jeff quadratic root 2

Percentage Accurate: 73.5% → 90.9%

Time: 21.6s

Alternatives: 7

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: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-b}{a}\\ t_1 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ t_2 := \frac{2 \cdot c}{b \cdot -2}\\ \mathbf{if}\;b \leq -2 \cdot 10^{+103}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array}\\ \mathbf{elif}\;b \leq -1.16 \cdot 10^{-294}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{+95}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(b + \sqrt{-4 \cdot \left(c \cdot a\right)}\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- b) a))
        (t_1 (sqrt (- (* b b) (* c (* a 4.0)))))
        (t_2 (/ (* 2.0 c) (* b -2.0))))
   (if (<= b -2e+103)
     (if (>= b 0.0) t_2 t_0)
     (if (<= b -1.16e-294)
       (if (>= b 0.0) t_2 (/ (- t_1 b) (* 2.0 a)))
       (if (<= b 2.45e+95)
         (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_1)) t_0)
         (if (>= b 0.0)
           t_2
           (* (/ 1.0 a) (* (+ b (sqrt (* -4.0 (* c a)))) 0.5))))))))

C

double code(double a, double b, double c) {
	double t_0 = -b / a;
	double t_1 = sqrt(((b * b) - (c * (a * 4.0))));
	double t_2 = (2.0 * c) / (b * -2.0);
	double tmp_1;
	if (b <= -2e+103) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_2;
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -1.16e-294) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_2;
		} else {
			tmp_3 = (t_1 - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 2.45e+95) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (2.0 * c) / (-b - t_1);
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = t_2;
	} else {
		tmp_1 = (1.0 / a) * ((b + sqrt((-4.0 * (c * a)))) * 0.5);
	}
	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) :: t_2
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    real(8) :: tmp_4
    t_0 = -b / a
    t_1 = sqrt(((b * b) - (c * (a * 4.0d0))))
    t_2 = (2.0d0 * c) / (b * (-2.0d0))
    if (b <= (-2d+103)) then
        if (b >= 0.0d0) then
            tmp_2 = t_2
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b <= (-1.16d-294)) then
        if (b >= 0.0d0) then
            tmp_3 = t_2
        else
            tmp_3 = (t_1 - b) / (2.0d0 * a)
        end if
        tmp_1 = tmp_3
    else if (b <= 2.45d+95) then
        if (b >= 0.0d0) then
            tmp_4 = (2.0d0 * c) / (-b - t_1)
        else
            tmp_4 = t_0
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = t_2
    else
        tmp_1 = (1.0d0 / a) * ((b + sqrt(((-4.0d0) * (c * a)))) * 0.5d0)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -b / a;
	double t_1 = Math.sqrt(((b * b) - (c * (a * 4.0))));
	double t_2 = (2.0 * c) / (b * -2.0);
	double tmp_1;
	if (b <= -2e+103) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_2;
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -1.16e-294) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_2;
		} else {
			tmp_3 = (t_1 - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 2.45e+95) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (2.0 * c) / (-b - t_1);
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = t_2;
	} else {
		tmp_1 = (1.0 / a) * ((b + Math.sqrt((-4.0 * (c * a)))) * 0.5);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = -b / a
	t_1 = math.sqrt(((b * b) - (c * (a * 4.0))))
	t_2 = (2.0 * c) / (b * -2.0)
	tmp_1 = 0
	if b <= -2e+103:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_2
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b <= -1.16e-294:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = t_2
		else:
			tmp_3 = (t_1 - b) / (2.0 * a)
		tmp_1 = tmp_3
	elif b <= 2.45e+95:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = (2.0 * c) / (-b - t_1)
		else:
			tmp_4 = t_0
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = t_2
	else:
		tmp_1 = (1.0 / a) * ((b + math.sqrt((-4.0 * (c * a)))) * 0.5)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-b) / a)
	t_1 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	t_2 = Float64(Float64(2.0 * c) / Float64(b * -2.0))
	tmp_1 = 0.0
	if (b <= -2e+103)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_2;
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b <= -1.16e-294)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_2;
		else
			tmp_3 = Float64(Float64(t_1 - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (b <= 2.45e+95)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_1));
		else
			tmp_4 = t_0;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = t_2;
	else
		tmp_1 = Float64(Float64(1.0 / a) * Float64(Float64(b + sqrt(Float64(-4.0 * Float64(c * a)))) * 0.5));
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = -b / a;
	t_1 = sqrt(((b * b) - (c * (a * 4.0))));
	t_2 = (2.0 * c) / (b * -2.0);
	tmp_2 = 0.0;
	if (b <= -2e+103)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_2;
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b <= -1.16e-294)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = t_2;
		else
			tmp_4 = (t_1 - b) / (2.0 * a);
		end
		tmp_2 = tmp_4;
	elseif (b <= 2.45e+95)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = (2.0 * c) / (-b - t_1);
		else
			tmp_5 = t_0;
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = t_2;
	else
		tmp_2 = (1.0 / a) * ((b + sqrt((-4.0 * (c * a)))) * 0.5);
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) / a), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * c), $MachinePrecision] / N[(b * -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2e+103], If[GreaterEqual[b, 0.0], t$95$2, t$95$0], If[LessEqual[b, -1.16e-294], If[GreaterEqual[b, 0.0], t$95$2, N[(N[(t$95$1 - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 2.45e+95], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$1), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], t$95$2, N[(N[(1.0 / a), $MachinePrecision] * N[(N[(b + N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2e103

