jeff quadratic root 2

Percentage Accurate: 72.2% → 90.6%

Time: 17.2s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ \mathbf{if}\;b \leq -1.05 \cdot 10^{+151}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+81}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0))))))
   (if (<= b -1.05e+151)
     (- (/ b a))
     (if (<= b 1.6e+81)
       (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_0)) (/ (- t_0 b) (* a 2.0)))
       (if (>= b 0.0)
         (/ (* 2.0 c) (* 2.0 (- (* a (/ c b)) b)))
         (/ (- (sqrt (* (* a c) -4.0)) b) (* a 2.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double tmp;
	if (b <= -1.05e+151) {
		tmp = -(b / a);
	} else if (b <= 1.6e+81) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_1 = (t_0 - b) / (a * 2.0);
		}
		tmp = tmp_1;
	} else if (b >= 0.0) {
		tmp = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
	} else {
		tmp = (sqrt(((a * c) * -4.0)) - b) / (a * 2.0);
	}
	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
    real(8) :: tmp_1
    t_0 = sqrt(((b * b) - (c * (a * 4.0d0))))
    if (b <= (-1.05d+151)) then
        tmp = -(b / a)
    else if (b <= 1.6d+81) then
        if (b >= 0.0d0) then
            tmp_1 = (2.0d0 * c) / (-b - t_0)
        else
            tmp_1 = (t_0 - b) / (a * 2.0d0)
        end if
        tmp = tmp_1
    else if (b >= 0.0d0) then
        tmp = (2.0d0 * c) / (2.0d0 * ((a * (c / b)) - b))
    else
        tmp = (sqrt(((a * c) * (-4.0d0))) - b) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (c * (a * 4.0))));
	double tmp;
	if (b <= -1.05e+151) {
		tmp = -(b / a);
	} else if (b <= 1.6e+81) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_1 = (t_0 - b) / (a * 2.0);
		}
		tmp = tmp_1;
	} else if (b >= 0.0) {
		tmp = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
	} else {
		tmp = (Math.sqrt(((a * c) * -4.0)) - b) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (c * (a * 4.0))))
	tmp = 0
	if b <= -1.05e+151:
		tmp = -(b / a)
	elif b <= 1.6e+81:
		tmp_1 = 0
		if b >= 0.0:
			tmp_1 = (2.0 * c) / (-b - t_0)
		else:
			tmp_1 = (t_0 - b) / (a * 2.0)
		tmp = tmp_1
	elif b >= 0.0:
		tmp = (2.0 * c) / (2.0 * ((a * (c / b)) - b))
	else:
		tmp = (math.sqrt(((a * c) * -4.0)) - b) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	tmp = 0.0
	if (b <= -1.05e+151)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 1.6e+81)
		tmp_1 = 0.0
		if (b >= 0.0)
			tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0));
		else
			tmp_1 = Float64(Float64(t_0 - b) / Float64(a * 2.0));
		end
		tmp = tmp_1;
	elseif (b >= 0.0)
		tmp = Float64(Float64(2.0 * c) / Float64(2.0 * Float64(Float64(a * Float64(c / b)) - b)));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(a * c) * -4.0)) - b) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	tmp = 0.0;
	if (b <= -1.05e+151)
		tmp = -(b / a);
	elseif (b <= 1.6e+81)
		tmp_2 = 0.0;
		if (b >= 0.0)
			tmp_2 = (2.0 * c) / (-b - t_0);
		else
			tmp_2 = (t_0 - b) / (a * 2.0);
		end
		tmp = tmp_2;
	elseif (b >= 0.0)
		tmp = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
	else
		tmp = (sqrt(((a * c) * -4.0)) - b) / (a * 2.0);
	end
	tmp_3 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.05e+151], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 1.6e+81], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.05e151

