4.0 * (c * a))))) / a;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9.5e-8], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.65e-133], N[(N[(-0.5 * N[(b + N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.50000000000000036e-8

   1. Initial program 65.7%

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

   3. Simplified 65.7%

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

      1. flip-- N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 7.9%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 21.9%

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

   9. Simplified 21.9%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right)}{a}} \]

   10. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

          \[\leadsto \frac{c}{-1 \cdot \color{blue}{b}} \]

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

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

       5. mul-1-neg N/A

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

       6. neg-sub0 N/A

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

       7. --lowering--.f64 86.7%

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

   12. Simplified 86.7%

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

       1. sub0-neg N/A

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

       2. neg-lowering-neg.f64 86.7%

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

   14. Applied egg-rr 86.7%

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

   if -9.50000000000000036e-8 < b < 1.65000000000000005e-133

   1. Initial program 83.9%

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

   3. Simplified 83.9%

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

      1. flip-- N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 76.5%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 77.0%

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

   9. Simplified 77.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right)}{a}} \]

   10. Taylor expanded in a around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right)}\right)\right)\right), a\right) \]
   11. Step-by-step derivation

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

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

       2. *-lowering-*.f64 74.2%

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

   12. Simplified 74.2%

       \[\leadsto \frac{-0.5 \cdot \left(b + \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}\right)}{a} \]

   if 1.65000000000000005e-133 < b

   1. Initial program 78.0%

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

   3. Simplified 78.0%

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

      1. flip-- N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 78.0%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 78.0%

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

   9. Simplified 78.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right)}{a}} \]

   10. Taylor expanded in c around 0

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

       1. +-commutative N/A

          \[\leadsto \frac{c}{b} + \color{blue}{-1 \cdot \frac{b}{a}} \]

       2. mul-1-neg N/A

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

       3. unsub-neg N/A

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

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

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

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

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

       6. /-lowering-/.f64 90.6%

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

   12. Simplified 90.6%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-8}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-133}:\\ \;\;\;\;\frac{-0.5 \cdot \left(b + \sqrt{-4 \cdot \left(c \cdot a\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-8}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-133}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{-4 \cdot \left(c \cdot a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.5e-8)
   (- 0.0 (/ c b))
   (if (<= b 1.65e-133)
     (* (/ -0.5 a) (+ b (sqrt (* -4.0 (* c a)))))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.5e-8) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1.65e-133) {
		tmp = (-0.5 / a) * (b + sqrt((-4.0 * (c * a))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-9.5d-8)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 1.65d-133) then
        tmp = ((-0.5d0) / a) * (b + sqrt(((-4.0d0) * (c * a))))
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.5e-8) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1.65e-133) {
		tmp = (-0.5 / a) * (b + Math.sqrt((-4.0 * (c * a))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9.5e-8:
		tmp = 0.0 - (c / b)
	elif b <= 1.65e-133:
		tmp = (-0.5 / a) * (b + math.sqrt((-4.0 * (c * a))))
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9.5e-8)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 1.65e-133)
		tmp = Float64(Float64(-0.5 / a) * Float64(b + sqrt(Float64(-4.0 * Float64(c * a)))));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -9.5e-8)
		tmp = 0.0 - (c / b);
	elseif (b <= 1.65e-133)
		tmp = (-0.5 / a) * (b + sqrt((-4.0 * (c * a))));
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9.5e-8], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.65e-133], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.50000000000000036e-8

   1. Initial program 65.7%

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

   3. Simplified 65.7%

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

      1. flip-- N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 7.9%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 21.9%

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

   9. Simplified 21.9%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right)}{a}} \]

   10. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

          \[\leadsto \frac{c}{-1 \cdot \color{blue}{b}} \]

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

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

       5. mul-1-neg N/A

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

       6. neg-sub0 N/A

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

       7. --lowering--.f64 86.7%

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

   12. Simplified 86.7%

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

       1. sub0-neg N/A

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

       2. neg-lowering-neg.f64 86.7%

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

   14. Applied egg-rr 86.7%

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

   if -9.50000000000000036e-8 < b < 1.65000000000000005e-133

   1. Initial program 83.9%

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

   3. Simplified 83.9%

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

      1. flip-- N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 76.5%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 77.0%

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

   9. Simplified 77.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right)}{a}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. associate-/l* N/A

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

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

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

   11. Applied egg-rr 76.9%

       \[\leadsto \color{blue}{\left(b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}\right) \cdot \frac{-0.5}{a}} \]

   12. Taylor expanded in b around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right)}\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, a\right)\right) \]
   13. Step-by-step derivation

