jeff quadratic root 1

Percentage Accurate: 72.1% → 89.3%

Time: 19.3s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ t_1 := \frac{-c}{b}\\ \mathbf{if}\;b \leq -6.6 \cdot 10^{+100}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{+20}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, b \cdot 2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{\left(2 \cdot \left(a \cdot t_1\right) - b\right) - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0))))) (t_1 (/ (- c) b)))
   (if (<= b -6.6e+100)
     (if (>= b 0.0) (- (/ b a)) t_1)
     (if (<= b 3.9e+20)
       (if (>= b 0.0) (/ (- (- b) t_0) (* a 2.0)) (/ (* c 2.0) (- t_0 b)))
       (if (>= b 0.0)
         (* -0.5 (/ (fma -2.0 (/ a (/ b c)) (* b 2.0)) a))
         (/ (* c -2.0) (- (- (* 2.0 (* a t_1)) b) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double t_1 = -c / b;
	double tmp_1;
	if (b <= -6.6e+100) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -(b / a);
		} else {
			tmp_2 = t_1;
		}
		tmp_1 = tmp_2;
	} else if (b <= 3.9e+20) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_0) / (a * 2.0);
		} else {
			tmp_3 = (c * 2.0) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -0.5 * (fma(-2.0, (a / (b / c)), (b * 2.0)) / a);
	} else {
		tmp_1 = (c * -2.0) / (((2.0 * (a * t_1)) - b) - b);
	}
	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(-c) / b)
	tmp_1 = 0.0
	if (b <= -6.6e+100)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-Float64(b / a));
		else
			tmp_2 = t_1;
		end
		tmp_1 = tmp_2;
	elseif (b <= 3.9e+20)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(-b) - t_0) / Float64(a * 2.0));
		else
			tmp_3 = Float64(Float64(c * 2.0) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(-0.5 * Float64(fma(-2.0, Float64(a / Float64(b / c)), Float64(b * 2.0)) / a));
	else
		tmp_1 = Float64(Float64(c * -2.0) / Float64(Float64(Float64(2.0 * Float64(a * t_1)) - b) - b));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-c) / b), $MachinePrecision]}, If[LessEqual[b, -6.6e+100], If[GreaterEqual[b, 0.0], (-N[(b / a), $MachinePrecision]), t$95$1], If[LessEqual[b, 3.9e+20], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(-0.5 * N[(N[(-2.0 * N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(b * 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -2.0), $MachinePrecision] / N[(N[(N[(2.0 * N[(a * t$95$1), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.6000000000000002e100

   1. Initial program 51.6%

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

      1. sqr-neg 51.6%

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

      2. sqr-neg 51.6%

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

      3. associate-*l* 51.6%

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

      4. *-commutative 51.6%

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

      5. associate-/l* 51.5%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in b around inf 51.5%

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

      1. associate-*r/ 51.5%

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

      2. mul-1-neg 51.5%

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

   6. Simplified 51.5%

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

   7. Taylor expanded in b around -inf 94.0%

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

      1. mul-1-neg 94.0%

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

      2. distribute-neg-frac 94.0%

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

   9. Simplified 94.0%

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

   if -6.6000000000000002e100 < b < 3.9e20

   1. Initial program 88.8%

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

   if 3.9e20 < b

   1. Initial program 61.9%

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

   3. Simplified 61.9%

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

   4. Taylor expanded in b around -inf 61.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{\color{blue}{2}}{\left(-1 \cdot b + 2 \cdot \frac{a \cdot c}{b}\right) - b}\\ \end{array} \]
   5. Step-by-step derivation

      1. fma-def 61.9%

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

      2. associate-/l* 61.9%

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

   6. Simplified 61.9%

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

   7. Taylor expanded in b around inf 85.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b} + 2 \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\mathsf{fma}\left(-1, b, 2 \cdot \frac{a}{\frac{b}{c}}\right) - b}\\ \end{array} \]
   8. Step-by-step derivation