   1. Initial program 58.2%

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

   2. Taylor expanded in b around inf 58.2%

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

      1. *-commutative 58.2%

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

   4. Simplified 58.2%

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

   5. Taylor expanded in b around -inf 97.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-*r/ 97.0%

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

      2. mul-1-neg 97.0%

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

   7. Simplified 97.0%

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

   if -2e103 < b < -1.16000000000000006e-294

   1. Initial program 88.3%

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

   2. Taylor expanded in b around inf 88.3%

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

      1. *-commutative 88.3%

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

   4. Simplified 88.3%

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

   if -1.16000000000000006e-294 < b < 2.4499999999999999e95

   1. Initial program 82.4%

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

   2. Taylor expanded in b around -inf 82.4%

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

      1. associate-*r/ 44.3%

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

      2. mul-1-neg 44.3%

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

   4. Simplified 82.4%

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

   if 2.4499999999999999e95 < b

   1. Initial program 65.5%

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

   2. Taylor expanded in b around inf 96.9%

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

      1. *-commutative 96.9%

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

   4. Simplified 96.9%

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

   5. Taylor expanded in b around 0 96.9%

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

      1. associate-*r* 96.9%

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

      2. metadata-eval 96.9%

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

      3. distribute-lft-neg-in 96.9%

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

      4. *-commutative 96.9%

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

      5. distribute-lft-neg-in 96.9%

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

      6. metadata-eval 96.9%

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

      7. *-commutative 96.9%

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

   7. Simplified 96.9%

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

      1. *-un-lft-identity 96.9%

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

      2. *-commutative 96.9%

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

      3. times-frac 96.9%

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

      4. div-inv 96.9%

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

      5. add-sqr-sqrt 96.9%

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

      6. sqrt-unprod 96.9%

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

      7. sqr-neg 96.9%

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

      8. sqrt-prod 96.9%

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

      9. add-sqr-sqrt 96.9%

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

      10. *-commutative 96.9%

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

      11. *-commutative 96.9%

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

      12. associate-*r* 96.9%

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

      13. *-commutative 96.9%

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

      14. metadata-eval 96.9%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \color{blue}{\left(\left(b + \sqrt{-4 \cdot \left(c \cdot a\right)}\right) \cdot 0.5\right)}\\ \end{array} \]

