jeff quadratic root 1

Percentage Accurate: 73.1% → 89.7%

Time: 18.6s

Alternatives: 10

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: 73.1% 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: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -2 \cdot \frac{c}{\frac{b}{a}}\\ \mathbf{if}\;b \leq -5 \cdot 10^{+113}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \left(b + t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \end{array}\\ \mathbf{elif}\;b \leq -1.6 \cdot 10^{-278}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \mathsf{fma}\left(b, 2, t_0\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-2}{b - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}\\ \end{array}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+126}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{b \cdot -2}{c}}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* -2.0 (/ c (/ b a)))))
   (if (<= b -5e+113)
     (if (>= b 0.0) (* (/ -0.5 a) (+ b (+ b t_0))) (/ (* -2.0 c) (+ b b)))
     (if (<= b -1.6e-278)
       (if (>= b 0.0)
         (* (/ -0.5 a) (fma b 2.0 t_0))
         (* c (/ -2.0 (- b (sqrt (+ (* b b) (* c (* a -4.0))))))))
       (if (<= b 2e+126)
         (if (>= b 0.0)
           (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* a 2.0))
           (/ 2.0 (/ (* b -2.0) c)))
         (if (>= b 0.0) (- (/ c b) (/ b a)) (* c (/ -2.0 (+ b b)))))))))

C

double code(double a, double b, double c) {
	double t_0 = -2.0 * (c / (b / a));
	double tmp_1;
	if (b <= -5e+113) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-0.5 / a) * (b + (b + t_0));
		} else {
			tmp_2 = (-2.0 * c) / (b + b);
		}
		tmp_1 = tmp_2;
	} else if (b <= -1.6e-278) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-0.5 / a) * fma(b, 2.0, t_0);
		} else {
			tmp_3 = c * (-2.0 / (b - sqrt(((b * b) + (c * (a * -4.0))))));
		}
		tmp_1 = tmp_3;
	} else if (b <= 2e+126) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (a * 2.0);
		} else {
			tmp_4 = 2.0 / ((b * -2.0) / c);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = c * (-2.0 / (b + b));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(-2.0 * Float64(c / Float64(b / a)))
	tmp_1 = 0.0
	if (b <= -5e+113)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-0.5 / a) * Float64(b + Float64(b + t_0)));
		else
			tmp_2 = Float64(Float64(-2.0 * c) / Float64(b + b));
		end
		tmp_1 = tmp_2;
	elseif (b <= -1.6e-278)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-0.5 / a) * fma(b, 2.0, t_0));
		else
			tmp_3 = Float64(c * Float64(-2.0 / Float64(b - sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0)))))));
		end
		tmp_1 = tmp_3;
	elseif (b <= 2e+126)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(a * 2.0));
		else
			tmp_4 = Float64(2.0 / Float64(Float64(b * -2.0) / c));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp_1 = Float64(c * Float64(-2.0 / Float64(b + b)));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-2.0 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -5e+113], If[GreaterEqual[b, 0.0], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[(b + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * c), $MachinePrecision] / N[(b + b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -1.6e-278], If[GreaterEqual[b, 0.0], N[(N[(-0.5 / a), $MachinePrecision] * N[(b * 2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(c * N[(-2.0 / N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 2e+126], If[GreaterEqual[b, 0.0], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(b * -2.0), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-2.0 / N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -5e113

   1. Initial program 46.9%

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

   3. Simplified 46.8%

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

   4. Taylor expanded in b around -inf 92.7%

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

   5. Taylor expanded in b around inf 92.7%

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

      1. associate-/l* 92.7%

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

   7. Simplified 92.7%

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

      1. associate-*r/ 93.0%

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

      2. cancel-sign-sub-inv 93.0%

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

      3. metadata-eval 93.0%

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

      4. *-un-lft-identity 93.0%

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

   9. Applied egg-rr 93.0%

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

   if -5e113 < b < -1.60000000000000009e-278

   1. Initial program 89.3%

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

   3. Simplified 89.1%

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

   4. Taylor expanded in b around inf 89.1%

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

      1. +-commutative 89.1%

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

      2. *-commutative 89.1%

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

      3. fma-def 89.1%

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

      4. associate-/l* 89.1%

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

   6. Simplified 89.1%

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

      1. fma-udef 89.1%

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

      2. *-commutative 89.1%

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

      3. associate-*l* 89.1%

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

      4. *-commutative 89.1%

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

   8. Applied egg-rr 89.1%

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

   if -1.60000000000000009e-278 < b < 1.99999999999999985e126

   1. Initial program 79.8%

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

      1. associate-*l* 79.8%

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

      2. *-commutative 79.8%

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

      3. associate-/l* 79.8%

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

      4. associate-*l* 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in b around -inf 80.9%

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

      1. associate-*r/ 80.9%

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

      2. *-commutative 80.9%

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

   6. Simplified 80.9%

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

   if 1.99999999999999985e126 < b

   1. Initial program 39.6%

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

   3. Simplified 39.5%

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

   4. Taylor expanded in b around -inf 39.5%

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

   5. Taylor expanded in a around 0 98.1%

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

      1. mul-1-neg 98.1%

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

      2. unsub-neg 98.1%

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

   7. Simplified 98.1%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+113}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \left(b + -2 \cdot \frac{c}{\frac{b}{a}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \end{array}\\ \mathbf{elif}\;b \leq -1.6 \cdot 10^{-278}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \mathsf{fma}\left(b, 2, -2 \cdot \frac{c}{\frac{b}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-2}{b - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}\\ \end{array}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+126}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{b \cdot -2}{c}}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \end{array} \]

