jeff quadratic root 1

Percentage Accurate: 71.8% → 89.7%

Time: 18.1s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

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: 71.8% 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: 89.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\\ \mathbf{if}\;b \leq -5.8 \cdot 10^{+168}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;b \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{+137}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + t\_0}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t\_0 - b}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* b b) (* c (* a -4.0))))))
   (if (<= b -5.8e+168)
     (if (>= b 0.0) (* b (/ -2.0 a)) (/ c (- 0.0 b)))
     (if (<= b 1e+137)
       (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_1;
	if (b <= -5.8e+168) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b * (-2.0 / a);
		} else {
			tmp_2 = c / (0.0 - b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1e+137) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = ((b + t_0) / -2.0) / a;
		} else {
			tmp_3 = (c * 2.0) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = 0.0 - (b / a);
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = sqrt(((b * b) + (c * (a * (-4.0d0)))))
    if (b <= (-5.8d+168)) then
        if (b >= 0.0d0) then
            tmp_2 = b * ((-2.0d0) / a)
        else
            tmp_2 = c / (0.0d0 - b)
        end if
        tmp_1 = tmp_2
    else if (b <= 1d+137) then
        if (b >= 0.0d0) then
            tmp_3 = ((b + t_0) / (-2.0d0)) / a
        else
            tmp_3 = (c * 2.0d0) / (t_0 - b)
        end if
        tmp_1 = tmp_3
    else
        tmp_1 = 0.0d0 - (b / a)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) + (c * (a * -4.0))));
	double tmp_1;
	if (b <= -5.8e+168) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b * (-2.0 / a);
		} else {
			tmp_2 = c / (0.0 - b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1e+137) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = ((b + t_0) / -2.0) / a;
		} else {
			tmp_3 = (c * 2.0) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = 0.0 - (b / a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) + (c * (a * -4.0))))
	tmp_1 = 0
	if b <= -5.8e+168:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = b * (-2.0 / a)
		else:
			tmp_2 = c / (0.0 - b)
		tmp_1 = tmp_2
	elif b <= 1e+137:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = ((b + t_0) / -2.0) / a
		else:
			tmp_3 = (c * 2.0) / (t_0 - b)
		tmp_1 = tmp_3
	else:
		tmp_1 = 0.0 - (b / a)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0))))
	tmp_1 = 0.0
	if (b <= -5.8e+168)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(b * Float64(-2.0 / a));
		else
			tmp_2 = Float64(c / Float64(0.0 - b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1e+137)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(b + t_0) / -2.0) / a);
		else
			tmp_3 = Float64(Float64(c * 2.0) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	else
		tmp_1 = Float64(0.0 - Float64(b / a));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) + (c * (a * -4.0))));
	tmp_2 = 0.0;
	if (b <= -5.8e+168)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = b * (-2.0 / a);
		else
			tmp_3 = c / (0.0 - b);
		end
		tmp_2 = tmp_3;
	elseif (b <= 1e+137)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = ((b + t_0) / -2.0) / a;
		else
			tmp_4 = (c * 2.0) / (t_0 - b);
		end
		tmp_2 = tmp_4;
	else
		tmp_2 = 0.0 - (b / a);
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -5.8e+168], If[GreaterEqual[b, 0.0], N[(b * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision], N[(c / N[(0.0 - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1e+137], 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 < -5.8e168

   1. Initial program 30.4%

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

   3. Simplified 30.4%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ } \end{array}} \]
   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 30.4%

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

   8. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 100.0%

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

   10. Simplified 100.0%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + b}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
   11. Step-by-step derivation

       1. flip-+ N/A

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

       2. +-inverses N/A

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

       3. +-inverses N/A

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

       4. associate-/l/ N/A

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

       5. metadata-eval N/A

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

       6. metadata-eval N/A

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

       7. distribute-neg-frac N/A

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

       8. +-inverses N/A

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

       9. +-inverses N/A

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

       10. flip-+ N/A

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

       11. neg-mul-1 N/A

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

       12. distribute-lft-out N/A

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

       13. distribute-rgt-out N/A

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

       14. metadata-eval N/A

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

       15. *-lowering-*.f64 100.0%

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

   12. Applied egg-rr 100.0%

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

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

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

       4. /-lowering-/.f64 100.0%

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

   14. Applied egg-rr 100.0%

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

   if -5.8e168 < b < 1e137

   1. Initial program 89.9%

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

   3. Simplified 89.9%

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

   if 1e137 < b

   1. Initial program 46.8%

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

   3. Simplified 46.8%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ } \end{array}} \]
   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 98.2%

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

      1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

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

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

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

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

      12. *-lowering-*.f64 98.2%

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

   9. Applied egg-rr 98.2%

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

   10. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(b, b\right), -2\right), a\right)\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\mathsf{/.f64}\left(1, \left(-1 \cdot \frac{a}{b}\right)\right)}\\ \end{array} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