   9. Applied egg-rr 96.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(b + \sqrt{-4 \cdot \left(c \cdot a\right)}\right) \cdot 0.5\right)\\ \end{array} \]
3. Recombined 4 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+103}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq -1.16 \cdot 10^{-294}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{+95}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(b + \sqrt{-4 \cdot \left(c \cdot a\right)}\right) \cdot 0.5\right)\\ \end{array} \]

Alternative 2: 91.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ t_1 := \frac{2 \cdot c}{b \cdot -2}\\ \mathbf{if}\;b \leq -2.1 \cdot 10^{+103}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{+90}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(b + \sqrt{-4 \cdot \left(c \cdot a\right)}\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0)))))
        (t_1 (/ (* 2.0 c) (* b -2.0))))
   (if (<= b -2.1e+103)
     (if (>= b 0.0) t_1 (/ (- b) a))
     (if (<= b 1e+90)
       (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_0)) (/ (- t_0 b) (* 2.0 a)))
       (if (>= b 0.0)
         t_1
         (* (/ 1.0 a) (* (+ b (sqrt (* -4.0 (* c a)))) 0.5)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double t_1 = (2.0 * c) / (b * -2.0);
	double tmp_1;
	if (b <= -2.1e+103) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = -b / a;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1e+90) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_3 = (t_0 - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = t_1;
	} else {
		tmp_1 = (1.0 / a) * ((b + sqrt((-4.0 * (c * a)))) * 0.5);
	}
	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 = (2.0d0 * c) / (b * (-2.0d0))
    if (b <= (-2.1d+103)) then
        if (b >= 0.0d0) then
            tmp_2 = t_1
        else
            tmp_2 = -b / a
        end if
        tmp_1 = tmp_2
    else if (b <= 1d+90) then
        if (b >= 0.0d0) then
            tmp_3 = (2.0d0 * c) / (-b - t_0)
        else
            tmp_3 = (t_0 - b) / (2.0d0 * a)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = t_1
    else
        tmp_1 = (1.0d0 / a) * ((b + sqrt(((-4.0d0) * (c * a)))) * 0.5d0)
    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 = (2.0 * c) / (b * -2.0);
	double tmp_1;
	if (b <= -2.1e+103) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = -b / a;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1e+90) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_3 = (t_0 - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = t_1;
	} else {
		tmp_1 = (1.0 / a) * ((b + Math.sqrt((-4.0 * (c * a)))) * 0.5);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (c * (a * 4.0))))
	t_1 = (2.0 * c) / (b * -2.0)
	tmp_1 = 0
	if b <= -2.1e+103:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_1
		else:
			tmp_2 = -b / a
		tmp_1 = tmp_2
	elif b <= 1e+90:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (2.0 * c) / (-b - t_0)
		else:
			tmp_3 = (t_0 - b) / (2.0 * a)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = t_1
	else:
		tmp_1 = (1.0 / a) * ((b + math.sqrt((-4.0 * (c * a)))) * 0.5)
	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(2.0 * c) / Float64(b * -2.0))
	tmp_1 = 0.0
	if (b <= -2.1e+103)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(Float64(-b) / a);
		end
		tmp_1 = tmp_2;
	elseif (b <= 1e+90)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0));
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = t_1;
	else
		tmp_1 = Float64(Float64(1.0 / a) * Float64(Float64(b + sqrt(Float64(-4.0 * Float64(c * a)))) * 0.5));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	t_1 = (2.0 * c) / (b * -2.0);
	tmp_2 = 0.0;
	if (b <= -2.1e+103)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = -b / a;
		end
		tmp_2 = tmp_3;
	elseif (b <= 1e+90)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (2.0 * c) / (-b - t_0);
		else
			tmp_4 = (t_0 - b) / (2.0 * a);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = t_1;
	else
		tmp_2 = (1.0 / a) * ((b + sqrt((-4.0 * (c * a)))) * 0.5);
	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[(2.0 * c), $MachinePrecision] / N[(b * -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.1e+103], If[GreaterEqual[b, 0.0], t$95$1, N[((-b) / a), $MachinePrecision]], If[LessEqual[b, 1e+90], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(1.0 / a), $MachinePrecision] * N[(N[(b + N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.1000000000000002e103