Alternative 2: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \frac{-2}{b + b}\\ \mathbf{if}\;b \leq -1.75 \cdot 10^{-99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{a \cdot \frac{c}{b} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-2}{\left(b + b\right) - \frac{c}{b} \cdot \left(a \cdot 2\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq -1.6 \cdot 10^{-278}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+125}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (/ -2.0 (+ b b)))))
   (if (<= b -1.75e-99)
     (if (>= b 0.0)
       (/ (- (* a (/ c b)) b) a)
       (* c (/ -2.0 (- (+ b b) (* (/ c b) (* a 2.0))))))
     (if (<= b -1.6e-278)
       (if (>= b 0.0)
         (/ (* b -2.0) (* a 2.0))
         (/ 2.0 (/ (- (sqrt (* (* a c) -4.0)) b) c)))
       (if (<= b 5.2e+125)
         (if (>= b 0.0)
           (* (/ -0.5 a) (+ b (sqrt (+ (* b b) (* c (* a -4.0))))))
           t_0)
         (if (>= b 0.0) (- (/ c b) (/ b a)) t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (-2.0 / (b + b));
	double tmp_1;
	if (b <= -1.75e-99) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = ((a * (c / b)) - b) / a;
		} else {
			tmp_2 = c * (-2.0 / ((b + b) - ((c / b) * (a * 2.0))));
		}
		tmp_1 = tmp_2;
	} else if (b <= -1.6e-278) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (b * -2.0) / (a * 2.0);
		} else {
			tmp_3 = 2.0 / ((sqrt(((a * c) * -4.0)) - b) / c);
		}
		tmp_1 = tmp_3;
	} else if (b <= 5.2e+125) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (-0.5 / a) * (b + sqrt(((b * b) + (c * (a * -4.0)))));
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    real(8) :: tmp_4
    t_0 = c * ((-2.0d0) / (b + b))
    if (b <= (-1.75d-99)) then
        if (b >= 0.0d0) then
            tmp_2 = ((a * (c / b)) - b) / a
        else
            tmp_2 = c * ((-2.0d0) / ((b + b) - ((c / b) * (a * 2.0d0))))
        end if
        tmp_1 = tmp_2
    else if (b <= (-1.6d-278)) then
        if (b >= 0.0d0) then
            tmp_3 = (b * (-2.0d0)) / (a * 2.0d0)
        else
            tmp_3 = 2.0d0 / ((sqrt(((a * c) * (-4.0d0))) - b) / c)
        end if
        tmp_1 = tmp_3
    else if (b <= 5.2d+125) then
        if (b >= 0.0d0) then
            tmp_4 = ((-0.5d0) / a) * (b + sqrt(((b * b) + (c * (a * (-4.0d0))))))
        else
            tmp_4 = t_0
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = (c / b) - (b / a)
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (-2.0 / (b + b));
	double tmp_1;
	if (b <= -1.75e-99) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = ((a * (c / b)) - b) / a;
		} else {
			tmp_2 = c * (-2.0 / ((b + b) - ((c / b) * (a * 2.0))));
		}
		tmp_1 = tmp_2;
	} else if (b <= -1.6e-278) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (b * -2.0) / (a * 2.0);
		} else {
			tmp_3 = 2.0 / ((Math.sqrt(((a * c) * -4.0)) - b) / c);
		}
		tmp_1 = tmp_3;
	} else if (b <= 5.2e+125) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (-0.5 / a) * (b + Math.sqrt(((b * b) + (c * (a * -4.0)))));
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = c * (-2.0 / (b + b))
	tmp_1 = 0
	if b <= -1.75e-99:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = ((a * (c / b)) - b) / a
		else:
			tmp_2 = c * (-2.0 / ((b + b) - ((c / b) * (a * 2.0))))
		tmp_1 = tmp_2
	elif b <= -1.6e-278:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (b * -2.0) / (a * 2.0)
		else:
			tmp_3 = 2.0 / ((math.sqrt(((a * c) * -4.0)) - b) / c)
		tmp_1 = tmp_3
	elif b <= 5.2e+125:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = (-0.5 / a) * (b + math.sqrt(((b * b) + (c * (a * -4.0)))))
		else:
			tmp_4 = t_0
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = (c / b) - (b / a)
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(-2.0 / Float64(b + b)))
	tmp_1 = 0.0
	if (b <= -1.75e-99)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(Float64(a * Float64(c / b)) - b) / a);
		else
			tmp_2 = Float64(c * Float64(-2.0 / Float64(Float64(b + b) - Float64(Float64(c / b) * Float64(a * 2.0)))));
		end
		tmp_1 = tmp_2;
	elseif (b <= -1.6e-278)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(b * -2.0) / Float64(a * 2.0));
		else
			tmp_3 = Float64(2.0 / Float64(Float64(sqrt(Float64(Float64(a * c) * -4.0)) - b) / c));
		end
		tmp_1 = tmp_3;
	elseif (b <= 5.2e+125)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(-0.5 / a) * Float64(b + sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0))))));
		else
			tmp_4 = t_0;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = c * (-2.0 / (b + b));
	tmp_2 = 0.0;
	if (b <= -1.75e-99)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = ((a * (c / b)) - b) / a;
		else
			tmp_3 = c * (-2.0 / ((b + b) - ((c / b) * (a * 2.0))));
		end
		tmp_2 = tmp_3;
	elseif (b <= -1.6e-278)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (b * -2.0) / (a * 2.0);
		else
			tmp_4 = 2.0 / ((sqrt(((a * c) * -4.0)) - b) / c);
		end
		tmp_2 = tmp_4;
	elseif (b <= 5.2e+125)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = (-0.5 / a) * (b + sqrt(((b * b) + (c * (a * -4.0)))));
		else
			tmp_5 = t_0;
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = (c / b) - (b / a);
	else
		tmp_2 = t_0;
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(-2.0 / N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.75e-99], If[GreaterEqual[b, 0.0], N[(N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision], N[(c * N[(-2.0 / N[(N[(b + b), $MachinePrecision] - N[(N[(c / b), $MachinePrecision] * N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -1.6e-278], If[GreaterEqual[b, 0.0], N[(N[(b * -2.0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 5.2e+125], If[GreaterEqual[b, 0.0], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.7499999999999999e-99