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

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

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

       4. /-lowering-/.f64 98.2%

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

   12. Simplified 98.2%

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

   13. Taylor expanded in b around 0

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

       1. if-same N/A

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

       2. mul-1-neg N/A

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

       3. neg-sub0 N/A

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

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

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

       5. /-lowering-/.f64 98.2%

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

   15. Simplified 98.2%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{+168}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;b \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{+137}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot 2}{0 - \left(b + b\right)}\\ t_1 := \sqrt{-4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b \leq -1.38 \cdot 10^{-62}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;b \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array}\\ \mathbf{elif}\;b \leq -8.2 \cdot 10^{-304}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + b}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t\_1 - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.75 \cdot 10^{-78}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + t\_1}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* c 2.0) (- 0.0 (+ b b)))) (t_1 (sqrt (* -4.0 (* a c)))))
   (if (<= b -1.38e-62)
     (if (>= b 0.0) (* b (/ -2.0 a)) (/ c (- 0.0 b)))
     (if (<= b -8.2e-304)
       (if (>= b 0.0) (/ (/ (+ b b) -2.0) a) (/ (* c 2.0) (- t_1 b)))
       (if (<= b 1.75e-78)
         (if (>= b 0.0) (/ (/ (+ b t_1) -2.0) a) t_0)
         (if (>= b 0.0) (- (/ c b) (/ b a)) t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = (c * 2.0) / (0.0 - (b + b));
	double t_1 = sqrt((-4.0 * (a * c)));
	double tmp_1;
	if (b <= -1.38e-62) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b * (-2.0 / a);
		} else {
			tmp_2 = c / (0.0 - b);
		}
		tmp_1 = tmp_2;
	} else if (b <= -8.2e-304) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = ((b + b) / -2.0) / a;
		} else {
			tmp_3 = (c * 2.0) / (t_1 - b);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.75e-78) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = ((b + t_1) / -2.0) / a;
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    real(8) :: tmp_4
    t_0 = (c * 2.0d0) / (0.0d0 - (b + b))
    t_1 = sqrt(((-4.0d0) * (a * c)))
    if (b <= (-1.38d-62)) then
        if (b >= 0.0d0) then
            tmp_2 = b * ((-2.0d0) / a)
        else
            tmp_2 = c / (0.0d0 - b)
        end if
        tmp_1 = tmp_2
    else if (b <= (-8.2d-304)) then
        if (b >= 0.0d0) then
            tmp_3 = ((b + b) / (-2.0d0)) / a
        else
            tmp_3 = (c * 2.0d0) / (t_1 - b)
        end if
        tmp_1 = tmp_3
    else if (b <= 1.75d-78) then
        if (b >= 0.0d0) then
            tmp_4 = ((b + t_1) / (-2.0d0)) / a
        else
            tmp_4 = t_0
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = (c / b) - (b / a)
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (c * 2.0) / (0.0 - (b + b));
	double t_1 = Math.sqrt((-4.0 * (a * c)));
	double tmp_1;
	if (b <= -1.38e-62) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b * (-2.0 / a);
		} else {
			tmp_2 = c / (0.0 - b);
		}
		tmp_1 = tmp_2;
	} else if (b <= -8.2e-304) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = ((b + b) / -2.0) / a;
		} else {
			tmp_3 = (c * 2.0) / (t_1 - b);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.75e-78) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = ((b + t_1) / -2.0) / a;
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (c * 2.0) / (0.0 - (b + b))
	t_1 = math.sqrt((-4.0 * (a * c)))
	tmp_1 = 0
	if b <= -1.38e-62:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = b * (-2.0 / a)
		else:
			tmp_2 = c / (0.0 - b)
		tmp_1 = tmp_2
	elif b <= -8.2e-304:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = ((b + b) / -2.0) / a
		else:
			tmp_3 = (c * 2.0) / (t_1 - b)
		tmp_1 = tmp_3
	elif b <= 1.75e-78:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = ((b + t_1) / -2.0) / a
		else:
			tmp_4 = t_0
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = (c / b) - (b / a)
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * 2.0) / Float64(0.0 - Float64(b + b)))
	t_1 = sqrt(Float64(-4.0 * Float64(a * c)))
	tmp_1 = 0.0
	if (b <= -1.38e-62)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(b * Float64(-2.0 / a));
		else
			tmp_2 = Float64(c / Float64(0.0 - b));
		end
		tmp_1 = tmp_2;
	elseif (b <= -8.2e-304)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(b + b) / -2.0) / a);
		else
			tmp_3 = Float64(Float64(c * 2.0) / Float64(t_1 - b));
		end
		tmp_1 = tmp_3;
	elseif (b <= 1.75e-78)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(Float64(b + t_1) / -2.0) / a);
		else
			tmp_4 = t_0;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = (c * 2.0) / (0.0 - (b + b));
	t_1 = sqrt((-4.0 * (a * c)));
	tmp_2 = 0.0;
	if (b <= -1.38e-62)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = b * (-2.0 / a);
		else
			tmp_3 = c / (0.0 - b);
		end
		tmp_2 = tmp_3;
	elseif (b <= -8.2e-304)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = ((b + b) / -2.0) / a;
		else
			tmp_4 = (c * 2.0) / (t_1 - b);
		end
		tmp_2 = tmp_4;
	elseif (b <= 1.75e-78)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = ((b + t_1) / -2.0) / a;
		else
			tmp_5 = t_0;
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = (c / b) - (b / a);
	else
		tmp_2 = t_0;
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * 2.0), $MachinePrecision] / N[(0.0 - N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.38e-62], If[GreaterEqual[b, 0.0], N[(b * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision], N[(c / N[(0.0 - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -8.2e-304], If[GreaterEqual[b, 0.0], N[(N[(N[(b + b), $MachinePrecision] / -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(t$95$1 - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.75e-78], If[GreaterEqual[b, 0.0], N[(N[(N[(b + t$95$1), $MachinePrecision] / -2.0), $MachinePrecision] / a), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.38e-62