   1. Initial program 58.2%

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

   2. Taylor expanded in b around inf 58.2%

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

      1. *-commutative 58.2%

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

   4. Simplified 58.2%

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

   5. Taylor expanded in b around -inf 97.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-*r/ 97.0%

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

      2. mul-1-neg 97.0%

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

   7. Simplified 97.0%

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

   if -2.1000000000000002e103 < b < 9.99999999999999966e89

   1. Initial program 85.1%

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

   if 9.99999999999999966e89 < b

   1. Initial program 65.5%

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

   2. Taylor expanded in b around inf 96.9%

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

      1. *-commutative 96.9%

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

   4. Simplified 96.9%

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

   5. Taylor expanded in b around 0 96.9%

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

      1. associate-*r* 96.9%

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

      2. metadata-eval 96.9%

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

      3. distribute-lft-neg-in 96.9%

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

      4. *-commutative 96.9%

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

      5. distribute-lft-neg-in 96.9%

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

      6. metadata-eval 96.9%

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

      7. *-commutative 96.9%

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

   7. Simplified 96.9%

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

      1. *-un-lft-identity 96.9%

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

      2. *-commutative 96.9%

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

      3. times-frac 96.9%

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

      4. div-inv 96.9%

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

      5. add-sqr-sqrt 96.9%

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

      6. sqrt-unprod 96.9%

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

      7. sqr-neg 96.9%

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

      8. sqrt-prod 96.9%

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

      9. add-sqr-sqrt 96.9%

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

      10. *-commutative 96.9%

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

      11. *-commutative 96.9%

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

      12. associate-*r* 96.9%

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

      13. *-commutative 96.9%

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

      14. metadata-eval 96.9%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \color{blue}{\left(\left(b + \sqrt{-4 \cdot \left(c \cdot a\right)}\right) \cdot 0.5\right)}\\ \end{array} \]

   9. Applied egg-rr 96.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(b + \sqrt{-4 \cdot \left(c \cdot a\right)}\right) \cdot 0.5\right)\\ \end{array} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{+103}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{+90}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(b + \sqrt{-4 \cdot \left(c \cdot a\right)}\right) \cdot 0.5\right)\\ \end{array} \]

Alternative 3: 86.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2 \cdot c}{b \cdot -2}\\ t_1 := \mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \mathbf{if}\;b \leq -3.2 \cdot 10^{+103}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq -1.16 \cdot 10^{-294}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-64}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + -2 \cdot \left(a \cdot \frac{c}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* 2.0 c) (* b -2.0))) (t_1 (fma -1.0 (/ b a) (/ c b))))
   (if (<= b -3.2e+103)
     (if (>= b 0.0) t_0 (/ (- b) a))
     (if (<= b -1.16e-294)
       (if (>= b 0.0)
         t_0
         (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a)))
       (if (<= b 1.5e-64)
         (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (* c (* a -4.0))))) t_1)
         (if (>= b 0.0)
           (/ (* 2.0 c) (- (- b) (+ b (* -2.0 (* a (/ c b))))))
           t_1))))))