   1. Initial program 69.9%

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

   3. Simplified 69.7%

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

   4. Taylor expanded in b around inf 69.7%

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

      1. +-commutative 69.7%

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

      2. *-commutative 69.7%

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

      3. fma-def 69.7%

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

      4. associate-/l* 69.7%

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

   6. Simplified 69.7%

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

   7. Taylor expanded in b around -inf 80.3%

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

      1. fma-def 80.3%

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

      2. associate-/l* 85.2%

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

      3. mul-1-neg 85.2%

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

   9. Simplified 85.2%

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

       1. associate-*r/ 85.5%

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

   11. Applied egg-rr 85.5%

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

       1. associate-*r/ 85.2%

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

       2. fma-udef 85.2%

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

       3. associate-/l* 80.3%

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

       4. neg-mul-1 80.3%

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

       5. +-commutative 80.3%

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

       6. associate--r+ 80.3%

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

       7. cancel-sign-sub-inv 80.3%

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

       8. metadata-eval 80.3%

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

       9. *-lft-identity 80.3%

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

       10. *-commutative 80.3%

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

       11. associate-/l* 85.2%

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

       12. associate-/r/ 85.2%

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

       13. associate-*l* 85.2%

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

   13. Simplified 85.2%

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

       1. associate-*l/ 85.2%

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

   15. Applied egg-rr 85.2%

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

       1. fma-udef 85.2%

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

       2. *-commutative 85.2%

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

       3. count-2 85.2%

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

       4. distribute-rgt-in 85.2%

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

       5. count-2 85.2%

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

       6. *-commutative 85.2%

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

       7. associate-*l* 85.2%

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

       8. metadata-eval 85.2%

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

       9. *-commutative 85.2%

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

       10. neg-mul-1 85.2%

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

       11. associate-/l* 85.2%

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

       12. *-commutative 85.2%

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

       13. associate-*l* 85.2%

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

       14. associate-/l* 85.2%

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

       15. associate-/r/ 85.2%

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

       16. metadata-eval 85.2%

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

   17. Simplified 85.2%

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

   if -1.7499999999999999e-99 < b < -1.60000000000000009e-278

   1. Initial program 78.6%

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

      1. associate-*l* 78.6%

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

      2. *-commutative 78.6%

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

      3. associate-/l* 78.5%

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

      4. associate-*l* 78.5%

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

   3. Simplified 78.5%

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

   4. Taylor expanded in b around inf 78.5%

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

      1. *-commutative 78.5%

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

   6. Simplified 78.5%

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

   7. Taylor expanded in b around 0 67.4%

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

      1. *-commutative 67.4%

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

   9. Simplified 67.4%

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

   if -1.60000000000000009e-278 < b < 5.20000000000000006e125

   1. Initial program 79.8%

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

   3. Simplified 79.7%

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

   4. Taylor expanded in b around -inf 80.7%

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

      1. fma-udef 46.6%

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

      2. *-commutative 46.6%

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

      3. associate-*l* 46.6%

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

      4. *-commutative 46.6%

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

   6. Applied egg-rr 80.7%

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

   if 5.20000000000000006e125 < b

   1. Initial program 39.6%

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

   3. Simplified 39.5%

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

   4. Taylor expanded in b around -inf 39.5%

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

   5. Taylor expanded in a around 0 98.1%

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

      1. mul-1-neg 98.1%

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

      2. unsub-neg 98.1%

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

   7. Simplified 98.1%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{-99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{a \cdot \frac{c}{b} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-2}{\left(b + b\right) - \frac{c}{b} \cdot \left(a \cdot 2\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq -1.6 \cdot 10^{-278}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+125}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \end{array} \]