   1. Initial program 73.2%

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

   3. Simplified 73.2%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ } \end{array}} \]
   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 73.2%

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

   8. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 91.5%

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

   10. Simplified 91.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + b}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
   11. Step-by-step derivation

       1. flip-+ N/A

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

       2. +-inverses N/A

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

       3. +-inverses N/A

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

       4. associate-/l/ N/A

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

       5. metadata-eval N/A

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

       6. metadata-eval N/A

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

       7. distribute-neg-frac N/A

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

       8. +-inverses N/A

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

       9. +-inverses N/A

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

       10. flip-+ N/A

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

       11. neg-mul-1 N/A

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

       12. distribute-lft-out N/A

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

       13. distribute-rgt-out N/A

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

       14. metadata-eval N/A

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

       15. *-lowering-*.f64 91.5%

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

   12. Applied egg-rr 91.5%

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

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

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

       4. /-lowering-/.f64 91.5%

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

   14. Applied egg-rr 91.5%

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

   if -1.38e-62 < b < -8.20000000000000005e-304

   1. Initial program 84.4%

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

   3. Simplified 84.4%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ } \end{array}} \]
   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 84.4%

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

   8. Taylor expanded in b around 0

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

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

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

      2. *-lowering-*.f64 72.9%

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

   10. Simplified 72.9%

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

   if -8.20000000000000005e-304 < b < 1.75e-78

   1. Initial program 73.4%

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

   3. Simplified 73.4%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 73.4%

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

   7. Simplified 73.4%

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

   8. Taylor expanded in b around 0

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

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

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

      2. *-lowering-*.f64 70.5%

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

   10. Simplified 70.5%

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

   if 1.75e-78 < b

   1. Initial program 68.5%

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

   3. Simplified 68.5%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 68.5%

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

   7. Simplified 68.5%

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

   8. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. /-lowering-/.f64 90.8%

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

   10. Simplified 90.8%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.38 \cdot 10^{-62}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;b \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array}\\ \mathbf{elif}\;b \leq -8.2 \cdot 10^{-304}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + b}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.75 \cdot 10^{-78}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + \sqrt{-4 \cdot \left(a \cdot c\right)}}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{0 - \left(b + b\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{0 - \left(b + b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot 2}{0 - \left(b + b\right)}\\ \mathbf{if}\;b \leq -1.88 \cdot 10^{-62}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;b \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array}\\ \mathbf{elif}\;b \leq -8.2 \cdot 10^{-304}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + b}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-81}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* c 2.0) (- 0.0 (+ b b)))))
   (if (<= b -1.88e-62)
     (if (>= b 0.0) (* b (/ -2.0 a)) (/ c (- 0.0 b)))
     (if (<= b -8.2e-304)
       (if (>= b 0.0)
         (/ (/ (+ b b) -2.0) a)
         (/ (* c 2.0) (- (sqrt (* -4.0 (* a c))) b)))
       (if (<= b 1.45e-81)
         (if (>= b 0.0) (* (/ -0.5 a) (+ b (sqrt (* a (* c -4.0))))) t_0)
         (if (>= b 0.0) (- (/ c b) (/ b a)) t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = (c * 2.0) / (0.0 - (b + b));
	double tmp_1;
	if (b <= -1.88e-62) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b * (-2.0 / a);
		} else {
			tmp_2 = c / (0.0 - b);
		}
		tmp_1 = tmp_2;
	} else if (b <= -8.2e-304) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = ((b + b) / -2.0) / a;
		} else {
			tmp_3 = (c * 2.0) / (sqrt((-4.0 * (a * c))) - b);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.45e-81) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (-0.5 / a) * (b + sqrt((a * (c * -4.0))));
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    real(8) :: tmp_4
    t_0 = (c * 2.0d0) / (0.0d0 - (b + b))