   1. Initial program 33.7%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around 0

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

       1. if-same N/A

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

       2. mul-1-neg N/A

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

       3. distribute-neg-frac2 N/A

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

       4. mul-1-neg N/A

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

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

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

       6. mul-1-neg N/A

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

       7. neg-lowering-neg.f64 100.0

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

   11. Simplified 100.0%

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

   if -1.05e151 < b < 1.6e81

   1. Initial program 87.5%

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

   if 1.6e81 < b

   1. Initial program 44.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. *-lowering-*.f64 98.4

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

   5. Simplified 98.4%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

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

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

      3. *-lowering-*.f64 98.4

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

   8. Simplified 98.4%

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

   9. Taylor expanded in c around 0

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

       1. distribute-lft-out-- N/A

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

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

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

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

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

       4. associate-/l* N/A

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

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

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

       6. /-lowering-/.f64 98.4

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

   11. Simplified 98.4%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{+151}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+81}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left(a \cdot c\right) \cdot -4}\\ t_1 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a \cdot 2}\\ \end{array}\\ \mathbf{if}\;b \leq -5 \cdot 10^{-76}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \mathsf{fma}\left(\frac{c}{b}, a \cdot -2, b\right)}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(b, -\frac{c}{b \cdot b}, \frac{b}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-92}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{a \cdot \left(-2\right)}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* (* a c) -4.0)))
        (t_1
         (if (>= b 0.0)
           (/ (* 2.0 c) (* 2.0 (- (* a (/ c b)) b)))
           (/ (- t_0 b) (* a 2.0)))))
   (if (<= b -5e-76)
     (if (>= b 0.0)
       (/ (* 2.0 c) (- (- b) (fma (/ c b) (* a -2.0) b)))
       (- (fma b (- (/ c (* b b))) (/ b a))))
     (if (<= b -5e-310)
       t_1
       (if (<= b 9.5e-92)
         (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_0)) (/ (+ b b) (* a (- 2.0))))
         t_1)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((a * c) * -4.0));
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
	} else {
		tmp = (t_0 - b) / (a * 2.0);
	}
	double t_1 = tmp;
	double tmp_2;
	if (b <= -5e-76) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * c) / (-b - fma((c / b), (a * -2.0), b));
		} else {
			tmp_3 = -fma(b, -(c / (b * b)), (b / a));
		}
		tmp_2 = tmp_3;
	} else if (b <= -5e-310) {
		tmp_2 = t_1;
	} else if (b <= 9.5e-92) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_4 = (b + b) / (a * -2.0);
		}
		tmp_2 = tmp_4;
	} else {
		tmp_2 = t_1;
	}
	return tmp_2;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(a * c) * -4.0))
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(2.0 * c) / Float64(2.0 * Float64(Float64(a * Float64(c / b)) - b)));
	else
		tmp = Float64(Float64(t_0 - b) / Float64(a * 2.0));
	end
	t_1 = tmp
	tmp_2 = 0.0
	if (b <= -5e-76)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - fma(Float64(c / b), Float64(a * -2.0), b)));
		else
			tmp_3 = Float64(-fma(b, Float64(-Float64(c / Float64(b * b))), Float64(b / a)));
		end
		tmp_2 = tmp_3;
	elseif (b <= -5e-310)
		tmp_2 = t_1;
	elseif (b <= 9.5e-92)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0));
		else
			tmp_4 = Float64(Float64(b + b) / Float64(a * Float64(-2.0)));
		end
		tmp_2 = tmp_4;
	else
		tmp_2 = t_1;
	end
	return tmp_2
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]}, If[LessEqual[b, -5e-76], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[(N[(c / b), $MachinePrecision] * N[(a * -2.0), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(b * (-N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]) + N[(b / a), $MachinePrecision]), $MachinePrecision])], If[LessEqual[b, -5e-310], t$95$1, If[LessEqual[b, 9.5e-92], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(b + b), $MachinePrecision] / N[(a * (-2.0)), $MachinePrecision]), $MachinePrecision]], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.9999999999999998e-76

   1. Initial program 62.3%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. *-lowering-*.f64 62.3

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

   5. Simplified 62.3%

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

      1. clear-num N/A

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. unsub-neg N/A

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

   7. Applied egg-rr 62.3%

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

   8. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. distribute-lft-in N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. mul-1-neg N/A

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

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

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

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

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

      10. unpow2 N/A

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

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

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

      12. /-lowering-/.f64 90.6

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

   10. Simplified 90.6%

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

   if -4.9999999999999998e-76 < b < -4.999999999999985e-310 or 9.49999999999999946e-92 < b