      1. fma-def 85.1%

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

      2. associate-/l* 95.7%

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

      3. *-commutative 95.7%

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

   9. Simplified 95.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\color{blue}{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, b \cdot 2\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\mathsf{fma}\left(-1, b, 2 \cdot \frac{a}{\frac{b}{c}}\right) - b}\\ \end{array} \]
   10. Step-by-step derivation

       1. associate-*r/ 95.7%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, b \cdot 2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(-1, b, 2 \cdot \frac{a}{\frac{b}{c}}\right) - b}\\ \end{array} \]

       2. frac-2neg 95.7%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, b \cdot 2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c \cdot 2}{-\left(\mathsf{fma}\left(-1, b, 2 \cdot \frac{a}{\frac{b}{c}}\right) - b\right)}\\ \end{array} \]

       3. sub-neg 95.7%

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

       4. fma-udef 95.7%

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

       5. neg-mul-1 95.7%

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

       6. add-sqr-sqrt 95.7%

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

       7. sqrt-unprod 95.7%

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

       8. sqr-neg 95.7%

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

       9. sqrt-prod 95.7%

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

       10. add-sqr-sqrt 95.7%

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

       11. associate-/r/ 95.7%

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

       12. add-sqr-sqrt 95.7%

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

       13. sqrt-unprod 95.7%

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

       14. sqr-neg 95.7%

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

       15. sqrt-prod 95.7%

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

       16. add-sqr-sqrt 95.7%

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

   11. Applied egg-rr 95.7%

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

       1. distribute-rgt-neg-in 95.7%

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

       2. metadata-eval 95.7%

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

       3. distribute-neg-in 95.7%

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

       4. unsub-neg 95.7%

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

       5. associate-*l/ 95.7%

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

       6. *-lft-identity 95.7%

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

       7. times-frac 95.7%

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

       8. /-rgt-identity 95.7%

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

   13. Simplified 95.7%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{+100}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{+20}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, b \cdot 2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{\left(2 \cdot \left(a \cdot \frac{-c}{b}\right) - b\right) - b}\\ \end{array} \]

Alternative 2: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\\ t_1 := \frac{-c}{b}\\ \mathbf{if}\;b \leq -1.02 \cdot 10^{+99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{+20}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t_0 - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, b \cdot 2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{\left(2 \cdot \left(a \cdot t_1\right) - b\right) - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* 4.0 (* a c))))) (t_1 (/ (- c) b)))
   (if (<= b -1.02e+99)
     (if (>= b 0.0) (- (/ b a)) t_1)
     (if (<= b 3.9e+20)
       (if (>= b 0.0) (/ (- (- b) t_0) (* a 2.0)) (/ 2.0 (/ (- t_0 b) c)))
       (if (>= b 0.0)
         (* -0.5 (/ (fma -2.0 (/ a (/ b c)) (* b 2.0)) a))
         (/ (* c -2.0) (- (- (* 2.0 (* a t_1)) b) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (4.0 * (a * c))));
	double t_1 = -c / b;
	double tmp_1;
	if (b <= -1.02e+99) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -(b / a);
		} else {
			tmp_2 = t_1;
		}
		tmp_1 = tmp_2;
	} else if (b <= 3.9e+20) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_0) / (a * 2.0);
		} else {
			tmp_3 = 2.0 / ((t_0 - b) / c);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -0.5 * (fma(-2.0, (a / (b / c)), (b * 2.0)) / a);
	} else {
		tmp_1 = (c * -2.0) / (((2.0 * (a * t_1)) - b) - b);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))
	t_1 = Float64(Float64(-c) / b)
	tmp_1 = 0.0
	if (b <= -1.02e+99)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-Float64(b / a));
		else
			tmp_2 = t_1;
		end
		tmp_1 = tmp_2;
	elseif (b <= 3.9e+20)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(-b) - t_0) / Float64(a * 2.0));
		else
			tmp_3 = Float64(2.0 / Float64(Float64(t_0 - b) / c));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(-0.5 * Float64(fma(-2.0, Float64(a / Float64(b / c)), Float64(b * 2.0)) / a));
	else
		tmp_1 = Float64(Float64(c * -2.0) / Float64(Float64(Float64(2.0 * Float64(a * t_1)) - b) - b));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-c) / b), $MachinePrecision]}, If[LessEqual[b, -1.02e+99], If[GreaterEqual[b, 0.0], (-N[(b / a), $MachinePrecision]), t$95$1], If[LessEqual[b, 3.9e+20], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$0 - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(-0.5 * N[(N[(-2.0 * N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(b * 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -2.0), $MachinePrecision] / N[(N[(N[(2.0 * N[(a * t$95$1), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.01999999999999998e99

