jeff quadratic root 2

Percentage Accurate: 72.3% → 88.5%

Time: 33.4s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.3% 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: 88.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ t_1 := \frac{b \cdot -2}{a \cdot 2}\\ \mathbf{if}\;b \leq -8.6 \cdot 10^{+159}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\\ \mathbf{elif}\;b \leq -7 \cdot 10^{-288}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{+42}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{c}{b} - b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0)))))
        (t_1 (/ (* b -2.0) (* a 2.0))))
   (if (<= b -8.6e+159)
     (if (>= b 0.0) (/ b a) t_1)
     (if (<= b -7e-288)
       (if (>= b 0.0) (/ b a) (/ (- t_0 b) (* a 2.0)))
       (if (<= b 1e+42)
         (if (>= b 0.0) (/ (* c 2.0) (- (- b) t_0)) t_1)
         (if (>= b 0.0) (/ c (- (* a (/ c b)) b)) t_1))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double t_1 = (b * -2.0) / (a * 2.0);
	double tmp_1;
	if (b <= -8.6e+159) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b / a;
		} else {
			tmp_2 = t_1;
		}
		tmp_1 = tmp_2;
	} else if (b <= -7e-288) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = b / a;
		} else {
			tmp_3 = (t_0 - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1e+42) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c * 2.0) / (-b - t_0);
		} else {
			tmp_4 = t_1;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = c / ((a * (c / b)) - b);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    real(8) :: tmp_4
    t_0 = sqrt(((b * b) - (c * (a * 4.0d0))))
    t_1 = (b * (-2.0d0)) / (a * 2.0d0)
    if (b <= (-8.6d+159)) then
        if (b >= 0.0d0) then
            tmp_2 = b / a
        else
            tmp_2 = t_1
        end if
        tmp_1 = tmp_2
    else if (b <= (-7d-288)) then
        if (b >= 0.0d0) then
            tmp_3 = b / a
        else
            tmp_3 = (t_0 - b) / (a * 2.0d0)
        end if
        tmp_1 = tmp_3
    else if (b <= 1d+42) then
        if (b >= 0.0d0) then
            tmp_4 = (c * 2.0d0) / (-b - t_0)
        else
            tmp_4 = t_1
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = c / ((a * (c / b)) - b)
    else
        tmp_1 = t_1
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (c * (a * 4.0))));
	double t_1 = (b * -2.0) / (a * 2.0);
	double tmp_1;
	if (b <= -8.6e+159) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b / a;
		} else {
			tmp_2 = t_1;
		}
		tmp_1 = tmp_2;
	} else if (b <= -7e-288) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = b / a;
		} else {
			tmp_3 = (t_0 - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1e+42) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c * 2.0) / (-b - t_0);
		} else {
			tmp_4 = t_1;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = c / ((a * (c / b)) - b);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (c * (a * 4.0))))
	t_1 = (b * -2.0) / (a * 2.0)
	tmp_1 = 0
	if b <= -8.6e+159:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = b / a
		else:
			tmp_2 = t_1
		tmp_1 = tmp_2
	elif b <= -7e-288:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = b / a
		else:
			tmp_3 = (t_0 - b) / (a * 2.0)
		tmp_1 = tmp_3
	elif b <= 1e+42:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = (c * 2.0) / (-b - t_0)
		else:
			tmp_4 = t_1
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = c / ((a * (c / b)) - b)
	else:
		tmp_1 = t_1
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	t_1 = Float64(Float64(b * -2.0) / Float64(a * 2.0))
	tmp_1 = 0.0
	if (b <= -8.6e+159)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(b / a);
		else
			tmp_2 = t_1;
		end
		tmp_1 = tmp_2;
	elseif (b <= -7e-288)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(b / a);
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(a * 2.0));
		end
		tmp_1 = tmp_3;
	elseif (b <= 1e+42)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(c * 2.0) / Float64(Float64(-b) - t_0));
		else
			tmp_4 = t_1;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(c / Float64(Float64(a * Float64(c / b)) - b));
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	t_1 = (b * -2.0) / (a * 2.0);
	tmp_2 = 0.0;
	if (b <= -8.6e+159)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = b / a;
		else
			tmp_3 = t_1;
		end
		tmp_2 = tmp_3;
	elseif (b <= -7e-288)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = b / a;
		else
			tmp_4 = (t_0 - b) / (a * 2.0);
		end
		tmp_2 = tmp_4;
	elseif (b <= 1e+42)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = (c * 2.0) / (-b - t_0);
		else
			tmp_5 = t_1;
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = c / ((a * (c / b)) - b);
	else
		tmp_2 = t_1;
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * -2.0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -8.6e+159], If[GreaterEqual[b, 0.0], N[(b / a), $MachinePrecision], t$95$1], If[LessEqual[b, -7e-288], If[GreaterEqual[b, 0.0], N[(b / a), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1e+42], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1], If[GreaterEqual[b, 0.0], N[(c / N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -8.6000000000000004e159