C

double code(double a, double b, double c) {
	double t_0 = (2.0 * c) / (b * -2.0);
	double t_1 = fma(-1.0, (b / a), (c / b));
	double tmp_1;
	if (b <= -3.2e+103) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = -b / a;
		}
		tmp_1 = tmp_2;
	} else if (b <= -1.16e-294) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_0;
		} else {
			tmp_3 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.5e-64) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (2.0 * c) / (-b - sqrt((c * (a * -4.0))));
		} else {
			tmp_4 = t_1;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-b - (b + (-2.0 * (a * (c / b)))));
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(2.0 * c) / Float64(b * -2.0))
	t_1 = fma(-1.0, Float64(b / a), Float64(c / b))
	tmp_1 = 0.0
	if (b <= -3.2e+103)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(Float64(-b) / a);
		end
		tmp_1 = tmp_2;
	elseif (b <= -1.16e-294)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_0;
		else
			tmp_3 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (b <= 1.5e-64)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - sqrt(Float64(c * Float64(a * -4.0)))));
		else
			tmp_4 = t_1;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - Float64(b + Float64(-2.0 * Float64(a * Float64(c / b))))));
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(2.0 * c), $MachinePrecision] / N[(b * -2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 * N[(b / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.2e+103], If[GreaterEqual[b, 0.0], t$95$0, N[((-b) / a), $MachinePrecision]], If[LessEqual[b, -1.16e-294], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.5e-64], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[(b + N[(-2.0 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -3.19999999999999993e103

   1. Initial program 58.2%

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

   2. Taylor expanded in b around inf 58.2%

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

      1. *-commutative 58.2%

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

   4. Simplified 58.2%

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

   5. Taylor expanded in b around -inf 97.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-*r/ 97.0%

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

      2. mul-1-neg 97.0%

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

   7. Simplified 97.0%

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

   if -3.19999999999999993e103 < b < -1.16000000000000006e-294

   1. Initial program 88.3%

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

   2. Taylor expanded in b around inf 88.3%

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

      1. *-commutative 88.3%

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

   4. Simplified 88.3%

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

   if -1.16000000000000006e-294 < b < 1.5e-64

   1. Initial program 79.8%

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

   2. Taylor expanded in b around -inf 79.8%

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

      1. fma-def 79.8%

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

   4. Simplified 79.8%

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

   5. Taylor expanded in b around 0 65.6%

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

      1. associate-*r* 23.9%

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

      2. metadata-eval 23.9%

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

      3. distribute-lft-neg-in 23.9%

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

      4. *-commutative 23.9%

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

      5. distribute-lft-neg-in 23.9%

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

      6. metadata-eval 23.9%

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

      7. *-commutative 23.9%

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

   7. Simplified 65.6%

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

   if 1.5e-64 < b

   1. Initial program 72.4%

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

   2. Taylor expanded in b around -inf 72.4%

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

      1. fma-def 72.4%

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

   4. Simplified 72.4%

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

   5. Taylor expanded in b around inf 85.9%

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

      1. associate-*r/ 90.3%

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

   7. Simplified 90.3%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{+103}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq -1.16 \cdot 10^{-294}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-64}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + -2 \cdot \left(a \cdot \frac{c}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \end{array} \]