   1. Initial program 51.6%

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

      1. sqr-neg 51.6%

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

      2. sqr-neg 51.6%

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

      3. associate-*l* 51.6%

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

      4. *-commutative 51.6%

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

      5. associate-/l* 51.5%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in b around inf 51.5%

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

      1. associate-*r/ 51.5%

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

      2. mul-1-neg 51.5%

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

   6. Simplified 51.5%

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

   7. Taylor expanded in b around -inf 94.0%

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

      1. mul-1-neg 94.0%

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

      2. distribute-neg-frac 94.0%

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

   9. Simplified 94.0%

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

   if -1.01999999999999998e99 < b < 3.9e20

   1. Initial program 88.8%

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

      1. sqr-neg 88.8%

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

      2. sqr-neg 88.8%

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

      3. associate-*l* 88.8%

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

      4. *-commutative 88.8%

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

      5. associate-/l* 88.5%

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

   3. Simplified 88.5%

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

   if 3.9e20 < b

   1. Initial program 61.9%

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

   3. Simplified 61.9%

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

   4. Taylor expanded in b around -inf 61.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{\color{blue}{2}}{\left(-1 \cdot b + 2 \cdot \frac{a \cdot c}{b}\right) - b}\\ \end{array} \]
   5. Step-by-step derivation

      1. fma-def 61.9%

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

      2. associate-/l* 61.9%

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

   6. Simplified 61.9%

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

   7. Taylor expanded in b around inf 85.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b} + 2 \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\mathsf{fma}\left(-1, b, 2 \cdot \frac{a}{\frac{b}{c}}\right) - b}\\ \end{array} \]
   8. Step-by-step derivation

      1. fma-def 85.1%

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

      2. associate-/l* 95.7%

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

      3. *-commutative 95.7%

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

   9. Simplified 95.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\color{blue}{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, b \cdot 2\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\mathsf{fma}\left(-1, b, 2 \cdot \frac{a}{\frac{b}{c}}\right) - b}\\ \end{array} \]
   10. Step-by-step derivation

       1. associate-*r/ 95.7%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, b \cdot 2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(-1, b, 2 \cdot \frac{a}{\frac{b}{c}}\right) - b}\\ \end{array} \]

       2. frac-2neg 95.7%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, b \cdot 2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c \cdot 2}{-\left(\mathsf{fma}\left(-1, b, 2 \cdot \frac{a}{\frac{b}{c}}\right) - b\right)}\\ \end{array} \]