   1. Initial program 44.9%

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

   3. Taylor expanded in b around -inf 98.0%

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

      1. *-commutative 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in a around 0 98.0%

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

      1. distribute-lft-out-- 98.0%

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

      2. associate-/l* 98.0%

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

   8. Simplified 98.0%

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

   9. Taylor expanded in c around inf 98.0%

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

   if -8.6000000000000004e159 < b < -7.0000000000000006e-288

   1. Initial program 93.1%

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

   3. Taylor expanded in a around 0 93.1%

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

      1. distribute-lft-out-- 93.1%

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

      2. associate-/l* 93.1%

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

      3. fma-neg 93.1%

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

   5. Simplified 93.1%

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

   6. Taylor expanded in c around inf 93.1%

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

   if -7.0000000000000006e-288 < b < 1.00000000000000004e42

   1. Initial program 80.7%

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

   3. Taylor expanded in b around -inf 81.1%

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

      1. *-commutative 81.1%

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

   5. Simplified 81.1%

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

   if 1.00000000000000004e42 < b

   1. Initial program 56.4%

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

   3. Taylor expanded in b around -inf 56.4%

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

      1. *-commutative 56.4%

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

   5. Simplified 56.4%

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

   6. Taylor expanded in a around 0 88.9%

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

      1. distribute-lft-out-- 88.9%

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

      2. associate-/l* 95.2%

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

   8. Simplified 95.2%

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

      1. associate-/l* 95.1%

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

      2. fma-neg 95.1%

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

   10. Applied egg-rr 95.1%

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

       1. associate-*r/ 95.2%

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

       2. times-frac 95.2%

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

       3. metadata-eval 95.2%

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

       4. *-lft-identity 95.2%

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

       5. fma-neg 95.2%

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

   12. Simplified 95.2%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.6 \cdot 10^{+159}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \leq -7 \cdot 10^{-288}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{+42}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{c}{b} - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ \mathbf{if}\;b \leq -8.6 \cdot 10^{+159}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq 1.22 \cdot 10^{+41}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{c}{b} - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0))))))
   (if (<= b -8.6e+159)
     (if (>= b 0.0)
       (* c (/ -2.0 (+ b (sqrt (fma c (* a -4.0) (* b b))))))
       (fma -1.0 (/ b a) (/ c b)))
     (if (<= b 1.22e+41)
       (if (>= b 0.0) (/ (* c 2.0) (- (- b) t_0)) (/ (- t_0 b) (* a 2.0)))
       (if (>= b 0.0) (/ c (- (* a (/ c b)) b)) (/ (* b -2.0) (* a 2.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -8.6e+159) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / (b + sqrt(fma(c, (a * -4.0), (b * b)))));
		} else {
			tmp_2 = fma(-1.0, (b / a), (c / b));
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.22e+41) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c * 2.0) / (-b - t_0);
		} else {
			tmp_3 = (t_0 - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = c / ((a * (c / b)) - b);
	} else {
		tmp_1 = (b * -2.0) / (a * 2.0);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	tmp_1 = 0.0
	if (b <= -8.6e+159)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * Float64(-2.0 / Float64(b + sqrt(fma(c, Float64(a * -4.0), Float64(b * b))))));
		else
			tmp_2 = fma(-1.0, Float64(b / a), Float64(c / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.22e+41)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(c * 2.0) / Float64(Float64(-b) - t_0));
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(a * 2.0));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(c / Float64(Float64(a * Float64(c / b)) - b));
	else
		tmp_1 = Float64(Float64(b * -2.0) / Float64(a * 2.0));
	end
	return tmp_1
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, -8.6e+159], 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[(-1.0 * N[(b / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.22e+41], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(c / N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[(b * -2.0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.6000000000000004e159