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

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

       2. *-lowering-*.f64 74.1%

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

   14. Simplified 74.1%

       \[\leadsto \left(b + \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}\right) \cdot \frac{-0.5}{a} \]

   if 1.65000000000000005e-133 < b

   1. Initial program 78.0%

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

   3. Simplified 78.0%

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

      1. flip-- N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 78.0%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 78.0%

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

   9. Simplified 78.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right)}{a}} \]

   10. Taylor expanded in c around 0

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

       1. +-commutative N/A

          \[\leadsto \frac{c}{b} + \color{blue}{-1 \cdot \frac{b}{a}} \]

       2. mul-1-neg N/A

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

       3. unsub-neg N/A

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

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

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

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

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

       6. /-lowering-/.f64 90.6%

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

   12. Simplified 90.6%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-8}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-133}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{-4 \cdot \left(c \cdot a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.4%

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

3. Simplified 76.4%

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

5. Taylor expanded in b around inf

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

7. Simplified 76.1%

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

8. Taylor expanded in b around -inf

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

   1. associate-*r* N/A

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

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

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

   3. mul-1-neg N/A

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

   4. neg-sub0 N/A

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

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

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

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

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

   7. associate-*r/ N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 66.4%

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

10. Simplified 66.4%

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

11. Final simplification 66.4%

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

Alternative 9: 67.2% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (- 0.0 (/ c b)) (- 0.0 (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = 0.0 - (c / b);
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = 0.0d0 - (c / b)
    else
        tmp = 0.0d0 - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = 0.0 - (c / b);
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = 0.0 - (c / b)
	else:
		tmp = 0.0 - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(0.0 - Float64(c / b));
	else
		tmp = Float64(0.0 - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = 0.0 - (c / b);
	else
		tmp = 0.0 - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 75.1%

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

   3. Simplified 75.1%

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

      1. flip-- N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 37.8%

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

   7. Taylor expanded in b around 0

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

      1. if-same N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(-4 \cdot \left(a \cdot c\right)\right), \left({b}^{2}\right)\right)\right)\right)\right), a\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \left(a \cdot c\right)\right), \left({b}^{2}\right)\right)\right)\right)\right), a\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, c\right)\right), \left({b}^{2}\right)\right)\right)\right)\right), a\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, c\right)\right), \left(b \cdot b\right)\right)\right)\right)\right), a\right) \]

      11. *-lowering-*.f64 46.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, c\right)\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), a\right) \]

   9. Simplified 46.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right)}{a}} \]

   10. Taylor expanded in b around -inf

       \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

       2. distribute-neg-frac2 N/A

          \[\leadsto \frac{c}{\color{blue}{\mathsf{neg}\left(b\right)}} \]

       3. mul-1-neg N/A

          \[\leadsto \frac{c}{-1 \cdot \color{blue}{b}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(c, \color{blue}{\left(-1 \cdot b\right)}\right) \]

       5. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(c, \left(\mathsf{neg}\left(b\right)\right)\right) \]

       6. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(c, \left(0 - \color{blue}{b}\right)\right) \]

       7. --lowering--.f64 56.8%

          \[\leadsto \mathsf{/.f64}\left(c, \mathsf{\_.f64}\left(0, \color{blue}{b}\right)\right) \]

   12. Simplified 56.8%

       \[\leadsto \color{blue}{\frac{c}{0 - b}} \]
   13. Step-by-step derivation

       1. sub0-neg N/A

          \[\leadsto \mathsf{/.f64}\left(c, \left(\mathsf{neg}\left(b\right)\right)\right) \]

       2. neg-lowering-neg.f64 56.8%

          \[\leadsto \mathsf{/.f64}\left(c, \mathsf{neg.f64}\left(b\right)\right) \]

   14. Applied egg-rr 56.8%

       \[\leadsto \frac{c}{\color{blue}{-b}} \]

   if -4.999999999999985e-310 < b

   1. Initial program 77.7%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Step-by-step derivation

   3. Simplified 77.7%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ } \end{array}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right), -2\right), a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} \cdot \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b \cdot b}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} + b}}\\ \end{array} \]

      2. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right), -2\right), a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} \cdot \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b \cdot b}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}}\\ \end{array} \]

      3. associate-/r/ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right), -2\right), a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} \cdot \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b \cdot b} \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right)\\ \end{array} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right), -2\right), a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(\left(\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} \cdot \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b \cdot b}\right), \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right)\right)\\ \end{array} \]

   6. Applied egg-rr 77.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right) - b \cdot b} \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right)\\ \end{array} \]

   7. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}{a}\\ } \end{array}} \]
   8. Step-by-step derivation