   1. Initial program 64.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. *-lowering-*.f64 81.5

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

   5. Simplified 81.5%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

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

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

      3. *-lowering-*.f64 78.3

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

   8. Simplified 78.3%

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

   9. Taylor expanded in c around 0

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

       1. distribute-lft-out-- N/A

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

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

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

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

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

       4. associate-/l* N/A

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

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

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

       6. /-lowering-/.f64 78.3

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

   11. Simplified 78.3%

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

   if -4.999999999999985e-310 < b < 9.49999999999999946e-92

   1. Initial program 90.2%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 90.2

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

   5. Simplified 90.2%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

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

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

      3. *-lowering-*.f64 85.5

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

   8. Simplified 85.5%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-76}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \mathsf{fma}\left(\frac{c}{b}, a \cdot -2, b\right)}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(b, -\frac{c}{b \cdot b}, \frac{b}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-92}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\left(a \cdot c\right) \cdot -4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{a \cdot \left(-2\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left(a \cdot c\right) \cdot -4}\\ t_1 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a \cdot 2}\\ \end{array}\\ \mathbf{if}\;b \leq -4.8 \cdot 10^{-76}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-79}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{a \cdot \left(-2\right)}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* (* a c) -4.0)))
        (t_1
         (if (>= b 0.0)
           (/ (* 2.0 c) (* 2.0 (- (* a (/ c b)) b)))
           (/ (- t_0 b) (* a 2.0)))))
   (if (<= b -4.8e-76)
     (- (/ b a))
     (if (<= b -5e-310)
       t_1
       (if (<= b 3.5e-79)
         (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_0)) (/ (+ b b) (* a (- 2.0))))
         t_1)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((a * c) * -4.0));
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
	} else {
		tmp = (t_0 - b) / (a * 2.0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b <= -4.8e-76) {
		tmp_1 = -(b / a);
	} else if (b <= -5e-310) {
		tmp_1 = t_1;
	} else if (b <= 3.5e-79) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_2 = (b + b) / (a * -2.0);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = sqrt(((a * c) * (-4.0d0)))
    if (b >= 0.0d0) then
        tmp = (2.0d0 * c) / (2.0d0 * ((a * (c / b)) - b))
    else
        tmp = (t_0 - b) / (a * 2.0d0)
    end if
    t_1 = tmp
    if (b <= (-4.8d-76)) then
        tmp_1 = -(b / a)
    else if (b <= (-5d-310)) then
        tmp_1 = t_1
    else if (b <= 3.5d-79) then
        if (b >= 0.0d0) then
            tmp_2 = (2.0d0 * c) / (-b - t_0)
        else
            tmp_2 = (b + b) / (a * -2.0d0)
        end if
        tmp_1 = tmp_2
    else
        tmp_1 = t_1
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((a * c) * -4.0));
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
	} else {
		tmp = (t_0 - b) / (a * 2.0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b <= -4.8e-76) {
		tmp_1 = -(b / a);
	} else if (b <= -5e-310) {
		tmp_1 = t_1;
	} else if (b <= 3.5e-79) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_2 = (b + b) / (a * -2.0);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((a * c) * -4.0))
	tmp = 0
	if b >= 0.0:
		tmp = (2.0 * c) / (2.0 * ((a * (c / b)) - b))
	else:
		tmp = (t_0 - b) / (a * 2.0)
	t_1 = tmp
	tmp_1 = 0
	if b <= -4.8e-76:
		tmp_1 = -(b / a)
	elif b <= -5e-310:
		tmp_1 = t_1
	elif b <= 3.5e-79:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (2.0 * c) / (-b - t_0)
		else:
			tmp_2 = (b + b) / (a * -2.0)
		tmp_1 = tmp_2
	else:
		tmp_1 = t_1
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(a * c) * -4.0))
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(2.0 * c) / Float64(2.0 * Float64(Float64(a * Float64(c / b)) - b)));
	else
		tmp = Float64(Float64(t_0 - b) / Float64(a * 2.0));
	end
	t_1 = tmp
	tmp_1 = 0.0
	if (b <= -4.8e-76)
		tmp_1 = Float64(-Float64(b / a));
	elseif (b <= -5e-310)
		tmp_1 = t_1;
	elseif (b <= 3.5e-79)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0));
		else
			tmp_2 = Float64(Float64(b + b) / Float64(a * Float64(-2.0)));
		end
		tmp_1 = tmp_2;
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = sqrt(((a * c) * -4.0));
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
	else
		tmp = (t_0 - b) / (a * 2.0);
	end
	t_1 = tmp;
	tmp_2 = 0.0;
	if (b <= -4.8e-76)
		tmp_2 = -(b / a);
	elseif (b <= -5e-310)
		tmp_2 = t_1;
	elseif (b <= 3.5e-79)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (2.0 * c) / (-b - t_0);
		else
			tmp_3 = (b + b) / (a * -2.0);
		end
		tmp_2 = tmp_3;
	else
		tmp_2 = t_1;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]}, If[LessEqual[b, -4.8e-76], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, -5e-310], t$95$1, If[LessEqual[b, 3.5e-79], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(b + b), $MachinePrecision] / N[(a * (-2.0)), $MachinePrecision]), $MachinePrecision]], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.80000000000000026e-76

   1. Initial program 62.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 90.3

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

   5. Simplified 90.3%

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

   6. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 90.3

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

   8. Simplified 90.3%

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

   9. Taylor expanded in b around 0

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

       1. if-same N/A

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

       2. mul-1-neg N/A

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

       3. distribute-neg-frac2 N/A

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

       4. mul-1-neg N/A

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

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

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

       6. mul-1-neg N/A

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

       7. neg-lowering-neg.f64 90.3

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

   11. Simplified 90.3%

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

   if -4.80000000000000026e-76 < b < -4.999999999999985e-310 or 3.5000000000000003e-79 < b