   1. Initial program 44.9%

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

   3. Simplified 45.0%

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

   5. Taylor expanded in b around -inf 97.8%

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

      1. mul-1-neg 97.8%

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

      2. *-commutative 97.8%

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

      3. distribute-rgt-neg-in 97.8%

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

      4. +-commutative 97.8%

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

      5. mul-1-neg 97.8%

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

      6. unsub-neg 97.8%

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

   7. Simplified 97.8%

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

   8. Taylor expanded in a around inf 98.0%

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

      1. fma-define 98.0%

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

   10. Simplified 98.0%

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

   if -8.6000000000000004e159 < b < 1.22e41

   1. Initial program 87.3%

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

   if 1.22e41 < b

   1. Initial program 56.4%

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

   3. Taylor expanded in b around -inf 56.4%

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

      1. *-commutative 56.4%

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

   5. Simplified 56.4%

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

   6. Taylor expanded in a around 0 88.9%

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

      1. distribute-lft-out-- 88.9%

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

      2. associate-/l* 95.2%

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

   8. Simplified 95.2%

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

      1. associate-/l* 95.1%

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

      2. fma-neg 95.1%

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

   10. Applied egg-rr 95.1%

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

       1. associate-*r/ 95.2%

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

       2. times-frac 95.2%

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

       3. metadata-eval 95.2%

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

       4. *-lft-identity 95.2%

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

       5. fma-neg 95.2%

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

   12. Simplified 95.2%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.6 \cdot 10^{+159}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq 1.22 \cdot 10^{+41}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{c}{b} - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot -2}{a \cdot 2}\\ \mathbf{if}\;b \leq -8.6 \cdot 10^{+159}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \leq -7 \cdot 10^{-288}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-81}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* b -2.0) (* a 2.0))))
   (if (<= b -8.6e+159)
     (if (>= b 0.0) (/ b a) t_0)
     (if (<= b -7e-288)
       (if (>= b 0.0)
         (/ b a)
         (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)))
       (if (<= b 9e-81)
         (if (>= b 0.0) (/ (* c 2.0) (- (- b) (sqrt (* c (* a -4.0))))) t_0)
         (if (>= b 0.0)
           (/ (* c 2.0) (* 2.0 (fma a (/ c b) (- b))))
           (* b (+ (/ c (pow b 2.0)) (/ -1.0 a)))))))))

C

double code(double a, double b, double c) {
	double t_0 = (b * -2.0) / (a * 2.0);
	double tmp_1;
	if (b <= -8.6e+159) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b / a;
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -7e-288) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = b / a;
		} else {
			tmp_3 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b <= 9e-81) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c * 2.0) / (-b - sqrt((c * (a * -4.0))));
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c * 2.0) / (2.0 * fma(a, (c / b), -b));
	} else {
		tmp_1 = b * ((c / pow(b, 2.0)) + (-1.0 / a));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * -2.0) / Float64(a * 2.0))
	tmp_1 = 0.0
	if (b <= -8.6e+159)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(b / a);
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b <= -7e-288)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(b / a);
		else
			tmp_3 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
		end
		tmp_1 = tmp_3;
	elseif (b <= 9e-81)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(c * 2.0) / Float64(Float64(-b) - sqrt(Float64(c * Float64(a * -4.0)))));
		else
			tmp_4 = t_0;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c * 2.0) / Float64(2.0 * fma(a, Float64(c / b), Float64(-b))));
	else
		tmp_1 = Float64(b * Float64(Float64(c / (b ^ 2.0)) + Float64(-1.0 / a)));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * -2.0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -8.6e+159], If[GreaterEqual[b, 0.0], N[(b / a), $MachinePrecision], t$95$0], If[LessEqual[b, -7e-288], If[GreaterEqual[b, 0.0], N[(b / a), $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[LessEqual[b, 9e-81], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[(2.0 * N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -8.6000000000000004e159

   1. Initial program 44.9%

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

   3. Taylor expanded in b around -inf 98.0%

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

      1. *-commutative 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in a around 0 98.0%