      1. if-same N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}{\color{blue}{a}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)\right), \color{blue}{a}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)\right), a\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(b, \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)\right)\right), a\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(-4 \cdot \left(a \cdot c\right) + {b}^{2}\right)\right)\right)\right), a\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(-4 \cdot \left(a \cdot c\right)\right), \left({b}^{2}\right)\right)\right)\right)\right), a\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \left(a \cdot c\right)\right), \left({b}^{2}\right)\right)\right)\right)\right), a\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, c\right)\right), \left({b}^{2}\right)\right)\right)\right)\right), a\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, c\right)\right), \left(b \cdot b\right)\right)\right)\right)\right), a\right) \]

      11. *-lowering-*.f64 77.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, c\right)\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), a\right) \]

   9. Simplified 77.7%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right)}{a}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot b\right)}, a\right) \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(b\right)\right), a\right) \]

       2. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(0 - b\right), a\right) \]

       3. --lowering--.f64 77.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, b\right), a\right) \]

   12. Simplified 77.0%

       \[\leadsto \frac{\color{blue}{0 - b}}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.4% accurate, 24.2× speedup

Math

\[\begin{array}{l} \\ 0 - \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (- 0.0 (/ c b)))

C

double code(double a, double b, double c) {
	return 0.0 - (c / b);
}

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 - (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return 0.0 - (c / b);
}

Python

def code(a, b, c):
	return 0.0 - (c / b)

Julia

function code(a, b, c)
	return Float64(0.0 - Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 - (c / b);
end

Wolfram

code[a_, b_, c_] := N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.4%

   \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
2. Step-by-step derivation

3. Simplified 76.4%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ } \end{array}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. flip-- N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right), -2\right), a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} \cdot \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b \cdot b}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} + b}}\\ \end{array} \]

   2. +-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right), -2\right), a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} \cdot \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b \cdot b}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}}\\ \end{array} \]

   3. associate-/r/ N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right), -2\right), a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} \cdot \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b \cdot b} \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right)\\ \end{array} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right), -2\right), a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(\left(\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} \cdot \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b \cdot b}\right), \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right)\right)\\ \end{array} \]

6. Applied egg-rr 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right) - b \cdot b} \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right)\\ \end{array} \]

7. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}{a}\\ } \end{array}} \]
8. Step-by-step derivation

   1. if-same N/A

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}{a}} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{\frac{-1}{2} \cdot \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}{\color{blue}{a}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)\right), \color{blue}{a}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)\right), a\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(b, \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)\right)\right), a\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(-4 \cdot \left(a \cdot c\right) + {b}^{2}\right)\right)\right)\right), a\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(-4 \cdot \left(a \cdot c\right)\right), \left({b}^{2}\right)\right)\right)\right)\right), a\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \left(a \cdot c\right)\right), \left({b}^{2}\right)\right)\right)\right)\right), a\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, c\right)\right), \left({b}^{2}\right)\right)\right)\right)\right), a\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, c\right)\right), \left(b \cdot b\right)\right)\right)\right)\right), a\right) \]

   11. *-lowering-*.f64 62.3%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(a, c\right)\right), \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), a\right) \]

9. Simplified 62.3%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}\right)}{a}} \]

10. Taylor expanded in b around -inf

    \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
11. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

    2. distribute-neg-frac2 N/A

       \[\leadsto \frac{c}{\color{blue}{\mathsf{neg}\left(b\right)}} \]

    3. mul-1-neg N/A

       \[\leadsto \frac{c}{-1 \cdot \color{blue}{b}} \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(c, \color{blue}{\left(-1 \cdot b\right)}\right) \]

    5. mul-1-neg N/A

       \[\leadsto \mathsf{/.f64}\left(c, \left(\mathsf{neg}\left(b\right)\right)\right) \]

    6. neg-sub0 N/A

       \[\leadsto \mathsf{/.f64}\left(c, \left(0 - \color{blue}{b}\right)\right) \]

    7. --lowering--.f64 28.7%

       \[\leadsto \mathsf{/.f64}\left(c, \mathsf{\_.f64}\left(0, \color{blue}{b}\right)\right) \]

12. Simplified 28.7%

    \[\leadsto \color{blue}{\frac{c}{0 - b}} \]
13. Step-by-step derivation

    1. sub0-neg N/A

       \[\leadsto \mathsf{/.f64}\left(c, \left(\mathsf{neg}\left(b\right)\right)\right) \]

    2. neg-lowering-neg.f64 28.7%

       \[\leadsto \mathsf{/.f64}\left(c, \mathsf{neg.f64}\left(b\right)\right) \]

14. Applied egg-rr 28.7%

    \[\leadsto \frac{c}{\color{blue}{-b}} \]

15. Final simplification 28.7%

    \[\leadsto 0 - \frac{c}{b} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  :precision binary64
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))