    if (b <= (-1.88d-62)) then
        if (b >= 0.0d0) then
            tmp_2 = b * ((-2.0d0) / a)
        else
            tmp_2 = c / (0.0d0 - b)
        end if
        tmp_1 = tmp_2
    else if (b <= (-8.2d-304)) then
        if (b >= 0.0d0) then
            tmp_3 = ((b + b) / (-2.0d0)) / a
        else
            tmp_3 = (c * 2.0d0) / (sqrt(((-4.0d0) * (a * c))) - b)
        end if
        tmp_1 = tmp_3
    else if (b <= 1.45d-81) then
        if (b >= 0.0d0) then
            tmp_4 = ((-0.5d0) / a) * (b + sqrt((a * (c * (-4.0d0)))))
        else
            tmp_4 = t_0
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = (c / b) - (b / a)
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (c * 2.0) / (0.0 - (b + b));
	double tmp_1;
	if (b <= -1.88e-62) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b * (-2.0 / a);
		} else {
			tmp_2 = c / (0.0 - b);
		}
		tmp_1 = tmp_2;
	} else if (b <= -8.2e-304) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = ((b + b) / -2.0) / a;
		} else {
			tmp_3 = (c * 2.0) / (Math.sqrt((-4.0 * (a * c))) - b);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.45e-81) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (-0.5 / a) * (b + Math.sqrt((a * (c * -4.0))));
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (c * 2.0) / (0.0 - (b + b))
	tmp_1 = 0
	if b <= -1.88e-62:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = b * (-2.0 / a)
		else:
			tmp_2 = c / (0.0 - b)
		tmp_1 = tmp_2
	elif b <= -8.2e-304:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = ((b + b) / -2.0) / a
		else:
			tmp_3 = (c * 2.0) / (math.sqrt((-4.0 * (a * c))) - b)
		tmp_1 = tmp_3
	elif b <= 1.45e-81:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = (-0.5 / a) * (b + math.sqrt((a * (c * -4.0))))
		else:
			tmp_4 = t_0
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = (c / b) - (b / a)
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * 2.0) / Float64(0.0 - Float64(b + b)))
	tmp_1 = 0.0
	if (b <= -1.88e-62)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(b * Float64(-2.0 / a));
		else
			tmp_2 = Float64(c / Float64(0.0 - b));
		end
		tmp_1 = tmp_2;
	elseif (b <= -8.2e-304)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(b + b) / -2.0) / a);
		else
			tmp_3 = Float64(Float64(c * 2.0) / Float64(sqrt(Float64(-4.0 * Float64(a * c))) - b));
		end
		tmp_1 = tmp_3;
	elseif (b <= 1.45e-81)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(-0.5 / a) * Float64(b + sqrt(Float64(a * Float64(c * -4.0)))));
		else
			tmp_4 = t_0;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = (c * 2.0) / (0.0 - (b + b));
	tmp_2 = 0.0;
	if (b <= -1.88e-62)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = b * (-2.0 / a);
		else
			tmp_3 = c / (0.0 - b);
		end
		tmp_2 = tmp_3;
	elseif (b <= -8.2e-304)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = ((b + b) / -2.0) / a;
		else
			tmp_4 = (c * 2.0) / (sqrt((-4.0 * (a * c))) - b);
		end
		tmp_2 = tmp_4;
	elseif (b <= 1.45e-81)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = (-0.5 / a) * (b + sqrt((a * (c * -4.0))));
		else
			tmp_5 = t_0;
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = (c / b) - (b / a);
	else
		tmp_2 = t_0;
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * 2.0), $MachinePrecision] / N[(0.0 - N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.88e-62], If[GreaterEqual[b, 0.0], N[(b * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision], N[(c / N[(0.0 - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -8.2e-304], If[GreaterEqual[b, 0.0], N[(N[(N[(b + b), $MachinePrecision] / -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.45e-81], If[GreaterEqual[b, 0.0], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.88e-62

   1. Initial program 73.2%

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

   3. Simplified 73.2%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ } \end{array}} \]
   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 73.2%

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

   8. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 91.5%

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

   10. Simplified 91.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + b}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
   11. Step-by-step derivation

       1. flip-+ N/A

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

       2. +-inverses N/A

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

       3. +-inverses N/A

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

       4. associate-/l/ N/A

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

       5. metadata-eval N/A

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

       6. metadata-eval N/A

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

       7. distribute-neg-frac N/A

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

       8. +-inverses N/A

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

       9. +-inverses N/A

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

       10. flip-+ N/A

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

       11. neg-mul-1 N/A

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

       12. distribute-lft-out N/A

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

       13. distribute-rgt-out N/A

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

       14. metadata-eval N/A

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

       15. *-lowering-*.f64 91.5%

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

   12. Applied egg-rr 91.5%

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

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

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

       4. /-lowering-/.f64 91.5%

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

   14. Applied egg-rr 91.5%

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

   if -1.88e-62 < b < -8.20000000000000005e-304

   1. Initial program 84.4%

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

   3. Simplified 84.4%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ } \end{array}} \]
   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 84.4%