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

      1. distribute-lft-out-- 98.0%

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

      2. associate-/l* 98.0%

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

   8. Simplified 98.0%

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

   9. Taylor expanded in c around inf 98.0%

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

   if -8.6000000000000004e159 < b < -7.0000000000000006e-288

   1. Initial program 93.1%

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

   3. Taylor expanded in a around 0 93.1%

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

      1. distribute-lft-out-- 93.1%

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

      2. associate-/l* 93.1%

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

      3. fma-neg 93.1%

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

   5. Simplified 93.1%

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

   6. Taylor expanded in c around inf 93.1%

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

   if -7.0000000000000006e-288 < b < 9.000000000000001e-81

   1. Initial program 74.2%

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

   3. Taylor expanded in b around -inf 74.8%

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

      1. *-commutative 74.8%

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

   5. Simplified 74.8%

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

   6. Taylor expanded in b around 0 68.7%

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

      1. associate-*r* 68.7%

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

      2. metadata-eval 68.7%

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

      3. distribute-lft-neg-in 68.7%

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

      4. *-commutative 68.7%

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

      5. distribute-lft-neg-in 68.7%

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

      6. metadata-eval 68.7%

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

      7. *-commutative 68.7%

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

   8. Simplified 68.7%

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

   if 9.000000000000001e-81 < b

   1. Initial program 64.2%

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

   3. Taylor expanded in a around 0 85.0%

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

      1. distribute-lft-out-- 85.0%

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

      2. associate-/l* 90.0%

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

      3. fma-neg 90.0%

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

   5. Simplified 90.0%

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

   6. Taylor expanded in b around -inf 90.0%

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

      1. associate-*r* 90.0%

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

      2. mul-1-neg 90.0%

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

      3. +-commutative 90.0%

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

      4. mul-1-neg 90.0%

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

      5. unsub-neg 90.0%

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

   8. Simplified 90.0%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.6 \cdot 10^{+159}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \leq -7 \cdot 10^{-288}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-81}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot 2}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ t_1 := \frac{b \cdot -2}{a \cdot 2}\\ t_2 := \sqrt{c \cdot \left(a \cdot -4\right)}\\ \mathbf{if}\;b \leq -3.3 \cdot 10^{-56}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\\ \mathbf{elif}\;b \leq -7 \cdot 10^{-288}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2 - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-81}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - t\_2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* c 2.0) (* 2.0 (fma a (/ c b) (- b)))))
        (t_1 (/ (* b -2.0) (* a 2.0)))
        (t_2 (sqrt (* c (* a -4.0)))))
   (if (<= b -3.3e-56)
     (if (>= b 0.0) (/ b a) t_1)
     (if (<= b -7e-288)
       (if (>= b 0.0) t_0 (/ (- t_2 b) (* a 2.0)))
       (if (<= b 9.5e-81)
         (if (>= b 0.0) (/ (* c 2.0) (- (- b) t_2)) t_1)
         (if (>= b 0.0) t_0 (* b (+ (/ c (pow b 2.0)) (/ -1.0 a)))))))))

C

double code(double a, double b, double c) {
	double t_0 = (c * 2.0) / (2.0 * fma(a, (c / b), -b));
	double t_1 = (b * -2.0) / (a * 2.0);
	double t_2 = sqrt((c * (a * -4.0)));
	double tmp_1;
	if (b <= -3.3e-56) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b / a;
		} else {
			tmp_2 = t_1;
		}
		tmp_1 = tmp_2;
	} else if (b <= -7e-288) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_0;
		} else {
			tmp_3 = (t_2 - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b <= 9.5e-81) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c * 2.0) / (-b - t_2);
		} else {
			tmp_4 = t_1;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = b * ((c / pow(b, 2.0)) + (-1.0 / a));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * 2.0) / Float64(2.0 * fma(a, Float64(c / b), Float64(-b))))
	t_1 = Float64(Float64(b * -2.0) / Float64(a * 2.0))
	t_2 = sqrt(Float64(c * Float64(a * -4.0)))
	tmp_1 = 0.0
	if (b <= -3.3e-56)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(b / a);
		else
			tmp_2 = t_1;
		end
		tmp_1 = tmp_2;
	elseif (b <= -7e-288)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_0;
		else
			tmp_3 = Float64(Float64(t_2 - b) / Float64(a * 2.0));
		end
		tmp_1 = tmp_3;
	elseif (b <= 9.5e-81)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(c * 2.0) / Float64(Float64(-b) - t_2));
		else
			tmp_4 = t_1;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = t_0;
	else
		tmp_1 = Float64(b * Float64(Float64(c / (b ^ 2.0)) + Float64(-1.0 / a)));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * 2.0), $MachinePrecision] / N[(2.0 * N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * -2.0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -3.3e-56], If[GreaterEqual[b, 0.0], N[(b / a), $MachinePrecision], t$95$1], If[LessEqual[b, -7e-288], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(t$95$2 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 9.5e-81], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - t$95$2), $MachinePrecision]), $MachinePrecision], t$95$1], If[GreaterEqual[b, 0.0], t$95$0, N[(b * N[(N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -3.29999999999999984e-56