   1. Initial program 64.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. *-lowering-*.f64 81.5

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

   5. Simplified 81.5%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

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

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

      3. *-lowering-*.f64 78.3

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

   8. Simplified 78.3%

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

   9. Taylor expanded in c around 0

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

       1. distribute-lft-out-- N/A

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

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

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

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

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

       4. associate-/l* N/A

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

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

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

       6. /-lowering-/.f64 78.3

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

   11. Simplified 78.3%

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

   if -4.999999999999985e-310 < b < 3.5000000000000003e-79

   1. Initial program 90.2%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 90.2

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

   5. Simplified 90.2%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

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

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

      3. *-lowering-*.f64 85.5

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

   8. Simplified 85.5%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{-76}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-79}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\left(a \cdot c\right) \cdot -4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{a \cdot \left(-2\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{if}\;b \leq -5 \cdot 10^{-76}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-77}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{a \cdot \left(-2\right)}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0
         (if (>= b 0.0)
           (/ (* 2.0 c) (* 2.0 (- (* a (/ c b)) b)))
           (/ (- (sqrt (* (* a c) -4.0)) b) (* a 2.0)))))
   (if (<= b -5e-76)
     (- (/ b a))
     (if (<= b -5e-310)
       t_0
       (if (<= b 6e-77)
         (if (>= b 0.0)
           (* c (/ 2.0 (- (- b) (sqrt (* c (* a -4.0))))))
           (/ (+ b b) (* a (- 2.0))))
         t_0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
	} else {
		tmp = (sqrt(((a * c) * -4.0)) - b) / (a * 2.0);
	}
	double t_0 = tmp;
	double tmp_1;
	if (b <= -5e-76) {
		tmp_1 = -(b / a);
	} else if (b <= -5e-310) {
		tmp_1 = t_0;
	} else if (b <= 6e-77) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (2.0 / (-b - sqrt((c * (a * -4.0)))));
		} else {
			tmp_2 = (b + b) / (a * -2.0);
		}
		tmp_1 = tmp_2;
	} 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
    if (b >= 0.0d0) then
        tmp = (2.0d0 * c) / (2.0d0 * ((a * (c / b)) - b))
    else
        tmp = (sqrt(((a * c) * (-4.0d0))) - b) / (a * 2.0d0)
    end if
    t_0 = tmp
    if (b <= (-5d-76)) then
        tmp_1 = -(b / a)
    else if (b <= (-5d-310)) then
        tmp_1 = t_0
    else if (b <= 6d-77) then
        if (b >= 0.0d0) then
            tmp_2 = c * (2.0d0 / (-b - sqrt((c * (a * (-4.0d0))))))
        else
            tmp_2 = (b + b) / (a * -2.0d0)
        end if
        tmp_1 = tmp_2
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
	} else {
		tmp = (Math.sqrt(((a * c) * -4.0)) - b) / (a * 2.0);
	}
	double t_0 = tmp;
	double tmp_1;
	if (b <= -5e-76) {
		tmp_1 = -(b / a);
	} else if (b <= -5e-310) {
		tmp_1 = t_0;
	} else if (b <= 6e-77) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (2.0 / (-b - Math.sqrt((c * (a * -4.0)))));
		} else {
			tmp_2 = (b + b) / (a * -2.0);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = (2.0 * c) / (2.0 * ((a * (c / b)) - b))
	else:
		tmp = (math.sqrt(((a * c) * -4.0)) - b) / (a * 2.0)
	t_0 = tmp
	tmp_1 = 0
	if b <= -5e-76:
		tmp_1 = -(b / a)
	elif b <= -5e-310:
		tmp_1 = t_0
	elif b <= 6e-77:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = c * (2.0 / (-b - math.sqrt((c * (a * -4.0)))))
		else:
			tmp_2 = (b + b) / (a * -2.0)
		tmp_1 = tmp_2
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(2.0 * c) / Float64(2.0 * Float64(Float64(a * Float64(c / b)) - b)));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(a * c) * -4.0)) - b) / Float64(a * 2.0));
	end
	t_0 = tmp
	tmp_1 = 0.0
	if (b <= -5e-76)
		tmp_1 = Float64(-Float64(b / a));
	elseif (b <= -5e-310)
		tmp_1 = t_0;
	elseif (b <= 6e-77)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * Float64(2.0 / Float64(Float64(-b) - sqrt(Float64(c * Float64(a * -4.0))))));
		else
			tmp_2 = Float64(Float64(b + b) / Float64(a * Float64(-2.0)));
		end
		tmp_1 = tmp_2;
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
	else
		tmp = (sqrt(((a * c) * -4.0)) - b) / (a * 2.0);
	end
	t_0 = tmp;
	tmp_2 = 0.0;
	if (b <= -5e-76)
		tmp_2 = -(b / a);
	elseif (b <= -5e-310)
		tmp_2 = t_0;
	elseif (b <= 6e-77)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = c * (2.0 / (-b - sqrt((c * (a * -4.0)))));
		else
			tmp_3 = (b + b) / (a * -2.0);
		end
		tmp_2 = tmp_3;
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]}, If[LessEqual[b, -5e-76], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, -5e-310], t$95$0, If[LessEqual[b, 6e-77], If[GreaterEqual[b, 0.0], N[(c * N[(2.0 / N[((-b) - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + b), $MachinePrecision] / N[(a * (-2.0)), $MachinePrecision]), $MachinePrecision]], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.9999999999999998e-76