Alternative 3: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -2 \cdot \frac{c}{\frac{b}{a}}\\ t_1 := \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\\ t_2 := c \cdot \frac{-2}{b + b}\\ \mathbf{if}\;b \leq -2.2 \cdot 10^{+116}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \left(b + t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \end{array}\\ \mathbf{elif}\;b \leq -1.6 \cdot 10^{-278}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \mathsf{fma}\left(b, 2, t_0\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-2}{b - t_1}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+125}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* -2.0 (/ c (/ b a))))
        (t_1 (sqrt (+ (* b b) (* c (* a -4.0)))))
        (t_2 (* c (/ -2.0 (+ b b)))))
   (if (<= b -2.2e+116)
     (if (>= b 0.0) (* (/ -0.5 a) (+ b (+ b t_0))) (/ (* -2.0 c) (+ b b)))
     (if (<= b -1.6e-278)
       (if (>= b 0.0) (* (/ -0.5 a) (fma b 2.0 t_0)) (* c (/ -2.0 (- b t_1))))
       (if (<= b 4.4e+125)
         (if (>= b 0.0) (* (/ -0.5 a) (+ b t_1)) t_2)
         (if (>= b 0.0) (- (/ c b) (/ b a)) t_2))))))

C

double code(double a, double b, double c) {
	double t_0 = -2.0 * (c / (b / a));
	double t_1 = sqrt(((b * b) + (c * (a * -4.0))));
	double t_2 = c * (-2.0 / (b + b));
	double tmp_1;
	if (b <= -2.2e+116) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-0.5 / a) * (b + (b + t_0));
		} else {
			tmp_2 = (-2.0 * c) / (b + b);
		}
		tmp_1 = tmp_2;
	} else if (b <= -1.6e-278) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-0.5 / a) * fma(b, 2.0, t_0);
		} else {
			tmp_3 = c * (-2.0 / (b - t_1));
		}
		tmp_1 = tmp_3;
	} else if (b <= 4.4e+125) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (-0.5 / a) * (b + t_1);
		} else {
			tmp_4 = t_2;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = t_2;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(-2.0 * Float64(c / Float64(b / a)))
	t_1 = sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0))))
	t_2 = Float64(c * Float64(-2.0 / Float64(b + b)))
	tmp_1 = 0.0
	if (b <= -2.2e+116)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-0.5 / a) * Float64(b + Float64(b + t_0)));
		else
			tmp_2 = Float64(Float64(-2.0 * c) / Float64(b + b));
		end
		tmp_1 = tmp_2;
	elseif (b <= -1.6e-278)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-0.5 / a) * fma(b, 2.0, t_0));
		else
			tmp_3 = Float64(c * Float64(-2.0 / Float64(b - t_1)));
		end
		tmp_1 = tmp_3;
	elseif (b <= 4.4e+125)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(-0.5 / a) * Float64(b + t_1));
		else
			tmp_4 = t_2;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp_1 = t_2;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-2.0 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(c * N[(-2.0 / N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.2e+116], If[GreaterEqual[b, 0.0], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[(b + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * c), $MachinePrecision] / N[(b + b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -1.6e-278], If[GreaterEqual[b, 0.0], N[(N[(-0.5 / a), $MachinePrecision] * N[(b * 2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(c * N[(-2.0 / N[(b - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 4.4e+125], If[GreaterEqual[b, 0.0], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2], If[GreaterEqual[b, 0.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.2e116

   1. Initial program 46.9%

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

   3. Simplified 46.8%

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

   4. Taylor expanded in b around -inf 92.7%

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

   5. Taylor expanded in b around inf 92.7%

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

      1. associate-/l* 92.7%

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

   7. Simplified 92.7%

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

      1. associate-*r/ 93.0%

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

      2. cancel-sign-sub-inv 93.0%

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

      3. metadata-eval 93.0%

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

      4. *-un-lft-identity 93.0%

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

   9. Applied egg-rr 93.0%

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

   if -2.2e116 < b < -1.60000000000000009e-278

   1. Initial program 89.3%

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

   3. Simplified 89.1%

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

   4. Taylor expanded in b around inf 89.1%

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

      1. +-commutative 89.1%

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

      2. *-commutative 89.1%

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

      3. fma-def 89.1%

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

      4. associate-/l* 89.1%

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

   6. Simplified 89.1%

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

      1. fma-udef 89.1%

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

      2. *-commutative 89.1%

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

      3. associate-*l* 89.1%

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

      4. *-commutative 89.1%

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

   8. Applied egg-rr 89.1%

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

   if -1.60000000000000009e-278 < b < 4.39999999999999982e125

   1. Initial program 79.8%

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

   3. Simplified 79.7%

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

   4. Taylor expanded in b around -inf 80.7%

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

      1. fma-udef 46.6%

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

      2. *-commutative 46.6%

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

      3. associate-*l* 46.6%

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

      4. *-commutative 46.6%

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

   6. Applied egg-rr 80.7%

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

   if 4.39999999999999982e125 < b

   1. Initial program 39.6%

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

   3. Simplified 39.5%

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

   4. Taylor expanded in b around -inf 39.5%

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

   5. Taylor expanded in a around 0 98.1%

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

      1. mul-1-neg 98.1%

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

      2. unsub-neg 98.1%

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

   7. Simplified 98.1%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{+116}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \left(b + -2 \cdot \frac{c}{\frac{b}{a}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \end{array}\\ \mathbf{elif}\;b \leq -1.6 \cdot 10^{-278}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \mathsf{fma}\left(b, 2, -2 \cdot \frac{c}{\frac{b}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-2}{b - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+125}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \end{array} \]