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

   8. Taylor expanded in b around 0

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

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

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

      2. *-lowering-*.f64 72.9%

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

   10. Simplified 72.9%

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

   if -8.20000000000000005e-304 < b < 1.44999999999999994e-81

   1. Initial program 73.4%

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

   3. Simplified 73.4%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 73.4%

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

   7. Simplified 73.4%

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

   8. Taylor expanded in b around 0

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

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

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

      2. *-lowering-*.f64 70.5%

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

   10. Simplified 70.5%

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

       1. div-inv N/A

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

       2. associate-/l* N/A

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

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

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

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

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. *-commutative N/A

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

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

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

       9. *-commutative N/A

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

       10. associate-*r* N/A

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

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

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

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

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

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

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

       14. metadata-eval 70.3%

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

   12. Applied egg-rr 70.3%

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

   if 1.44999999999999994e-81 < b

   1. Initial program 68.5%

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

   3. Simplified 68.5%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 68.5%

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

   7. Simplified 68.5%

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

   8. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. /-lowering-/.f64 90.8%

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

   10. Simplified 90.8%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.88 \cdot 10^{-62}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;b \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array}\\ \mathbf{elif}\;b \leq -8.2 \cdot 10^{-304}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + b}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-81}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{0 - \left(b + b\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{0 - \left(b + b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\\ \mathbf{if}\;b \leq -5.8 \cdot 10^{+168}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;b \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{+139}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(b + t\_0\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t\_0 - b}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* b b) (* c (* a -4.0))))))
   (if (<= b -5.8e+168)
     (if (>= b 0.0) (* b (/ -2.0 a)) (/ c (- 0.0 b)))
     (if (<= b 7.5e+139)
       (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_1;
	if (b <= -5.8e+168) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b * (-2.0 / a);
		} else {
			tmp_2 = c / (0.0 - b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 7.5e+139) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (b + t_0) * (-0.5 / a);
		} else {
			tmp_3 = (c * 2.0) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = 0.0 - (b / a);
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = sqrt(((b * b) + (c * (a * (-4.0d0)))))
    if (b <= (-5.8d+168)) then
        if (b >= 0.0d0) then
            tmp_2 = b * ((-2.0d0) / a)
        else
            tmp_2 = c / (0.0d0 - b)
        end if
        tmp_1 = tmp_2
    else if (b <= 7.5d+139) then
        if (b >= 0.0d0) then
            tmp_3 = (b + t_0) * ((-0.5d0) / a)
        else
            tmp_3 = (c * 2.0d0) / (t_0 - b)
        end if
        tmp_1 = tmp_3
    else
        tmp_1 = 0.0d0 - (b / a)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) + (c * (a * -4.0))));
	double tmp_1;
	if (b <= -5.8e+168) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b * (-2.0 / a);
		} else {
			tmp_2 = c / (0.0 - b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 7.5e+139) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (b + t_0) * (-0.5 / a);
		} else {
			tmp_3 = (c * 2.0) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = 0.0 - (b / a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) + (c * (a * -4.0))))
	tmp_1 = 0
	if b <= -5.8e+168:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = b * (-2.0 / a)
		else:
			tmp_2 = c / (0.0 - b)
		tmp_1 = tmp_2
	elif b <= 7.5e+139:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (b + t_0) * (-0.5 / a)
		else:
			tmp_3 = (c * 2.0) / (t_0 - b)
		tmp_1 = tmp_3
	else:
		tmp_1 = 0.0 - (b / a)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0))))
	tmp_1 = 0.0
	if (b <= -5.8e+168)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(b * Float64(-2.0 / a));
		else
			tmp_2 = Float64(c / Float64(0.0 - b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 7.5e+139)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(b + t_0) * Float64(-0.5 / a));
		else
			tmp_3 = Float64(Float64(c * 2.0) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	else
		tmp_1 = Float64(0.0 - Float64(b / a));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) + (c * (a * -4.0))));
	tmp_2 = 0.0;
	if (b <= -5.8e+168)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = b * (-2.0 / a);
		else
			tmp_3 = c / (0.0 - b);
		end
		tmp_2 = tmp_3;
	elseif (b <= 7.5e+139)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (b + t_0) * (-0.5 / a);
		else
			tmp_4 = (c * 2.0) / (t_0 - b);
		end
		tmp_2 = tmp_4;
	else
		tmp_2 = 0.0 - (b / a);
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -5.8e+168], If[GreaterEqual[b, 0.0], N[(b * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision], N[(c / N[(0.0 - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 7.5e+139], 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 < -5.8e168

   1. Initial program 30.4%

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

   3. Simplified 30.4%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ } \end{array}} \]
   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 30.4%

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

   8. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 100.0%

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

   10. Simplified 100.0%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + b}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
   11. Step-by-step derivation

       1. flip-+ N/A

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

       2. +-inverses N/A

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

       3. +-inverses N/A

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

       4. associate-/l/ N/A

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

       5. metadata-eval N/A

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

       6. metadata-eval N/A

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

       7. distribute-neg-frac N/A

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

       8. +-inverses N/A

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

       9. +-inverses N/A

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

       10. flip-+ N/A

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

       11. neg-mul-1 N/A

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

       12. distribute-lft-out N/A

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

       13. distribute-rgt-out N/A

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

       14. metadata-eval N/A

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

       15. *-lowering-*.f64 100.0%

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

   12. Applied egg-rr 100.0%

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

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

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

       4. /-lowering-/.f64 100.0%

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

   14. Applied egg-rr 100.0%

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

   if -5.8e168 < b < 7.49999999999999992e139

   1. Initial program 89.9%

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

   3. Simplified 89.9%

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

      1. associate-/l/ N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

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

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

      8. metadata-eval N/A

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

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

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

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

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

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

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

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

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

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

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

      14. *-lowering-*.f64 89.8%

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

   6. Applied egg-rr 89.8%

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

   if 7.49999999999999992e139 < b

   1. Initial program 46.8%

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

   3. Simplified 46.8%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ } \end{array}} \]
   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 98.2%