   1. Initial program 69.0%

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

   3. Taylor expanded in b around -inf 89.0%

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

      1. *-commutative 89.0%

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

   5. Simplified 89.0%

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

   6. Taylor expanded in a around 0 89.0%

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

      1. distribute-lft-out-- 89.0%

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

      2. associate-/l* 89.0%

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

   8. Simplified 89.0%

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

   9. Taylor expanded in c around inf 89.0%

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

   if -3.29999999999999984e-56 < b < -7.0000000000000006e-288

   1. Initial program 93.1%

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

   3. Taylor expanded in a around 0 93.1%

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

      1. distribute-lft-out-- 93.1%

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

      2. associate-/l* 93.1%

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

      3. fma-neg 93.1%

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

   5. Simplified 93.1%

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

   6. Taylor expanded in b around 0 87.3%

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

      1. associate-*r* 12.0%

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

      2. metadata-eval 12.0%

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

      3. distribute-lft-neg-in 12.0%

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

      4. *-commutative 12.0%

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

      5. distribute-lft-neg-in 12.0%

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

      6. metadata-eval 12.0%

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

      7. *-commutative 12.0%

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

   8. Simplified 87.3%

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

   if -7.0000000000000006e-288 < b < 9.49999999999999917e-81

   1. Initial program 74.2%

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

   3. Taylor expanded in b around -inf 74.8%

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

      1. *-commutative 74.8%

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

   5. Simplified 74.8%

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

   6. Taylor expanded in b around 0 68.7%

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

      1. associate-*r* 68.7%

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

      2. metadata-eval 68.7%

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

      3. distribute-lft-neg-in 68.7%

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

      4. *-commutative 68.7%

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

      5. distribute-lft-neg-in 68.7%

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

      6. metadata-eval 68.7%

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

      7. *-commutative 68.7%

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

   8. Simplified 68.7%

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

   if 9.49999999999999917e-81 < b

   1. Initial program 64.2%

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

   3. Taylor expanded in a around 0 85.0%

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

      1. distribute-lft-out-- 85.0%

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

      2. associate-/l* 90.0%

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

      3. fma-neg 90.0%

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

   5. Simplified 90.0%

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

   6. Taylor expanded in b around -inf 90.0%

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

      1. associate-*r* 90.0%

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

      2. mul-1-neg 90.0%

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

      3. +-commutative 90.0%

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

      4. mul-1-neg 90.0%

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

      5. unsub-neg 90.0%

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