       3. sub-neg 95.7%

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

       4. fma-udef 95.7%

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

       5. neg-mul-1 95.7%

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

       6. add-sqr-sqrt 95.7%

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

       7. sqrt-unprod 95.7%

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

       8. sqr-neg 95.7%

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

       9. sqrt-prod 95.7%

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

       10. add-sqr-sqrt 95.7%

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

       11. associate-/r/ 95.7%

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

       12. add-sqr-sqrt 95.7%

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

       13. sqrt-unprod 95.7%

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

       14. sqr-neg 95.7%

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

       15. sqrt-prod 95.7%

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

       16. add-sqr-sqrt 95.7%

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

   11. Applied egg-rr 95.7%

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

       1. distribute-rgt-neg-in 95.7%

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

       2. metadata-eval 95.7%

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

       3. distribute-neg-in 95.7%

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

       4. unsub-neg 95.7%

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

       5. associate-*l/ 95.7%

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

       6. *-lft-identity 95.7%

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

       7. times-frac 95.7%

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

       8. /-rgt-identity 95.7%

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

   13. Simplified 95.7%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{+99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{+20}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, b \cdot 2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{\left(2 \cdot \left(a \cdot \frac{-c}{b}\right) - b\right) - b}\\ \end{array} \]

Alternative 3: 78.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{b}{a}\\ \mathbf{if}\;b \leq -6.8 \cdot 10^{+99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (/ b a))))
   (if (<= b -6.8e+99)
     (if (>= b 0.0) t_0 (/ (- c) b))
     (if (>= b 0.0)
       t_0
       (/ 2.0 (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) c))))))

C

double code(double a, double b, double c) {
	double t_0 = -(b / a);
	double tmp_1;
	if (b <= -6.8e+99) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = -c / b;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = 2.0 / ((sqrt(((b * b) - (4.0 * (a * c)))) - b) / c);
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = -(b / a)
    if (b <= (-6.8d+99)) then
        if (b >= 0.0d0) then
            tmp_2 = t_0
        else
            tmp_2 = -c / b
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = t_0
    else
        tmp_1 = 2.0d0 / ((sqrt(((b * b) - (4.0d0 * (a * c)))) - b) / c)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -(b / a);
	double tmp_1;
	if (b <= -6.8e+99) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = -c / b;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = 2.0 / ((Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / c);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = -(b / a)
	tmp_1 = 0
	if b <= -6.8e+99:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_0
		else:
			tmp_2 = -c / b
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = t_0
	else:
		tmp_1 = 2.0 / ((math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / c)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(-Float64(b / a))
	tmp_1 = 0.0
	if (b <= -6.8e+99)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(Float64(-c) / b);
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = t_0;
	else
		tmp_1 = Float64(2.0 / Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / c));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = -(b / a);
	tmp_2 = 0.0;
	if (b <= -6.8e+99)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_0;
		else
			tmp_3 = -c / b;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = t_0;
	else
		tmp_2 = 2.0 / ((sqrt(((b * b) - (4.0 * (a * c)))) - b) / c);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = (-N[(b / a), $MachinePrecision])}, If[LessEqual[b, -6.8e+99], If[GreaterEqual[b, 0.0], t$95$0, N[((-c) / b), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$0, N[(2.0 / N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < -6.79999999999999968e99

   1. Initial program 51.6%

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

      1. sqr-neg 51.6%

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

      2. sqr-neg 51.6%

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

      3. associate-*l* 51.6%

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

      4. *-commutative 51.6%

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

      5. associate-/l* 51.5%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in b around inf 51.5%

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

      1. associate-*r/ 51.5%

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

      2. mul-1-neg 51.5%

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

   6. Simplified 51.5%

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

   7. Taylor expanded in b around -inf 94.0%

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

      1. mul-1-neg 94.0%

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

      2. distribute-neg-frac 94.0%

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

   9. Simplified 94.0%

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

   if -6.79999999999999968e99 < b

   1. Initial program 79.7%

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

      1. sqr-neg 79.7%

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

      2. sqr-neg 79.7%

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

      3. associate-*l* 79.7%

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

      4. *-commutative 79.7%

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

      5. associate-/l* 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in b around inf 77.4%

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

      1. associate-*r/ 77.4%

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

      2. mul-1-neg 77.4%

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

   6. Simplified 77.4%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.8 \cdot 10^{+99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{c}}\\ \end{array} \]