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

      1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

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

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

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

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

      12. *-lowering-*.f64 98.2%

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

   9. Applied egg-rr 98.2%

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

   10. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(b, b\right), -2\right), a\right)\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\mathsf{/.f64}\left(1, \left(-1 \cdot \frac{a}{b}\right)\right)}\\ \end{array} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

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

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

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

       4. /-lowering-/.f64 98.2%

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

   12. Simplified 98.2%

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

   13. Taylor expanded in b around 0

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

       1. if-same N/A

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

       2. mul-1-neg N/A

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

       3. neg-sub0 N/A

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

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

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

       5. /-lowering-/.f64 98.2%

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

   15. Simplified 98.2%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{+168}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;b \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{+139}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+76}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;b \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{+139}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.2e+76)
   (if (>= b 0.0) (* b (/ -2.0 a)) (/ c (- 0.0 b)))
   (if (<= b 4.5e+139)
     (if (>= b 0.0)
       (* (+ b (sqrt (+ (* b b) (* c (* a -4.0))))) (/ -0.5 a))
       (* c (/ 2.0 (- (sqrt (+ (* b b) (* a (* c -4.0)))) b))))
     (- 0.0 (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1.2e+76) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b * (-2.0 / a);
		} else {
			tmp_2 = c / (0.0 - b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 4.5e+139) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (b + sqrt(((b * b) + (c * (a * -4.0))))) * (-0.5 / a);
		} else {
			tmp_3 = c * (2.0 / (sqrt(((b * b) + (a * (c * -4.0)))) - b));
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = 0.0 - (b / a);
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    if (b <= (-1.2d+76)) then
        if (b >= 0.0d0) then
            tmp_2 = b * ((-2.0d0) / a)
        else
            tmp_2 = c / (0.0d0 - b)
        end if
        tmp_1 = tmp_2
    else if (b <= 4.5d+139) then
        if (b >= 0.0d0) then
            tmp_3 = (b + sqrt(((b * b) + (c * (a * (-4.0d0)))))) * ((-0.5d0) / a)
        else
            tmp_3 = c * (2.0d0 / (sqrt(((b * b) + (a * (c * (-4.0d0))))) - b))
        end if
        tmp_1 = tmp_3
    else
        tmp_1 = 0.0d0 - (b / a)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1.2e+76) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b * (-2.0 / a);
		} else {
			tmp_2 = c / (0.0 - b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 4.5e+139) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (b + Math.sqrt(((b * b) + (c * (a * -4.0))))) * (-0.5 / a);
		} else {
			tmp_3 = c * (2.0 / (Math.sqrt(((b * b) + (a * (c * -4.0)))) - b));
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = 0.0 - (b / a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if b <= -1.2e+76:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = b * (-2.0 / a)
		else:
			tmp_2 = c / (0.0 - b)
		tmp_1 = tmp_2
	elif b <= 4.5e+139:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (b + math.sqrt(((b * b) + (c * (a * -4.0))))) * (-0.5 / a)
		else:
			tmp_3 = c * (2.0 / (math.sqrt(((b * b) + (a * (c * -4.0)))) - b))
		tmp_1 = tmp_3
	else:
		tmp_1 = 0.0 - (b / a)
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -1.2e+76)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(b * Float64(-2.0 / a));
		else
			tmp_2 = Float64(c / Float64(0.0 - b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 4.5e+139)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(b + sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0))))) * Float64(-0.5 / a));
		else
			tmp_3 = Float64(c * Float64(2.0 / Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0)))) - b)));
		end
		tmp_1 = tmp_3;
	else
		tmp_1 = Float64(0.0 - Float64(b / a));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	tmp_2 = 0.0;
	if (b <= -1.2e+76)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = b * (-2.0 / a);
		else
			tmp_3 = c / (0.0 - b);
		end
		tmp_2 = tmp_3;
	elseif (b <= 4.5e+139)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (b + sqrt(((b * b) + (c * (a * -4.0))))) * (-0.5 / a);
		else
			tmp_4 = c * (2.0 / (sqrt(((b * b) + (a * (c * -4.0)))) - b));
		end
		tmp_2 = tmp_4;
	else
		tmp_2 = 0.0 - (b / a);
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.2e+76], If[GreaterEqual[b, 0.0], N[(b * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision], N[(c / N[(0.0 - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 4.5e+139], If[GreaterEqual[b, 0.0], N[(N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(2.0 / N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.2e76

   1. Initial program 61.1%

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

   3. Simplified 61.1%

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

   5. 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 61.1%

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

   8. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 100.0%

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

   10. Simplified 100.0%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + b}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
   11. Step-by-step derivation