   8. Simplified 90.0%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{-56}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \leq -7 \cdot 10^{-288}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-81}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ t_1 := \frac{b \cdot -2}{a \cdot 2}\\ \mathbf{if}\;b \leq -8.6 \cdot 10^{+159}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\\ \mathbf{elif}\;b \leq 1.22 \cdot 10^{+41}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{c}{b} - b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0)))))
        (t_1 (/ (* b -2.0) (* a 2.0))))
   (if (<= b -8.6e+159)
     (if (>= b 0.0) (/ b a) t_1)
     (if (<= b 1.22e+41)
       (if (>= b 0.0) (/ (* c 2.0) (- (- b) t_0)) (/ (- t_0 b) (* a 2.0)))
       (if (>= b 0.0) (/ c (- (* a (/ c b)) b)) t_1)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double t_1 = (b * -2.0) / (a * 2.0);
	double tmp_1;
	if (b <= -8.6e+159) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b / a;
		} else {
			tmp_2 = t_1;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.22e+41) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c * 2.0) / (-b - t_0);
		} else {
			tmp_3 = (t_0 - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = c / ((a * (c / b)) - b);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = sqrt(((b * b) - (c * (a * 4.0d0))))
    t_1 = (b * (-2.0d0)) / (a * 2.0d0)
    if (b <= (-8.6d+159)) then
        if (b >= 0.0d0) then
            tmp_2 = b / a
        else
            tmp_2 = t_1
        end if
        tmp_1 = tmp_2
    else if (b <= 1.22d+41) then
        if (b >= 0.0d0) then
            tmp_3 = (c * 2.0d0) / (-b - t_0)
        else
            tmp_3 = (t_0 - b) / (a * 2.0d0)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = c / ((a * (c / b)) - b)
    else
        tmp_1 = t_1
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (c * (a * 4.0))));
	double t_1 = (b * -2.0) / (a * 2.0);
	double tmp_1;
	if (b <= -8.6e+159) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b / a;
		} else {
			tmp_2 = t_1;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.22e+41) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c * 2.0) / (-b - t_0);
		} else {
			tmp_3 = (t_0 - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = c / ((a * (c / b)) - b);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (c * (a * 4.0))))
	t_1 = (b * -2.0) / (a * 2.0)
	tmp_1 = 0
	if b <= -8.6e+159:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = b / a
		else:
			tmp_2 = t_1
		tmp_1 = tmp_2
	elif b <= 1.22e+41:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (c * 2.0) / (-b - t_0)
		else:
			tmp_3 = (t_0 - b) / (a * 2.0)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = c / ((a * (c / b)) - b)
	else:
		tmp_1 = t_1
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	t_1 = Float64(Float64(b * -2.0) / Float64(a * 2.0))
	tmp_1 = 0.0
	if (b <= -8.6e+159)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(b / a);
		else
			tmp_2 = t_1;
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.22e+41)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(c * 2.0) / Float64(Float64(-b) - t_0));
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(a * 2.0));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(c / Float64(Float64(a * Float64(c / b)) - b));
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	t_1 = (b * -2.0) / (a * 2.0);
	tmp_2 = 0.0;
	if (b <= -8.6e+159)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = b / a;
		else
			tmp_3 = t_1;
		end
		tmp_2 = tmp_3;
	elseif (b <= 1.22e+41)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (c * 2.0) / (-b - t_0);
		else
			tmp_4 = (t_0 - b) / (a * 2.0);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = c / ((a * (c / b)) - b);
	else
		tmp_2 = t_1;
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * -2.0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -8.6e+159], If[GreaterEqual[b, 0.0], N[(b / a), $MachinePrecision], t$95$1], If[LessEqual[b, 1.22e+41], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(c / N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.6000000000000004e159

   1. Initial program 44.9%

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

   3. Taylor expanded in b around -inf 98.0%

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

      1. *-commutative 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in a around 0 98.0%