Alternative 4: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ t_1 := \frac{2 \cdot c}{b \cdot -2}\\ t_2 := \sqrt{c \cdot \left(a \cdot -4\right)}\\ \mathbf{if}\;b \leq -2.5 \cdot 10^{-105}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq -1.16 \cdot 10^{-294}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{b - t_2}{a} \cdot -0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{-63}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t_2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + -2 \cdot \left(a \cdot \frac{c}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -1.0 (/ b a) (/ c b)))
        (t_1 (/ (* 2.0 c) (* b -2.0)))
        (t_2 (sqrt (* c (* a -4.0)))))
   (if (<= b -2.5e-105)
     (if (>= b 0.0) t_1 (- (/ c b) (/ b a)))
     (if (<= b -1.16e-294)
       (if (>= b 0.0) t_1 (* (/ (- b t_2) a) -0.5))
       (if (<= b 1.02e-63)
         (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_2)) t_0)
         (if (>= b 0.0)
           (/ (* 2.0 c) (- (- b) (+ b (* -2.0 (* a (/ c b))))))
           t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-1.0, (b / a), (c / b));
	double t_1 = (2.0 * c) / (b * -2.0);
	double t_2 = sqrt((c * (a * -4.0)));
	double tmp_1;
	if (b <= -2.5e-105) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = (c / b) - (b / a);
		}
		tmp_1 = tmp_2;
	} else if (b <= -1.16e-294) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = ((b - t_2) / a) * -0.5;
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.02e-63) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (2.0 * c) / (-b - t_2);
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-b - (b + (-2.0 * (a * (c / b)))));
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = fma(-1.0, Float64(b / a), Float64(c / b))
	t_1 = Float64(Float64(2.0 * c) / Float64(b * -2.0))
	t_2 = sqrt(Float64(c * Float64(a * -4.0)))
	tmp_1 = 0.0
	if (b <= -2.5e-105)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(Float64(c / b) - Float64(b / a));
		end
		tmp_1 = tmp_2;
	elseif (b <= -1.16e-294)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = Float64(Float64(Float64(b - t_2) / a) * -0.5);
		end
		tmp_1 = tmp_3;
	elseif (b <= 1.02e-63)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_2));
		else
			tmp_4 = t_0;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - Float64(b + Float64(-2.0 * Float64(a * Float64(c / b))))));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-1.0 * N[(b / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * c), $MachinePrecision] / N[(b * -2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2.5e-105], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -1.16e-294], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(N[(b - t$95$2), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision]], If[LessEqual[b, 1.02e-63], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$2), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[(b + N[(-2.0 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.49999999999999982e-105

   1. Initial program 70.9%

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

   2. Taylor expanded in b around inf 70.9%

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

      1. *-commutative 70.9%

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

   4. Simplified 70.9%

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

   5. Taylor expanded in b around -inf 87.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a} + \frac{c}{b}\\ \end{array} \]
   6. Step-by-step derivation