Alternative 4: 67.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \left(b \cdot \frac{2}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\mathsf{fma}\left(-1, b, 2 \cdot \frac{a}{\frac{b}{c}}\right) - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0)
   (* -0.5 (* b (/ 2.0 a)))
   (* c (/ 2.0 (- (fma -1.0 b (* 2.0 (/ a (/ b c)))) b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -0.5 * (b * (2.0 / a));
	} else {
		tmp = c * (2.0 / (fma(-1.0, b, (2.0 * (a / (b / c)))) - b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(-0.5 * Float64(b * Float64(2.0 / a)));
	else
		tmp = Float64(c * Float64(2.0 / Float64(fma(-1.0, b, Float64(2.0 * Float64(a / Float64(b / c)))) - b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(-0.5 * N[(b * N[(2.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(2.0 / N[(N[(-1.0 * b + N[(2.0 * N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 72.8%

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

3. Simplified 72.7%

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

4. Taylor expanded in b around -inf 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{\color{blue}{2}}{\left(-1 \cdot b + 2 \cdot \frac{a \cdot c}{b}\right) - b}\\ \end{array} \]
5. Step-by-step derivation

   1. fma-def 67.7%

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

   2. associate-/l* 68.5%

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

6. Simplified 68.5%

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

7. Taylor expanded in b around inf 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b} + 2 \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\mathsf{fma}\left(-1, b, 2 \cdot \frac{a}{\frac{b}{c}}\right) - b}\\ \end{array} \]
8. Step-by-step derivation

   1. fma-def 64.3%

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

   2. associate-/l* 67.1%

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

   3. *-commutative 67.1%

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

9. Simplified 67.1%

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

10. Taylor expanded in a around 0 66.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \color{blue}{\left(2 \cdot \frac{b}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\mathsf{fma}\left(-1, b, 2 \cdot \frac{a}{\frac{b}{c}}\right) - b}\\ \end{array} \]
11. Step-by-step derivation

    1. associate-*r/ 66.9%

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

    2. *-commutative 66.9%

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

    3. *-lft-identity 66.9%

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

    4. times-frac 66.9%

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

    5. /-rgt-identity 66.9%

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

12. Simplified 66.9%

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

13. Final simplification 66.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \left(b \cdot \frac{2}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\mathsf{fma}\left(-1, b, 2 \cdot \frac{a}{\frac{b}{c}}\right) - b}\\ \end{array} \]

Alternative 5: 67.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \left(\mathsf{fma}\left(-2, a \cdot \frac{c}{b}, b \cdot 2\right) \cdot \frac{1}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\mathsf{fma}\left(-1, b, 2 \cdot \frac{a}{\frac{b}{c}}\right) - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0)
   (* -0.5 (* (fma -2.0 (* a (/ c b)) (* b 2.0)) (/ 1.0 a)))
   (* c (/ 2.0 (- (fma -1.0 b (* 2.0 (/ a (/ b c)))) b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -0.5 * (fma(-2.0, (a * (c / b)), (b * 2.0)) * (1.0 / a));
	} else {
		tmp = c * (2.0 / (fma(-1.0, b, (2.0 * (a / (b / c)))) - b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(-0.5 * Float64(fma(-2.0, Float64(a * Float64(c / b)), Float64(b * 2.0)) * Float64(1.0 / a)));
	else
		tmp = Float64(c * Float64(2.0 / Float64(fma(-1.0, b, Float64(2.0 * Float64(a / Float64(b / c)))) - b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(-0.5 * N[(N[(-2.0 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(b * 2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(2.0 / N[(N[(-1.0 * b + N[(2.0 * N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 72.8%

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

3. Simplified 72.7%

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

4. Taylor expanded in b around -inf 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{\color{blue}{2}}{\left(-1 \cdot b + 2 \cdot \frac{a \cdot c}{b}\right) - b}\\ \end{array} \]
5. Step-by-step derivation

   1. fma-def 67.7%

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

   2. associate-/l* 68.5%

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

6. Simplified 68.5%

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

7. Taylor expanded in b around inf 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b} + 2 \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\mathsf{fma}\left(-1, b, 2 \cdot \frac{a}{\frac{b}{c}}\right) - b}\\ \end{array} \]
8. Step-by-step derivation