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

      1. distribute-lft-out-- 98.0%

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

      2. associate-/l* 98.0%

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

   8. Simplified 98.0%

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

   9. Taylor expanded in c around inf 98.0%

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

   if -8.6000000000000004e159 < b < 1.22e41

   1. Initial program 87.3%

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

   if 1.22e41 < b

   1. Initial program 56.4%

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

   3. Taylor expanded in b around -inf 56.4%

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

      1. *-commutative 56.4%

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

   5. Simplified 56.4%

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

   6. Taylor expanded in a around 0 88.9%

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

      1. distribute-lft-out-- 88.9%

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

      2. associate-/l* 95.2%

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

   8. Simplified 95.2%

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

      1. associate-/l* 95.1%

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

      2. fma-neg 95.1%

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

   10. Applied egg-rr 95.1%

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

       1. associate-*r/ 95.2%

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

       2. times-frac 95.2%

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

       3. metadata-eval 95.2%

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

       4. *-lft-identity 95.2%

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

       5. fma-neg 95.2%

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

   12. Simplified 95.2%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.6 \cdot 10^{+159}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.22 \cdot 10^{+41}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{c}{b} - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-56}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.9e-56)
   (if (>= b 0.0) (/ b a) (/ (* b -2.0) (* a 2.0)))
   (if (>= b 0.0)
     (/ (* c 2.0) (* 2.0 (fma a (/ c b) (- b))))
     (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -2.9e-56) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b / a;
		} else {
			tmp_2 = (b * -2.0) / (a * 2.0);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (c * 2.0) / (2.0 * fma(a, (c / b), -b));
	} else {
		tmp_1 = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -2.9e-56)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(b / a);
		else
			tmp_2 = Float64(Float64(b * -2.0) / Float64(a * 2.0));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c * 2.0) / Float64(2.0 * fma(a, Float64(c / b), Float64(-b))));
	else
		tmp_1 = Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / Float64(a * 2.0));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.9e-56], If[GreaterEqual[b, 0.0], N[(b / a), $MachinePrecision], N[(N[(b * -2.0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[(2.0 * N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.89999999999999991e-56

   1. Initial program 69.0%

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

   3. Taylor expanded in b around -inf 89.0%

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

      1. *-commutative 89.0%

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

   5. Simplified 89.0%

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

   6. Taylor expanded in a around 0 89.0%

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

      1. distribute-lft-out-- 89.0%

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

      2. associate-/l* 89.0%

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

   8. Simplified 89.0%

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

   9. Taylor expanded in c around inf 89.0%

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

   if -2.89999999999999991e-56 < b

   1. Initial program 71.5%

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

   3. Taylor expanded in a around 0 69.4%

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

      1. distribute-lft-out-- 69.4%

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

      2. associate-/l* 72.2%

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

      3. fma-neg 72.2%

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

   5. Simplified 72.2%

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

   6. Taylor expanded in b around 0 71.3%

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

      1. associate-*r* 33.0%

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

      2. metadata-eval 33.0%

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

      3. distribute-lft-neg-in 33.0%

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

      4. *-commutative 33.0%

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

      5. distribute-lft-neg-in 33.0%

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

      6. metadata-eval 33.0%

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

      7. *-commutative 33.0%

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

   8. Simplified 71.3%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-56}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.4% accurate, 8.6× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = c / ((a * (c / b)) - b);
	} else {
		tmp = (b * -2.0) / (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 = c / ((a * (c / b)) - b)
    else
        tmp = (b * (-2.0d0)) / (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 = c / ((a * (c / b)) - b);
	} else {
		tmp = (b * -2.0) / (a * 2.0);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.7%

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

3. Taylor expanded in b around -inf 68.6%

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

   1. *-commutative 68.6%

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

5. Simplified 68.6%

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

6. Taylor expanded in a around 0 67.2%

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

   1. distribute-lft-out-- 67.2%

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

   2. associate-/l* 69.1%

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

8. Simplified 69.1%

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

   1. associate-/l* 69.1%

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

   2. fma-neg 69.1%

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

10. Applied egg-rr 69.1%

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

    1. associate-*r/ 69.1%

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

    2. times-frac 69.1%

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

    3. metadata-eval 69.1%

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

    4. *-lft-identity 69.1%

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

    5. fma-neg 69.1%

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

12. Simplified 69.1%

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

13. Final simplification 69.1%

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

Alternative 8: 67.4% accurate, 9.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (c * 2.0) / (-b - b);
	} else {
		tmp = (b * -2.0) / (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 = (c * 2.0d0) / (-b - b)
    else
        tmp = (b * (-2.0d0)) / (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 = (c * 2.0) / (-b - b);
	} else {
		tmp = (b * -2.0) / (a * 2.0);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.7%

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

3. Taylor expanded in b around -inf 68.6%

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

   1. *-commutative 68.6%

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

5. Simplified 68.6%

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

6. Taylor expanded in b around inf 68.7%

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

7. Final simplification 68.7%

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

Alternative 9: 36.1% accurate, 10.1× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = b / a;
	} else {
		tmp = (b * -2.0) / (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 = b / a
    else
        tmp = (b * (-2.0d0)) / (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 = b / a;
	} else {
		tmp = (b * -2.0) / (a * 2.0);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.7%

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

3. Taylor expanded in b around -inf 68.6%

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

   1. *-commutative 68.6%

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

5. Simplified 68.6%

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

6. Taylor expanded in a around 0 67.2%

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

   1. distribute-lft-out-- 67.2%

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

   2. associate-/l* 69.1%

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

8. Simplified 69.1%

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

9. Taylor expanded in c around inf 33.4%

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

10. Final simplification 33.4%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\ \end{array} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024086 
(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))))