      1. +-commutative 87.6%

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

      2. mul-1-neg 87.6%

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

      3. unsub-neg 87.6%

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

   7. Simplified 87.6%

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

   if -2.49999999999999982e-105 < b < -1.16000000000000006e-294

   1. Initial program 81.0%

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

   2. Taylor expanded in b around inf 81.0%

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

      1. *-commutative 81.0%

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

   4. Simplified 81.0%

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

   5. Taylor expanded in b around 0 71.2%

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

      1. associate-*r* 71.2%

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

      2. metadata-eval 71.2%

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

      3. distribute-lft-neg-in 71.2%

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

      4. *-commutative 71.2%

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

      5. distribute-lft-neg-in 71.2%

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

      6. metadata-eval 71.2%

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

      7. *-commutative 71.2%

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

   7. Simplified 71.2%

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

      1. frac-2neg 71.2%

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

      2. div-inv 71.1%

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

      3. distribute-neg-in 71.1%

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

      4. add-sqr-sqrt 71.1%

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

      5. sqrt-unprod 69.7%

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

      6. sqr-neg 69.7%

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

      7. sqrt-prod 0.0%

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

      8. add-sqr-sqrt 66.9%

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

      9. sub-neg 66.9%

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

      10. add-sqr-sqrt 66.9%

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

      11. sqrt-unprod 67.1%

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

      12. sqr-neg 67.1%

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

      13. sqrt-prod 0.0%

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

      14. add-sqr-sqrt 71.1%

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

      15. *-commutative 71.1%

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

      16. *-commutative 71.1%

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

      17. associate-*r* 71.1%

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

      18. *-commutative 71.1%

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

      19. *-commutative 71.1%

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

      20. distribute-rgt-neg-in 71.1%

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

   9. Applied egg-rr 71.1%

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

       1. associate-*r/ 71.2%

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

       2. times-frac 71.2%

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

       3. *-commutative 71.2%

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

       4. associate-*r* 71.2%

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

       5. *-commutative 71.2%

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

       6. metadata-eval 71.2%

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

   11. Simplified 71.2%

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

   if -1.16000000000000006e-294 < b < 1.01999999999999997e-63

   1. Initial program 79.8%

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

   2. Taylor expanded in b around -inf 79.8%

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

      1. fma-def 79.8%

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

   4. Simplified 79.8%

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

   5. Taylor expanded in b around 0 65.6%

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

      1. associate-*r* 23.9%

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

      2. metadata-eval 23.9%

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

      3. distribute-lft-neg-in 23.9%

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

      4. *-commutative 23.9%

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

      5. distribute-lft-neg-in 23.9%

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

      6. metadata-eval 23.9%

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

      7. *-commutative 23.9%

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

   7. Simplified 65.6%

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

   if 1.01999999999999997e-63 < b

   1. Initial program 72.4%

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

   2. Taylor expanded in b around -inf 72.4%

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

      1. fma-def 72.4%

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

   4. Simplified 72.4%

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

   5. Taylor expanded in b around inf 85.9%

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

      1. associate-*r/ 90.3%

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

   7. Simplified 90.3%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{-105}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq -1.16 \cdot 10^{-294}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \sqrt{c \cdot \left(a \cdot -4\right)}}{a} \cdot -0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{-63}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + -2 \cdot \left(a \cdot \frac{c}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \end{array} \]

Alternative 5: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2 \cdot c}{b \cdot -2}\\ \mathbf{if}\;b \leq -3.8 \cdot 10^{-105}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \sqrt{c \cdot \left(a \cdot -4\right)}}{a} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* 2.0 c) (* b -2.0))))
   (if (<= b -3.8e-105)
     (if (>= b 0.0) t_0 (- (/ c b) (/ b a)))
     (if (>= b 0.0) t_0 (* (/ (- b (sqrt (* c (* a -4.0)))) a) -0.5)))))

C

double code(double a, double b, double c) {
	double t_0 = (2.0 * c) / (b * -2.0);
	double tmp_1;
	if (b <= -3.8e-105) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = (c / b) - (b / a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = ((b - sqrt((c * (a * -4.0)))) / a) * -0.5;
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = (2.0d0 * c) / (b * (-2.0d0))
    if (b <= (-3.8d-105)) then
        if (b >= 0.0d0) then
            tmp_2 = t_0
        else
            tmp_2 = (c / b) - (b / a)
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = t_0
    else
        tmp_1 = ((b - sqrt((c * (a * (-4.0d0))))) / a) * (-0.5d0)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (2.0 * c) / (b * -2.0);
	double tmp_1;
	if (b <= -3.8e-105) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = (c / b) - (b / a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = ((b - Math.sqrt((c * (a * -4.0)))) / a) * -0.5;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (2.0 * c) / (b * -2.0)
	tmp_1 = 0
	if b <= -3.8e-105:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_0
		else:
			tmp_2 = (c / b) - (b / a)
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = t_0
	else:
		tmp_1 = ((b - math.sqrt((c * (a * -4.0)))) / a) * -0.5
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(2.0 * c) / Float64(b * -2.0))
	tmp_1 = 0.0
	if (b <= -3.8e-105)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(Float64(c / b) - Float64(b / a));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = t_0;
	else
		tmp_1 = Float64(Float64(Float64(b - sqrt(Float64(c * Float64(a * -4.0)))) / a) * -0.5);
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = (2.0 * c) / (b * -2.0);
	tmp_2 = 0.0;
	if (b <= -3.8e-105)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_0;
		else
			tmp_3 = (c / b) - (b / a);
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = t_0;
	else
		tmp_2 = ((b - sqrt((c * (a * -4.0)))) / a) * -0.5;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(2.0 * c), $MachinePrecision] / N[(b * -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.8e-105], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(N[(b - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.7999999999999998e-105