   1. Initial program 62.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 90.3

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

   5. Simplified 90.3%

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

   6. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 90.3

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

   8. Simplified 90.3%

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

   9. Taylor expanded in b around 0

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

       1. if-same N/A

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

       2. mul-1-neg N/A

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

       3. distribute-neg-frac2 N/A

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

       4. mul-1-neg N/A

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

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

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

       6. mul-1-neg N/A

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

       7. neg-lowering-neg.f64 90.3

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

   11. Simplified 90.3%

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

   if -4.9999999999999998e-76 < b < -4.999999999999985e-310 or 6.00000000000000033e-77 < b

   1. Initial program 64.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. *-lowering-*.f64 81.5

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

   5. Simplified 81.5%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

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

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

      3. *-lowering-*.f64 78.3

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

   8. Simplified 78.3%

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

   9. Taylor expanded in c around 0

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

       1. distribute-lft-out-- N/A

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

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

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

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

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

       4. associate-/l* N/A

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

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

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

       6. /-lowering-/.f64 78.3

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

   11. Simplified 78.3%

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

   if -4.999999999999985e-310 < b < 6.00000000000000033e-77

   1. Initial program 90.2%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 90.2

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

   5. Simplified 90.2%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

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

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

      3. *-lowering-*.f64 85.5

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

   8. Simplified 85.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 85.4

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

   10. Applied egg-rr 85.4%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-76}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-77}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{a \cdot \left(-2\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+154}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+80}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+154)
   (- (/ b a))
   (if (<= b 1.05e+80)
     (if (>= b 0.0)
       (* c (/ -2.0 (+ b (sqrt (fma c (* a -4.0) (* b b))))))
       (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)))
     (if (>= b 0.0)
       (/ (* 2.0 c) (* 2.0 (- (* a (/ c b)) b)))
       (/ (- (sqrt (* (* a c) -4.0)) b) (* a 2.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+154) {
		tmp = -(b / a);
	} else if (b <= 1.05e+80) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = c * (-2.0 / (b + sqrt(fma(c, (a * -4.0), (b * b)))));
		} else {
			tmp_1 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
		}
		tmp = tmp_1;
	} else if (b >= 0.0) {
		tmp = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
	} else {
		tmp = (sqrt(((a * c) * -4.0)) - b) / (a * 2.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e+154)
		tmp = Float64(-Float64(b / a));
	elseif (b <= 1.05e+80)
		tmp_1 = 0.0
		if (b >= 0.0)
			tmp_1 = Float64(c * Float64(-2.0 / Float64(b + sqrt(fma(c, Float64(a * -4.0), Float64(b * b))))));
		else
			tmp_1 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
		end
		tmp = tmp_1;
	elseif (b >= 0.0)
		tmp = Float64(Float64(2.0 * c) / Float64(2.0 * Float64(Float64(a * Float64(c / b)) - b)));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(a * c) * -4.0)) - b) / Float64(a * 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+154], (-N[(b / a), $MachinePrecision]), If[LessEqual[b, 1.05e+80], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.00000000000000004e154