   1. fma-def 64.3%

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

   2. associate-/l* 67.1%

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

   3. *-commutative 67.1%

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

9. Simplified 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\color{blue}{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, b \cdot 2\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\mathsf{fma}\left(-1, b, 2 \cdot \frac{a}{\frac{b}{c}}\right) - b}\\ \end{array} \]
10. Step-by-step derivation

    1. div-inv 67.0%

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

    2. div-inv 67.0%

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

    3. clear-num 67.0%

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

11. Applied egg-rr 67.0%

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

12. Final simplification 67.0%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \left(\mathsf{fma}\left(-2, a \cdot \frac{c}{b}, b \cdot 2\right) \cdot \frac{1}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\mathsf{fma}\left(-1, b, 2 \cdot \frac{a}{\frac{b}{c}}\right) - b}\\ \end{array} \]

Alternative 6: 67.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, b \cdot 2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\mathsf{fma}\left(-1, b, 2 \cdot \left(a \cdot \frac{c}{b}\right)\right) - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0)
   (* -0.5 (/ (fma -2.0 (/ a (/ b c)) (* b 2.0)) a))
   (* c (/ 2.0 (- (fma -1.0 b (* 2.0 (* a (/ c b)))) b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -0.5 * (fma(-2.0, (a / (b / c)), (b * 2.0)) / a);
	} else {
		tmp = c * (2.0 / (fma(-1.0, b, (2.0 * (a * (c / b)))) - b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(-0.5 * Float64(fma(-2.0, Float64(a / Float64(b / c)), Float64(b * 2.0)) / a));
	else
		tmp = Float64(c * Float64(2.0 / Float64(fma(-1.0, b, Float64(2.0 * Float64(a * Float64(c / b)))) - b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(-0.5 * N[(N[(-2.0 * N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(b * 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(2.0 / N[(N[(-1.0 * b + N[(2.0 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 72.8%

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

3. Simplified 72.7%

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

4. Taylor expanded in b around -inf 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{\color{blue}{2}}{\left(-1 \cdot b + 2 \cdot \frac{a \cdot c}{b}\right) - b}\\ \end{array} \]
5. Step-by-step derivation

   1. fma-def 67.7%

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

   2. associate-/l* 68.5%

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

6. Simplified 68.5%

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

7. Taylor expanded in b around inf 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b} + 2 \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\mathsf{fma}\left(-1, b, 2 \cdot \frac{a}{\frac{b}{c}}\right) - b}\\ \end{array} \]
8. Step-by-step derivation

   1. fma-def 64.3%

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

   2. associate-/l* 67.1%

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

   3. *-commutative 67.1%

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

9. Simplified 67.1%

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

10. Taylor expanded in a around 0 66.3%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, b \cdot 2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\mathsf{fma}\left(-1, b, 2 \cdot \frac{a \cdot c}{b}\right) - b}\\ \end{array} \]
11. Step-by-step derivation

    1. *-lft-identity 66.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, b \cdot 2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\mathsf{fma}\left(-1, b, 2 \cdot \frac{a \cdot c}{1 \cdot b}\right) - b}\\ \end{array} \]

    2. times-frac 67.0%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, b \cdot 2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\mathsf{fma}\left(-1, b, 2 \cdot \left(\frac{a}{1} \cdot \frac{c}{b}\right)\right) - b}\\ \end{array} \]

    3. /-rgt-identity 67.0%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, b \cdot 2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\mathsf{fma}\left(-1, b, 2 \cdot \left(a \cdot \frac{c}{b}\right)\right) - b}\\ \end{array} \]

12. Simplified 67.0%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, b \cdot 2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\mathsf{fma}\left(-1, b, 2 \cdot \left(a \cdot \frac{c}{b}\right)\right) - b}\\ \end{array} \]