Alternative 4: 91.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ \mathbf{if}\;b \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \left(b + -2 \cdot \frac{c}{\frac{b}{a}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \end{array}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+125}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0))))))
   (if (<= b -2e+125)
     (if (>= b 0.0)
       (* (/ -0.5 a) (+ b (+ b (* -2.0 (/ c (/ b a))))))
       (/ (* -2.0 c) (+ b b)))
     (if (<= b 4e+125)
       (if (>= b 0.0) (/ (- (- b) t_0) (* a 2.0)) (/ (* c 2.0) (- t_0 b)))
       (if (>= b 0.0) (- (/ c b) (/ b a)) (* c (/ -2.0 (+ b b))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -2e+125) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-0.5 / a) * (b + (b + (-2.0 * (c / (b / a)))));
		} else {
			tmp_2 = (-2.0 * c) / (b + b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 4e+125) {
		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 = (c / b) - (b / a);
	} else {
		tmp_1 = c * (-2.0 / (b + b));
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = sqrt(((b * b) - (c * (a * 4.0d0))))
    if (b <= (-2d+125)) then
        if (b >= 0.0d0) then
            tmp_2 = ((-0.5d0) / a) * (b + (b + ((-2.0d0) * (c / (b / a)))))
        else
            tmp_2 = ((-2.0d0) * c) / (b + b)
        end if
        tmp_1 = tmp_2
    else if (b <= 4d+125) then
        if (b >= 0.0d0) then
            tmp_3 = (-b - t_0) / (a * 2.0d0)
        else
            tmp_3 = (c * 2.0d0) / (t_0 - b)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = (c / b) - (b / a)
    else
        tmp_1 = c * ((-2.0d0) / (b + b))
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -2e+125) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-0.5 / a) * (b + (b + (-2.0 * (c / (b / a)))));
		} else {
			tmp_2 = (-2.0 * c) / (b + b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 4e+125) {
		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 = (c / b) - (b / a);
	} else {
		tmp_1 = c * (-2.0 / (b + b));
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (c * (a * 4.0))))
	tmp_1 = 0
	if b <= -2e+125:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (-0.5 / a) * (b + (b + (-2.0 * (c / (b / a)))))
		else:
			tmp_2 = (-2.0 * c) / (b + b)
		tmp_1 = tmp_2
	elif b <= 4e+125:
		tmp_3 = 0
		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
	elif b >= 0.0:
		tmp_1 = (c / b) - (b / a)
	else:
		tmp_1 = c * (-2.0 / (b + b))
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	tmp_1 = 0.0
	if (b <= -2e+125)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-0.5 / a) * Float64(b + Float64(b + Float64(-2.0 * Float64(c / Float64(b / a))))));
		else
			tmp_2 = Float64(Float64(-2.0 * c) / Float64(b + b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 4e+125)
		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(Float64(c / b) - Float64(b / a));
	else
		tmp_1 = Float64(c * Float64(-2.0 / Float64(b + b)));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	tmp_2 = 0.0;
	if (b <= -2e+125)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (-0.5 / a) * (b + (b + (-2.0 * (c / (b / a)))));
		else
			tmp_3 = (-2.0 * c) / (b + b);
		end
		tmp_2 = tmp_3;
	elseif (b <= 4e+125)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (-b - t_0) / (a * 2.0);
		else
			tmp_4 = (c * 2.0) / (t_0 - b);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = (c / b) - (b / a);
	else
		tmp_2 = c * (-2.0 / (b + b));
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2e+125], If[GreaterEqual[b, 0.0], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[(b + N[(-2.0 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * c), $MachinePrecision] / N[(b + b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 4e+125], 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[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-2.0 / N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.9999999999999998e125

   1. Initial program 42.6%

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

   3. Simplified 42.6%

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

   4. Taylor expanded in b around -inf 92.2%

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

   5. Taylor expanded in b around inf 92.2%

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

      1. associate-/l* 92.2%

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

   7. Simplified 92.2%

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

      1. associate-*r/ 92.4%

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

      2. cancel-sign-sub-inv 92.4%

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

      3. metadata-eval 92.4%

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

      4. *-un-lft-identity 92.4%

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

   9. Applied egg-rr 92.4%

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

   if -1.9999999999999998e125 < b < 3.9999999999999997e125

   1. Initial program 84.9%

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

   if 3.9999999999999997e125 < b

   1. Initial program 39.6%

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

   3. Simplified 39.5%

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

   4. Taylor expanded in b around -inf 39.5%

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

   5. Taylor expanded in a around 0 98.1%

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

      1. mul-1-neg 98.1%

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

      2. unsub-neg 98.1%

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

   7. Simplified 98.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c}{b} - \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-2}{b - -1 \cdot b}\\ \end{array} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \left(b + -2 \cdot \frac{c}{\frac{b}{a}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \end{array}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+125}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \end{array} \]