   1. Initial program 33.7%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around 0

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

       1. if-same N/A

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

       2. mul-1-neg N/A

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

       3. distribute-neg-frac2 N/A

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

       4. mul-1-neg N/A

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

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

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

       6. mul-1-neg N/A

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

       7. neg-lowering-neg.f64 100.0

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

   11. Simplified 100.0%

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

   if -1.00000000000000004e154 < b < 1.05000000000000001e80

   1. Initial program 87.5%

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

   4. Applied egg-rr 87.5%

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

   if 1.05000000000000001e80 < b

   1. Initial program 44.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. *-lowering-*.f64 98.4

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

   5. Simplified 98.4%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

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

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

      3. *-lowering-*.f64 98.4

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

   8. Simplified 98.4%

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

   9. Taylor expanded in c around 0

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

       1. distribute-lft-out-- N/A

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

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

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

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

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

       4. associate-/l* N/A

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

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

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

       6. /-lowering-/.f64 98.4

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

   11. Simplified 98.4%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+154}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+80}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+65}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \mathsf{fma}\left(\frac{c}{b}, a \cdot -2, b\right)}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(b, -\frac{c}{b \cdot b}, \frac{b}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{+81}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e+65)
   (if (>= b 0.0)
     (/ (* 2.0 c) (- (- b) (fma (/ c b) (* a -2.0) b)))
     (- (fma b (- (/ c (* b b))) (/ b a))))
   (if (<= b 1.55e+81)
     (if (>= b 0.0)
       (* c (/ -2.0 (+ b (sqrt (fma c (* a -4.0) (* b b))))))
       (* (/ 0.5 a) (- (sqrt (fma a (* c -4.0) (* b b))) b)))
     (if (>= b 0.0)
       (/ (* 2.0 c) (* 2.0 (- (* a (/ c b)) b)))
       (/ (- (sqrt (* (* a c) -4.0)) b) (* a 2.0))))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -4.5e+65) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (-b - fma((c / b), (a * -2.0), b));
		} else {
			tmp_2 = -fma(b, -(c / (b * b)), (b / a));
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.55e+81) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = c * (-2.0 / (b + sqrt(fma(c, (a * -4.0), (b * b)))));
		} else {
			tmp_3 = (0.5 / a) * (sqrt(fma(a, (c * -4.0), (b * b))) - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (2.0 * ((a * (c / b)) - b));
	} else {
		tmp_1 = (sqrt(((a * c) * -4.0)) - b) / (a * 2.0);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -4.5e+65)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - fma(Float64(c / b), Float64(a * -2.0), b)));
		else
			tmp_2 = Float64(-fma(b, Float64(-Float64(c / Float64(b * b))), Float64(b / a)));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.55e+81)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(c * Float64(-2.0 / Float64(b + sqrt(fma(c, Float64(a * -4.0), Float64(b * b))))));
		else
			tmp_3 = Float64(Float64(0.5 / a) * Float64(sqrt(fma(a, Float64(c * -4.0), Float64(b * b))) - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(2.0 * Float64(Float64(a * Float64(c / b)) - b)));
	else
		tmp_1 = Float64(Float64(sqrt(Float64(Float64(a * c) * -4.0)) - b) / Float64(a * 2.0));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e+65], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[(N[(c / b), $MachinePrecision] * N[(a * -2.0), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(b * (-N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]) + N[(b / a), $MachinePrecision]), $MachinePrecision])], If[LessEqual[b, 1.55e+81], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.5e65

   1. Initial program 52.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. *-lowering-*.f64 52.2

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

   5. Simplified 52.2%

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

      1. clear-num N/A

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. unsub-neg N/A

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

   7. Applied egg-rr 52.2%

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

   8. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. distribute-lft-in N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. mul-1-neg N/A

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

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

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

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

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

      10. unpow2 N/A

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

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

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

      12. /-lowering-/.f64 100.0

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

   10. Simplified 100.0%

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

   if -4.5e65 < b < 1.55e81

   1. Initial program 85.7%

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

   4. Applied egg-rr 85.6%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

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

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

      7. +-commutative N/A

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

      8. unsub-neg N/A

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

   6. Applied egg-rr 85.5%

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

   if 1.55e81 < b

   1. Initial program 44.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. *-lowering-*.f64 98.4

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

   5. Simplified 98.4%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

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

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

      3. *-lowering-*.f64 98.4

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

   8. Simplified 98.4%

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

   9. Taylor expanded in c around 0

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

       1. distribute-lft-out-- N/A

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

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

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

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

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

       4. associate-/l* N/A

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

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

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

       6. /-lowering-/.f64 98.4

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

   11. Simplified 98.4%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+65}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \mathsf{fma}\left(\frac{c}{b}, a \cdot -2, b\right)}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(b, -\frac{c}{b \cdot b}, \frac{b}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{+81}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b + b}{a \cdot \left(-2\right)}\\ \mathbf{if}\;b \leq 4.4 \cdot 10^{-78}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (+ b b) (* a (- 2.0)))))
   (if (<= b 4.4e-78)
     (if (>= b 0.0) (* c (/ 2.0 (- (- b) (sqrt (* c (* a -4.0)))))) t_0)
     (if (>= b 0.0) (/ (* 2.0 c) (- (- b) b)) t_0))))

C

double code(double a, double b, double c) {
	double t_0 = (b + b) / (a * -2.0);
	double tmp_1;
	if (b <= 4.4e-78) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (2.0 / (-b - sqrt((c * (a * -4.0)))));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-b - b);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = (b + b) / (a * -2.0d0)
    if (b <= 4.4d-78) then
        if (b >= 0.0d0) then
            tmp_2 = c * (2.0d0 / (-b - sqrt((c * (a * (-4.0d0))))))
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (2.0d0 * c) / (-b - b)
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (b + b) / (a * -2.0);
	double tmp_1;
	if (b <= 4.4e-78) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (2.0 / (-b - Math.sqrt((c * (a * -4.0)))));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-b - b);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (b + b) / (a * -2.0)
	tmp_1 = 0
	if b <= 4.4e-78:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = c * (2.0 / (-b - math.sqrt((c * (a * -4.0)))))
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = (2.0 * c) / (-b - b)
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b + b) / Float64(a * Float64(-2.0)))
	tmp_1 = 0.0
	if (b <= 4.4e-78)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * Float64(2.0 / Float64(Float64(-b) - sqrt(Float64(c * Float64(a * -4.0))))));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - b));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = (b + b) / (a * -2.0);
	tmp_2 = 0.0;
	if (b <= 4.4e-78)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = c * (2.0 / (-b - sqrt((c * (a * -4.0)))));
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = (2.0 * c) / (-b - b);
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b + b), $MachinePrecision] / N[(a * (-2.0)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 4.4e-78], If[GreaterEqual[b, 0.0], N[(c * N[(2.0 / N[((-b) - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - b), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.3999999999999998e-78