13. Final simplification 67.0%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, b \cdot 2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\mathsf{fma}\left(-1, b, 2 \cdot \left(a \cdot \frac{c}{b}\right)\right) - b}\\ \end{array} \]

Alternative 7: 67.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{\frac{b}{c}}\\ \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\mathsf{fma}\left(-2, t_0, b \cdot 2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\mathsf{fma}\left(-1, b, 2 \cdot t_0\right) - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ a (/ b c))))
   (if (>= b 0.0)
     (* -0.5 (/ (fma -2.0 t_0 (* b 2.0)) a))
     (* c (/ 2.0 (- (fma -1.0 b (* 2.0 t_0)) b))))))

C

double code(double a, double b, double c) {
	double t_0 = a / (b / c);
	double tmp;
	if (b >= 0.0) {
		tmp = -0.5 * (fma(-2.0, t_0, (b * 2.0)) / a);
	} else {
		tmp = c * (2.0 / (fma(-1.0, b, (2.0 * t_0)) - b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(a / Float64(b / c))
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(-0.5 * Float64(fma(-2.0, t_0, Float64(b * 2.0)) / a));
	else
		tmp = Float64(c * Float64(2.0 / Float64(fma(-1.0, b, Float64(2.0 * t_0)) - b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision]}, If[GreaterEqual[b, 0.0], N[(-0.5 * N[(N[(-2.0 * t$95$0 + N[(b * 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(2.0 / N[(N[(-1.0 * b + N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 72.8%

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

3. Simplified 72.7%

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

4. Taylor expanded in b around -inf 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{\color{blue}{2}}{\left(-1 \cdot b + 2 \cdot \frac{a \cdot c}{b}\right) - b}\\ \end{array} \]
5. Step-by-step derivation

   1. fma-def 67.7%

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

   2. associate-/l* 68.5%

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

6. Simplified 68.5%

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

7. Taylor expanded in b around inf 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b} + 2 \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\mathsf{fma}\left(-1, b, 2 \cdot \frac{a}{\frac{b}{c}}\right) - b}\\ \end{array} \]
8. Step-by-step derivation

   1. fma-def 64.3%

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

   2. associate-/l* 67.1%

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

   3. *-commutative 67.1%

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

9. Simplified 67.1%

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

10. Final simplification 67.1%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, b \cdot 2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\mathsf{fma}\left(-1, b, 2 \cdot \frac{a}{\frac{b}{c}}\right) - b}\\ \end{array} \]

Alternative 8: 67.7% accurate, 19.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.8%

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

   1. sqr-neg 72.8%

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

   2. sqr-neg 72.8%

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

   3. associate-*l* 72.8%

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

   4. *-commutative 72.8%

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

   5. associate-/l* 72.6%

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

3. Simplified 72.6%

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

4. Taylor expanded in b around inf 71.0%

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

   1. associate-*r/ 71.0%

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

   2. mul-1-neg 71.0%

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

6. Simplified 71.0%

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

7. Taylor expanded in b around -inf 66.8%

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

   1. mul-1-neg 66.8%

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

   2. distribute-neg-frac 66.8%

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

9. Simplified 66.8%

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

10. Final simplification 66.8%

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

Alternative 9: 35.2% accurate, 29.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.8%

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

   1. sqr-neg 72.8%

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

   2. sqr-neg 72.8%

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

   3. associate-*l* 72.8%

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

   4. *-commutative 72.8%

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

   5. associate-/l* 72.6%

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

3. Simplified 72.6%

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

4. Taylor expanded in b around inf 71.0%

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

   1. associate-*r/ 71.0%

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

   2. mul-1-neg 71.0%

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

6. Simplified 71.0%

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

7. Taylor expanded in b around inf 32.4%

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

   1. associate-*r/ 32.4%

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

   2. mul-1-neg 32.4%

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

9. Simplified 32.4%

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

10. Final simplification 32.4%

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

Reproduce

herbie shell --seed 2023298 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  :precision binary64
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))