   1. Initial program 70.9%

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

   2. Taylor expanded in b around inf 70.9%

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

      1. *-commutative 70.9%

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

   4. Simplified 70.9%

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

   5. Taylor expanded in b around -inf 87.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a} + \frac{c}{b}\\ \end{array} \]
   6. Step-by-step derivation

      1. +-commutative 87.6%

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

      2. mul-1-neg 87.6%

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

      3. unsub-neg 87.6%

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

   7. Simplified 87.6%

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

   if -3.7999999999999998e-105 < b

   1. Initial program 75.9%

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

   2. Taylor expanded in b around inf 69.9%

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

      1. *-commutative 69.9%

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

   4. Simplified 69.9%

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

   5. Taylor expanded in b around 0 68.2%

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

      1. associate-*r* 68.2%

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

      2. metadata-eval 68.2%

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

      3. distribute-lft-neg-in 68.2%

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

      4. *-commutative 68.2%

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

      5. distribute-lft-neg-in 68.2%

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

      6. metadata-eval 68.2%

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

      7. *-commutative 68.2%

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

   7. Simplified 68.2%

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

      1. frac-2neg 68.2%

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

      2. div-inv 68.2%

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

      3. distribute-neg-in 68.2%

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

      4. add-sqr-sqrt 68.2%

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

      5. sqrt-unprod 68.0%

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

      6. sqr-neg 68.0%

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

      7. sqrt-prod 56.2%

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

      8. add-sqr-sqrt 67.5%

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

      9. sub-neg 67.5%

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

      10. add-sqr-sqrt 67.5%

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

      11. sqrt-unprod 67.5%

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

      12. sqr-neg 67.5%

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

      13. sqrt-prod 56.2%

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

      14. add-sqr-sqrt 68.2%

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

      15. *-commutative 68.2%

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

      16. *-commutative 68.2%

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

      17. associate-*r* 68.2%

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

      18. *-commutative 68.2%

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

      19. *-commutative 68.2%

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

      20. distribute-rgt-neg-in 68.2%

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

   9. Applied egg-rr 68.2%

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

       1. associate-*r/ 68.2%

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

       2. times-frac 68.2%

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

       3. *-commutative 68.2%

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

       4. associate-*r* 68.2%

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

       5. *-commutative 68.2%

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

       6. metadata-eval 68.2%

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

   11. Simplified 68.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \sqrt{\left(a \cdot -4\right) \cdot c}}{a} \cdot -0.5\\ \end{array} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-105}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \sqrt{c \cdot \left(a \cdot -4\right)}}{a} \cdot -0.5\\ \end{array} \]

Alternative 6: 67.5% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0)
   (/ (* 2.0 c) (- (- b) (+ b (* -2.0 (* a (/ c b))))))
   (fma -1.0 (/ b a) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (-b - (b + (-2.0 * (a * (c / b)))));
	} else {
		tmp = fma(-1.0, (b / a), (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) - Float64(b + Float64(-2.0 * Float64(a * Float64(c / b))))));
	else
		tmp = fma(-1.0, Float64(b / a), Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[(b + N[(-2.0 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(b / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 74.1%

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

2. Taylor expanded in b around -inf 74.5%

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

   1. fma-def 74.5%

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

4. Simplified 74.5%

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

5. Taylor expanded in b around inf 69.4%

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

   1. associate-*r/ 70.9%

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

7. Simplified 70.9%

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

8. Final simplification 70.9%

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

Alternative 7: 67.4% accurate, 13.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

2. Taylor expanded in b around inf 70.3%

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

   1. *-commutative 70.3%

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

4. Simplified 70.3%

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

5. Taylor expanded in b around -inf 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a} + \frac{c}{b}\\ \end{array} \]
6. Step-by-step derivation

   1. +-commutative 70.7%

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

   2. mul-1-neg 70.7%

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

   3. unsub-neg 70.7%

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

7. Simplified 70.7%

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

8. Final simplification 70.7%

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

Reproduce

herbie shell --seed 2023334 
(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))))