   1. Initial program 70.9%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 76.9

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

   5. Simplified 76.9%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

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

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

      3. *-lowering-*.f64 75.8

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

   8. Simplified 75.8%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 75.8

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

   10. Applied egg-rr 75.8%

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

   if 4.3999999999999998e-78 < b

   1. Initial program 61.8%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 61.8

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

   5. Simplified 61.8%

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

   6. Taylor expanded in b around inf

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

   8. Simplified 84.1%

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

      1. sub-neg N/A

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

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

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

      3. neg-lowering-neg.f64 84.1

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

   10. Applied egg-rr 84.1%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{-78}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{a \cdot \left(-2\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{a \cdot \left(-2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-262}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(c \cdot -2\right) \cdot \left(b + b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{a \cdot \left(-2\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b + b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.15e-262)
   (if (>= b 0.0) (* (* c -2.0) (+ b b)) (/ (+ b b) (* a (- 2.0))))
   (if (>= b 0.0) (/ (- c) (+ b b)) (- (/ (+ b b) a)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= 1.15e-262) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c * -2.0) * (b + b);
		} else {
			tmp_2 = (b + b) / (a * -2.0);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = -c / (b + b);
	} else {
		tmp_1 = -((b + 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 <= 1.15d-262) then
        if (b >= 0.0d0) then
            tmp_2 = (c * (-2.0d0)) * (b + b)
        else
            tmp_2 = (b + b) / (a * -2.0d0)
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = -c / (b + b)
    else
        tmp_1 = -((b + 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 <= 1.15e-262) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c * -2.0) * (b + b);
		} else {
			tmp_2 = (b + b) / (a * -2.0);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = -c / (b + b);
	} else {
		tmp_1 = -((b + b) / a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if b <= 1.15e-262:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (c * -2.0) * (b + b)
		else:
			tmp_2 = (b + b) / (a * -2.0)
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = -c / (b + b)
	else:
		tmp_1 = -((b + b) / a)
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= 1.15e-262)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(c * -2.0) * Float64(b + b));
		else
			tmp_2 = Float64(Float64(b + b) / Float64(a * Float64(-2.0)));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-c) / Float64(b + b));
	else
		tmp_1 = Float64(-Float64(Float64(b + b) / a));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp_2 = 0.0;
	if (b <= 1.15e-262)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (c * -2.0) * (b + b);
		else
			tmp_3 = (b + b) / (a * -2.0);
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = -c / (b + b);
	else
		tmp_2 = -((b + b) / a);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.15e-262], If[GreaterEqual[b, 0.0], N[(N[(c * -2.0), $MachinePrecision] * N[(b + b), $MachinePrecision]), $MachinePrecision], N[(N[(b + b), $MachinePrecision] / N[(a * (-2.0)), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[((-c) / N[(b + b), $MachinePrecision]), $MachinePrecision], (-N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if b < 1.15000000000000005e-262

   1. Initial program 66.5%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 73.8

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

   5. Simplified 73.8%

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

   6. Taylor expanded in b around inf

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

   8. Simplified 68.9%

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

      1. sub-neg N/A

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

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

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

      3. neg-lowering-neg.f64 68.9

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

   10. Applied egg-rr 68.9%

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

       1. frac-2neg N/A

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

       2. div-inv N/A

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

       3. sub-neg N/A

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

       4. distribute-neg-out N/A

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

       5. remove-double-neg N/A

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

       6. flip-+ N/A

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

       7. +-inverses N/A

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

       8. +-inverses N/A

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

       9. +-inverses N/A

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

       10. +-inverses N/A

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

       11. clear-num N/A

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

       12. flip-+ N/A

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

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

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

       14. *-commutative N/A

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

       15. distribute-rgt-neg-in N/A

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

       16. metadata-eval N/A

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

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

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

       18. +-lowering-+.f64 69.0

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

   12. Applied egg-rr 69.0%

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

   if 1.15000000000000005e-262 < b

   1. Initial program 69.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 69.1

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

   5. Simplified 69.1%

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

   6. Taylor expanded in b around inf

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

   8. Simplified 67.0%

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

      1. clear-num N/A

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

      2. associate-/r* N/A

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

      3. sub-neg N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. associate-/l/ N/A

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

      7. metadata-eval N/A

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

      8. +-inverses N/A

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

      9. flip-+ N/A

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

      10. sub-neg N/A

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

      11. clear-num N/A

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

      12. frac-2neg N/A

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

      13. sub-neg N/A

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

      14. flip-+ N/A

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

      15. +-inverses N/A

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

      16. distribute-neg-frac2 N/A

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

   10. Applied egg-rr 23.3%

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

       1. sub-neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \end{array} \]

       2. associate-/r* N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \end{array} \]

       3. frac-2neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \end{array} \]

       4. sub-neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \end{array} \]

       5. flip-+ N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \end{array} \]

       6. +-inverses N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \end{array} \]

       7. associate-/l/ N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \end{array} \]

       8. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \end{array} \]

       9. +-inverses N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \end{array} \]

       10. flip-+ N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \end{array} \]

       11. distribute-neg-out N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \end{array} \]

       12. remove-double-neg N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \end{array} \]

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

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \end{array} \]