       1. flip-+ N/A

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

       2. +-inverses N/A

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

       3. +-inverses N/A

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

       4. associate-/l/ N/A

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

       5. metadata-eval N/A

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

       6. metadata-eval N/A

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

       7. distribute-neg-frac N/A

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

       8. +-inverses N/A

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

       9. +-inverses N/A

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

       10. flip-+ N/A

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

       11. neg-mul-1 N/A

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

       12. distribute-lft-out N/A

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

       13. distribute-rgt-out N/A

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

       14. metadata-eval N/A

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

       15. *-lowering-*.f64 100.0%

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

   12. Applied egg-rr 100.0%

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

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

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

       4. /-lowering-/.f64 100.0%

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

   14. Applied egg-rr 100.0%

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

   if -1.2e76 < b < 4.4999999999999999e139

   1. Initial program 88.1%

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

   3. Simplified 88.1%

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

      1. associate-/l/ N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

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

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

      8. metadata-eval N/A

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

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

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

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

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

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

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

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

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

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

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

      14. *-lowering-*.f64 88.0%

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

   6. Applied egg-rr 88.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

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

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

      12. *-lowering-*.f64 87.9%

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

   8. Applied egg-rr 87.9%

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

   if 4.4999999999999999e139 < b

   1. Initial program 46.8%

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

   3. Simplified 46.8%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ } \end{array}} \]
   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 98.2%

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

      1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

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

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

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

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

      12. *-lowering-*.f64 98.2%

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

   9. Applied egg-rr 98.2%

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

   10. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(b, b\right), -2\right), a\right)\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\mathsf{/.f64}\left(1, \left(-1 \cdot \frac{a}{b}\right)\right)}\\ \end{array} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

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

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

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

       4. /-lowering-/.f64 98.2%

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

   12. Simplified 98.2%

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

   13. Taylor expanded in b around 0

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

       1. if-same N/A

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

       2. mul-1-neg N/A

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

       3. neg-sub0 N/A

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

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

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

       5. /-lowering-/.f64 98.2%

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

   15. Simplified 98.2%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+76}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;b \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{+139}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot 2}{0 - \left(b + b\right)}\\ \mathbf{if}\;b \leq 1.9 \cdot 10^{-82}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* c 2.0) (- 0.0 (+ b b)))))
   (if (<= b 1.9e-82)
     (if (>= b 0.0) (* (/ -0.5 a) (+ b (sqrt (* a (* c -4.0))))) t_0)
     (if (>= b 0.0) (- (/ c b) (/ b a)) t_0))))

C

double code(double a, double b, double c) {
	double t_0 = (c * 2.0) / (0.0 - (b + b));
	double tmp_1;
	if (b <= 1.9e-82) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-0.5 / a) * (b + sqrt((a * (c * -4.0))));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = (c * 2.0d0) / (0.0d0 - (b + b))
    if (b <= 1.9d-82) then
        if (b >= 0.0d0) then
            tmp_2 = ((-0.5d0) / a) * (b + sqrt((a * (c * (-4.0d0)))))
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (c / b) - (b / a)
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (c * 2.0) / (0.0 - (b + b));
	double tmp_1;
	if (b <= 1.9e-82) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-0.5 / a) * (b + Math.sqrt((a * (c * -4.0))));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (c * 2.0) / (0.0 - (b + b))
	tmp_1 = 0
	if b <= 1.9e-82:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (-0.5 / a) * (b + math.sqrt((a * (c * -4.0))))
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = (c / b) - (b / a)
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * 2.0) / Float64(0.0 - Float64(b + b)))
	tmp_1 = 0.0
	if (b <= 1.9e-82)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-0.5 / a) * Float64(b + sqrt(Float64(a * Float64(c * -4.0)))));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = (c * 2.0) / (0.0 - (b + b));
	tmp_2 = 0.0;
	if (b <= 1.9e-82)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (-0.5 / a) * (b + sqrt((a * (c * -4.0))));
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = (c / b) - (b / a);
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * 2.0), $MachinePrecision] / N[(0.0 - N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.9e-82], If[GreaterEqual[b, 0.0], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.9000000000000001e-82