Alternative 5: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b \leq -3.5 \cdot 10^{+103}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \left(b + -2 \cdot \frac{c}{\frac{b}{a}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \end{array}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+125}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t_0 - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* 4.0 (* a c))))))
   (if (<= b -3.5e+103)
     (if (>= b 0.0)
       (* (/ -0.5 a) (+ b (+ b (* -2.0 (/ c (/ b a))))))
       (/ (* -2.0 c) (+ b b)))
     (if (<= b 4e+125)
       (if (>= b 0.0) (/ (- (- b) t_0) (* a 2.0)) (/ 2.0 (/ (- t_0 b) c)))
       (if (>= b 0.0) (- (/ c b) (/ b a)) (* c (/ -2.0 (+ b b))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (4.0 * (a * c))));
	double tmp_1;
	if (b <= -3.5e+103) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-0.5 / a) * (b + (b + (-2.0 * (c / (b / a)))));
		} else {
			tmp_2 = (-2.0 * c) / (b + b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 4e+125) {
		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 = (c / b) - (b / a);
	} else {
		tmp_1 = c * (-2.0 / (b + b));
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = sqrt(((b * b) - (4.0d0 * (a * c))))
    if (b <= (-3.5d+103)) then
        if (b >= 0.0d0) then
            tmp_2 = ((-0.5d0) / a) * (b + (b + ((-2.0d0) * (c / (b / a)))))
        else
            tmp_2 = ((-2.0d0) * c) / (b + b)
        end if
        tmp_1 = tmp_2
    else if (b <= 4d+125) then
        if (b >= 0.0d0) then
            tmp_3 = (-b - t_0) / (a * 2.0d0)
        else
            tmp_3 = 2.0d0 / ((t_0 - b) / c)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = (c / b) - (b / a)
    else
        tmp_1 = c * ((-2.0d0) / (b + b))
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (4.0 * (a * c))));
	double tmp_1;
	if (b <= -3.5e+103) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-0.5 / a) * (b + (b + (-2.0 * (c / (b / a)))));
		} else {
			tmp_2 = (-2.0 * c) / (b + b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 4e+125) {
		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 = (c / b) - (b / a);
	} else {
		tmp_1 = c * (-2.0 / (b + b));
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (4.0 * (a * c))))
	tmp_1 = 0
	if b <= -3.5e+103:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (-0.5 / a) * (b + (b + (-2.0 * (c / (b / a)))))
		else:
			tmp_2 = (-2.0 * c) / (b + b)
		tmp_1 = tmp_2
	elif b <= 4e+125:
		tmp_3 = 0
		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
	elif b >= 0.0:
		tmp_1 = (c / b) - (b / a)
	else:
		tmp_1 = c * (-2.0 / (b + b))
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))
	tmp_1 = 0.0
	if (b <= -3.5e+103)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-0.5 / a) * Float64(b + Float64(b + Float64(-2.0 * Float64(c / Float64(b / a))))));
		else
			tmp_2 = Float64(Float64(-2.0 * c) / Float64(b + b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 4e+125)
		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(Float64(c / b) - Float64(b / a));
	else
		tmp_1 = Float64(c * Float64(-2.0 / Float64(b + b)));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) - (4.0 * (a * c))));
	tmp_2 = 0.0;
	if (b <= -3.5e+103)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (-0.5 / a) * (b + (b + (-2.0 * (c / (b / a)))));
		else
			tmp_3 = (-2.0 * c) / (b + b);
		end
		tmp_2 = tmp_3;
	elseif (b <= 4e+125)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (-b - t_0) / (a * 2.0);
		else
			tmp_4 = 2.0 / ((t_0 - b) / c);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = (c / b) - (b / a);
	else
		tmp_2 = c * (-2.0 / (b + b));
	end
	tmp_5 = tmp_2;
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]}, If[LessEqual[b, -3.5e+103], If[GreaterEqual[b, 0.0], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[(b + N[(-2.0 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * c), $MachinePrecision] / N[(b + b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 4e+125], 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[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-2.0 / N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.5e103

   1. Initial program 50.6%

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

   3. Simplified 50.6%

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

   4. Taylor expanded in b around -inf 93.2%

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

   5. Taylor expanded in b around inf 93.2%

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

      1. associate-/l* 93.2%

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

   7. Simplified 93.2%

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

      1. associate-*r/ 93.5%

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

      2. cancel-sign-sub-inv 93.5%

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

      3. metadata-eval 93.5%

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

      4. *-un-lft-identity 93.5%

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

   9. Applied egg-rr 93.5%

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

   if -3.5e103 < b < 3.9999999999999997e125

   1. Initial program 84.1%

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

      1. associate-*l* 84.1%

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

      2. *-commutative 84.1%

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

      3. associate-/l* 83.7%

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

      4. associate-*l* 83.7%

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

   3. Simplified 83.7%

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

   if 3.9999999999999997e125 < b

   1. Initial program 39.6%

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

   3. Simplified 39.5%

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

   4. Taylor expanded in b around -inf 39.5%

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

   5. Taylor expanded in a around 0 98.1%

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

      1. mul-1-neg 98.1%

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

      2. unsub-neg 98.1%

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

   7. Simplified 98.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c}{b} - \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-2}{b - -1 \cdot b}\\ \end{array} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+103}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \left(b + -2 \cdot \frac{c}{\frac{b}{a}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \end{array}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+125}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \end{array} \]