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

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \end{array} \]

       15. neg-lowering-neg.f64 23.3

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

   12. Applied egg-rr 23.3%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-262}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(c \cdot -2\right) \cdot \left(b + b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{a \cdot \left(-2\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b + b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{-229}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b + b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.05e-229)
   (- (/ b a))
   (if (>= b 0.0) (/ (- c) (+ b b)) (- (/ (+ b b) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.05e-229) {
		tmp = -(b / a);
	} else if (b >= 0.0) {
		tmp = -c / (b + b);
	} else {
		tmp = -((b + b) / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.05d-229)) then
        tmp = -(b / a)
    else if (b >= 0.0d0) then
        tmp = -c / (b + b)
    else
        tmp = -((b + b) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.05e-229) {
		tmp = -(b / a);
	} else if (b >= 0.0) {
		tmp = -c / (b + b);
	} else {
		tmp = -((b + b) / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.05e-229:
		tmp = -(b / a)
	elif b >= 0.0:
		tmp = -c / (b + b)
	else:
		tmp = -((b + b) / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.05e-229)
		tmp = Float64(-Float64(b / a));
	elseif (b >= 0.0)
		tmp = Float64(Float64(-c) / Float64(b + b));
	else
		tmp = Float64(-Float64(Float64(b + b) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.05e-229)
		tmp = -(b / a);
	elseif (b >= 0.0)
		tmp = -c / (b + b);
	else
		tmp = -((b + b) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.05e-229], (-N[(b / a), $MachinePrecision]), If[GreaterEqual[b, 0.0], N[((-c) / N[(b + b), $MachinePrecision]), $MachinePrecision], (-N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if b < -2.05e-229

   1. Initial program 64.9%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 78.2

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

   5. Simplified 78.2%

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

   6. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 78.2

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

   8. Simplified 78.2%

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

   9. Taylor expanded in b around 0

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

       1. if-same N/A

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

       2. mul-1-neg N/A

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

       3. distribute-neg-frac2 N/A

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

       4. mul-1-neg N/A

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

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

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

       6. mul-1-neg N/A

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

       7. neg-lowering-neg.f64 78.2

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

   11. Simplified 78.2%

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

   if -2.05e-229 < b

   1. Initial program 70.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 65.9

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

   5. Simplified 65.9%

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

   6. Taylor expanded in b around inf

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

   8. Simplified 59.2%

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

      1. clear-num N/A

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

      2. associate-/r* N/A

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

      3. sub-neg N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. associate-/l/ N/A

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

      7. metadata-eval N/A

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

      8. +-inverses N/A

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

      9. flip-+ N/A

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

      10. sub-neg N/A

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

      11. clear-num N/A

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

      12. frac-2neg N/A

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

      13. sub-neg N/A

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

      14. flip-+ N/A

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

      15. +-inverses N/A

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

      16. distribute-neg-frac2 N/A

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

   10. Applied egg-rr 20.9%

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

       1. sub-neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \end{array} \]

       2. associate-/r* N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \end{array} \]

       3. frac-2neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \end{array} \]

       4. sub-neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \end{array} \]

       5. flip-+ N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \end{array} \]

       6. +-inverses N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \end{array} \]

       7. associate-/l/ N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \end{array} \]

       8. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \end{array} \]

       9. +-inverses N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \end{array} \]

       10. flip-+ N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \end{array} \]

       11. distribute-neg-out N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \end{array} \]

       12. remove-double-neg N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \end{array} \]

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

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \end{array} \]

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

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{neg}\left(c\right)}{b + b}\\ \end{array} \]

       15. neg-lowering-neg.f64 20.9

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

   12. Applied egg-rr 20.9%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{-229}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b + b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (-b - b);
	} else {
		tmp = (b + b) / (a * -2.0);
	}
	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) :: tmp
    if (b >= 0.0d0) then
        tmp = (2.0d0 * c) / (-b - b)
    else
        tmp = (b + b) / (a * -2.0d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.7%

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

3. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 71.6

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

5. Simplified 71.6%

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

6. Taylor expanded in b around inf

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

8. Simplified 68.0%

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

   1. sub-neg N/A

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

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

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

   3. neg-lowering-neg.f64 68.0

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

10. Applied egg-rr 68.0%

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

11. Final simplification 68.0%

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

Alternative 11: 35.9% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.7%

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

3. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 71.6

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

5. Simplified 71.6%

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

6. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

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

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

   5. mul-1-neg N/A

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

   6. neg-lowering-neg.f64 38.0

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

8. Simplified 38.0%

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

9. Taylor expanded in b around 0

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

    1. if-same N/A

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

    2. mul-1-neg N/A

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

    3. distribute-neg-frac2 N/A

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

    4. mul-1-neg N/A

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

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

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

    6. mul-1-neg N/A

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

    7. neg-lowering-neg.f64 38.0

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

11. Simplified 38.0%

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

12. Final simplification 38.0%

    \[\leadsto -\frac{b}{a} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))