   1. Initial program 76.2%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 67.9%

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

   7. Simplified 67.9%

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

   8. Taylor expanded in b around 0

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

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

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

      2. *-lowering-*.f64 67.4%

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

   10. Simplified 67.4%

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

       1. div-inv N/A

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

       2. associate-/l* N/A

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

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

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

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

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. *-commutative N/A

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

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

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

       9. *-commutative N/A

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

       10. associate-*r* N/A

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

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

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

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

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

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

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

       14. metadata-eval 67.4%

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

   12. Applied egg-rr 67.4%

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

   if 1.9000000000000001e-82 < b

   1. Initial program 68.5%

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

   3. Simplified 68.5%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 68.5%

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

   7. Simplified 68.5%

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

   8. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. /-lowering-/.f64 90.8%

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

   10. Simplified 90.8%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.9 \cdot 10^{-82}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{0 - \left(b + b\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{0 - \left(b + b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.6% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-251}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;b \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.1e-251)
   (if (>= b 0.0) (* b (/ -2.0 a)) (/ c (- 0.0 b)))
   (- 0.0 (/ b a))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= 2.1e-251) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b * (-2.0 / a);
		} else {
			tmp_2 = c / (0.0 - b);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = 0.0 - (b / a);
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    if (b <= 2.1d-251) then
        if (b >= 0.0d0) then
            tmp_2 = b * ((-2.0d0) / a)
        else
            tmp_2 = c / (0.0d0 - b)
        end if
        tmp_1 = tmp_2
    else
        tmp_1 = 0.0d0 - (b / a)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if (b <= 2.1e-251) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b * (-2.0 / a);
		} else {
			tmp_2 = c / (0.0 - b);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = 0.0 - (b / a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if b <= 2.1e-251:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = b * (-2.0 / a)
		else:
			tmp_2 = c / (0.0 - b)
		tmp_1 = tmp_2
	else:
		tmp_1 = 0.0 - (b / a)
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= 2.1e-251)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(b * Float64(-2.0 / a));
		else
			tmp_2 = Float64(c / Float64(0.0 - b));
		end
		tmp_1 = tmp_2;
	else
		tmp_1 = Float64(0.0 - Float64(b / a));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp_2 = 0.0;
	if (b <= 2.1e-251)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = b * (-2.0 / a);
		else
			tmp_3 = c / (0.0 - b);
		end
		tmp_2 = tmp_3;
	else
		tmp_2 = 0.0 - (b / a);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.1e-251], If[GreaterEqual[b, 0.0], N[(b * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision], N[(c / N[(0.0 - b), $MachinePrecision]), $MachinePrecision]], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.09999999999999982e-251

   1. Initial program 76.6%

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

   3. Simplified 76.6%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ } \end{array}} \]
   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 71.4%

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

   8. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 62.8%

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

   10. Simplified 62.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + b}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]
   11. Step-by-step derivation

       1. flip-+ N/A

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

       2. +-inverses N/A

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

       3. +-inverses N/A

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

       4. associate-/l/ N/A

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

       5. metadata-eval N/A

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

       6. metadata-eval N/A

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

       7. distribute-neg-frac N/A

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

       8. +-inverses N/A

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

       9. +-inverses N/A

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

       10. flip-+ N/A

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

       11. neg-mul-1 N/A

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

       12. distribute-lft-out N/A

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

       13. distribute-rgt-out N/A

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

       14. metadata-eval N/A

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

       15. *-lowering-*.f64 62.8%

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

   12. Applied egg-rr 62.8%

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

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

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

       4. /-lowering-/.f64 62.8%

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

   14. Applied egg-rr 62.8%

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

   if 2.09999999999999982e-251 < b

   1. Initial program 69.2%

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

   3. Simplified 69.2%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ } \end{array}} \]
   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 79.9%

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

      1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

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

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

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

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

      12. *-lowering-*.f64 79.9%

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

   9. Applied egg-rr 79.9%

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

   10. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(b, b\right), -2\right), a\right)\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\mathsf{/.f64}\left(1, \left(-1 \cdot \frac{a}{b}\right)\right)}\\ \end{array} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

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

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

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

       4. /-lowering-/.f64 79.9%

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

   12. Simplified 79.9%

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

   13. Taylor expanded in b around 0

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

       1. if-same N/A

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

       2. mul-1-neg N/A

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

       3. neg-sub0 N/A

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

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

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

       5. /-lowering-/.f64 79.9%

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

   15. Simplified 79.9%

       \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-251}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;b \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.6% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

3. Simplified 73.5%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ } \end{array}} \]
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 75.0%

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

8. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

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

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

   4. /-lowering-/.f64 70.0%

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

10. Simplified 70.0%

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

11. Taylor expanded in b around 0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot b\right)}, a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, b\right)\right)\\ \end{array} \]
12. Step-by-step derivation

    1. mul-1-neg N/A

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

    2. neg-sub0 N/A

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

    3. --lowering--.f64 70.0%

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

13. Simplified 70.0%

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

14. Final simplification 70.0%

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

Alternative 9: 35.7% accurate, 24.2× speedup

Math

\[\begin{array}{l} \\ 0 - \frac{b}{a} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return 0.0 - (b / a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0 - (b / a)
end function

Java

public static double code(double a, double b, double c) {
	return 0.0 - (b / a);
}

Python

def code(a, b, c):
	return 0.0 - (b / a)

Julia

function code(a, b, c)
	return Float64(0.0 - Float64(b / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 - (b / a);
end

Wolfram

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

Derivation

1. Initial program 73.5%

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

3. Simplified 73.5%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{-2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ } \end{array}} \]
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 75.0%

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

   1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

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

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

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

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

   12. *-lowering-*.f64 74.9%

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

9. Applied egg-rr 74.9%

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

10. Taylor expanded in b around inf

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(b, b\right), -2\right), a\right)\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\mathsf{/.f64}\left(1, \left(-1 \cdot \frac{a}{b}\right)\right)}\\ \end{array} \]
11. Step-by-step derivation

    1. mul-1-neg N/A

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

    2. neg-sub0 N/A

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

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

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

    4. /-lowering-/.f64 35.4%

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

12. Simplified 35.4%

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

13. Taylor expanded in b around 0

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

    1. if-same N/A

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

    2. mul-1-neg N/A

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

    3. neg-sub0 N/A

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

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

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

    5. /-lowering-/.f64 35.4%

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

15. Simplified 35.4%

    \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(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)))))))