Alternative 6: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \frac{-2}{b + b}\\ \mathbf{if}\;b \leq -1.7 \cdot 10^{-94}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{a \cdot \frac{c}{b} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-2}{\left(b + b\right) - \frac{c}{b} \cdot \left(a \cdot 2\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq -1.6 \cdot 10^{-278}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \leq 8.4 \cdot 10^{-50}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (/ -2.0 (+ b b)))))
   (if (<= b -1.7e-94)
     (if (>= b 0.0)
       (/ (- (* a (/ c b)) b) a)
       (* c (/ -2.0 (- (+ b b) (* (/ c b) (* a 2.0))))))
     (if (<= b -1.6e-278)
       (if (>= b 0.0)
         (/ (* b -2.0) (* a 2.0))
         (/ 2.0 (/ (- (sqrt (* (* a c) -4.0)) b) c)))
       (if (<= b 8.4e-50)
         (if (>= b 0.0) (* (/ -0.5 a) (+ b (sqrt (* c (* a -4.0))))) t_0)
         (if (>= b 0.0) (- (/ c b) (/ b a)) t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (-2.0 / (b + b));
	double tmp_1;
	if (b <= -1.7e-94) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = ((a * (c / b)) - b) / a;
		} else {
			tmp_2 = c * (-2.0 / ((b + b) - ((c / b) * (a * 2.0))));
		}
		tmp_1 = tmp_2;
	} else if (b <= -1.6e-278) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (b * -2.0) / (a * 2.0);
		} else {
			tmp_3 = 2.0 / ((sqrt(((a * c) * -4.0)) - b) / c);
		}
		tmp_1 = tmp_3;
	} else if (b <= 8.4e-50) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (-0.5 / a) * (b + sqrt((c * (a * -4.0))));
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    real(8) :: tmp_4
    t_0 = c * ((-2.0d0) / (b + b))
    if (b <= (-1.7d-94)) then
        if (b >= 0.0d0) then
            tmp_2 = ((a * (c / b)) - b) / a
        else
            tmp_2 = c * ((-2.0d0) / ((b + b) - ((c / b) * (a * 2.0d0))))
        end if
        tmp_1 = tmp_2
    else if (b <= (-1.6d-278)) then
        if (b >= 0.0d0) then
            tmp_3 = (b * (-2.0d0)) / (a * 2.0d0)
        else
            tmp_3 = 2.0d0 / ((sqrt(((a * c) * (-4.0d0))) - b) / c)
        end if
        tmp_1 = tmp_3
    else if (b <= 8.4d-50) then
        if (b >= 0.0d0) then
            tmp_4 = ((-0.5d0) / a) * (b + sqrt((c * (a * (-4.0d0)))))
        else
            tmp_4 = t_0
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = (c / b) - (b / a)
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (-2.0 / (b + b));
	double tmp_1;
	if (b <= -1.7e-94) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = ((a * (c / b)) - b) / a;
		} else {
			tmp_2 = c * (-2.0 / ((b + b) - ((c / b) * (a * 2.0))));
		}
		tmp_1 = tmp_2;
	} else if (b <= -1.6e-278) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (b * -2.0) / (a * 2.0);
		} else {
			tmp_3 = 2.0 / ((Math.sqrt(((a * c) * -4.0)) - b) / c);
		}
		tmp_1 = tmp_3;
	} else if (b <= 8.4e-50) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (-0.5 / a) * (b + Math.sqrt((c * (a * -4.0))));
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = c * (-2.0 / (b + b))
	tmp_1 = 0
	if b <= -1.7e-94:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = ((a * (c / b)) - b) / a
		else:
			tmp_2 = c * (-2.0 / ((b + b) - ((c / b) * (a * 2.0))))
		tmp_1 = tmp_2
	elif b <= -1.6e-278:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (b * -2.0) / (a * 2.0)
		else:
			tmp_3 = 2.0 / ((math.sqrt(((a * c) * -4.0)) - b) / c)
		tmp_1 = tmp_3
	elif b <= 8.4e-50:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = (-0.5 / a) * (b + math.sqrt((c * (a * -4.0))))
		else:
			tmp_4 = t_0
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = (c / b) - (b / a)
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(-2.0 / Float64(b + b)))
	tmp_1 = 0.0
	if (b <= -1.7e-94)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(Float64(a * Float64(c / b)) - b) / a);
		else
			tmp_2 = Float64(c * Float64(-2.0 / Float64(Float64(b + b) - Float64(Float64(c / b) * Float64(a * 2.0)))));
		end
		tmp_1 = tmp_2;
	elseif (b <= -1.6e-278)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(b * -2.0) / Float64(a * 2.0));
		else
			tmp_3 = Float64(2.0 / Float64(Float64(sqrt(Float64(Float64(a * c) * -4.0)) - b) / c));
		end
		tmp_1 = tmp_3;
	elseif (b <= 8.4e-50)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(-0.5 / a) * Float64(b + sqrt(Float64(c * Float64(a * -4.0)))));
		else
			tmp_4 = t_0;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = c * (-2.0 / (b + b));
	tmp_2 = 0.0;
	if (b <= -1.7e-94)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = ((a * (c / b)) - b) / a;
		else
			tmp_3 = c * (-2.0 / ((b + b) - ((c / b) * (a * 2.0))));
		end
		tmp_2 = tmp_3;
	elseif (b <= -1.6e-278)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (b * -2.0) / (a * 2.0);
		else
			tmp_4 = 2.0 / ((sqrt(((a * c) * -4.0)) - b) / c);
		end
		tmp_2 = tmp_4;
	elseif (b <= 8.4e-50)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = (-0.5 / a) * (b + sqrt((c * (a * -4.0))));
		else
			tmp_5 = t_0;
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = (c / b) - (b / a);
	else
		tmp_2 = t_0;
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(-2.0 / N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.7e-94], If[GreaterEqual[b, 0.0], N[(N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision], N[(c * N[(-2.0 / N[(N[(b + b), $MachinePrecision] - N[(N[(c / b), $MachinePrecision] * N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -1.6e-278], If[GreaterEqual[b, 0.0], N[(N[(b * -2.0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 8.4e-50], If[GreaterEqual[b, 0.0], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.6999999999999999e-94

   1. Initial program 69.9%

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

   3. Simplified 69.7%

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

   4. Taylor expanded in b around inf 69.7%

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

      1. +-commutative 69.7%

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

      2. *-commutative 69.7%

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

      3. fma-def 69.7%

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

      4. associate-/l* 69.7%

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

   6. Simplified 69.7%

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

   7. Taylor expanded in b around -inf 80.3%

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

      1. fma-def 80.3%

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

      2. associate-/l* 85.2%

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

      3. mul-1-neg 85.2%

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

   9. Simplified 85.2%

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

       1. associate-*r/ 85.5%

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

   11. Applied egg-rr 85.5%

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

       1. associate-*r/ 85.2%

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

       2. fma-udef 85.2%

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

       3. associate-/l* 80.3%

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

       4. neg-mul-1 80.3%

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

       5. +-commutative 80.3%

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

       6. associate--r+ 80.3%

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

       7. cancel-sign-sub-inv 80.3%

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

       8. metadata-eval 80.3%

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

       9. *-lft-identity 80.3%

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

       10. *-commutative 80.3%

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

       11. associate-/l* 85.2%

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

       12. associate-/r/ 85.2%

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

       13. associate-*l* 85.2%

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

   13. Simplified 85.2%

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

       1. associate-*l/ 85.2%

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

   15. Applied egg-rr 85.2%

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

       1. fma-udef 85.2%

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

       2. *-commutative 85.2%

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

       3. count-2 85.2%

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

       4. distribute-rgt-in 85.2%

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

       5. count-2 85.2%

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

       6. *-commutative 85.2%

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

       7. associate-*l* 85.2%

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

       8. metadata-eval 85.2%

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

       9. *-commutative 85.2%

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

       10. neg-mul-1 85.2%

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

       11. associate-/l* 85.2%

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

       12. *-commutative 85.2%

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

       13. associate-*l* 85.2%

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

       14. associate-/l* 85.2%

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

       15. associate-/r/ 85.2%

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

       16. metadata-eval 85.2%

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

   17. Simplified 85.2%

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

   if -1.6999999999999999e-94 < b < -1.60000000000000009e-278

   1. Initial program 78.6%

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

      1. associate-*l* 78.6%

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

      2. *-commutative 78.6%

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

      3. associate-/l* 78.5%

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

      4. associate-*l* 78.5%

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

   3. Simplified 78.5%

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

   4. Taylor expanded in b around inf 78.5%

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

      1. *-commutative 78.5%

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

   6. Simplified 78.5%

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

   7. Taylor expanded in b around 0 67.4%

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

      1. *-commutative 67.4%

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

   9. Simplified 67.4%

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

   if -1.60000000000000009e-278 < b < 8.4000000000000003e-50

   1. Initial program 72.1%

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

   3. Simplified 72.0%

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

   4. Taylor expanded in b around -inf 73.8%

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

      1. fma-udef 22.4%

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

      2. *-commutative 22.4%

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

      3. associate-*l* 22.4%

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

      4. *-commutative 22.4%

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

   6. Applied egg-rr 73.8%

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

   7. Taylor expanded in b around 0 63.8%

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

      1. *-commutative 63.8%

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

      2. associate-*r* 63.8%

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

   9. Simplified 63.8%

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

   if 8.4000000000000003e-50 < b

   1. Initial program 59.8%

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

   3. Simplified 59.7%

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

   4. Taylor expanded in b around -inf 59.7%

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

   5. Taylor expanded in a around 0 91.7%

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

      1. mul-1-neg 91.7%

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

      2. unsub-neg 91.7%

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

   7. Simplified 91.7%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{-94}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{a \cdot \frac{c}{b} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-2}{\left(b + b\right) - \frac{c}{b} \cdot \left(a \cdot 2\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq -1.6 \cdot 10^{-278}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \leq 8.4 \cdot 10^{-50}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \end{array} \]

Alternative 7: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \frac{-2}{b + b}\\ \mathbf{if}\;b \leq 7.5 \cdot 10^{-56}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (/ -2.0 (+ b b)))))
   (if (<= b 7.5e-56)
     (if (>= b 0.0) (* (/ -0.5 a) (+ b (sqrt (* c (* a -4.0))))) t_0)
     (if (>= b 0.0) (- (/ c b) (/ b a)) t_0))))

C

double code(double a, double b, double c) {
	double t_0 = c * (-2.0 / (b + b));
	double tmp_1;
	if (b <= 7.5e-56) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-0.5 / a) * (b + sqrt((c * (a * -4.0))));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = c * ((-2.0d0) / (b + b))
    if (b <= 7.5d-56) then
        if (b >= 0.0d0) then
            tmp_2 = ((-0.5d0) / a) * (b + sqrt((c * (a * (-4.0d0)))))
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (c / b) - (b / a)
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (-2.0 / (b + b));
	double tmp_1;
	if (b <= 7.5e-56) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-0.5 / a) * (b + Math.sqrt((c * (a * -4.0))));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = c * (-2.0 / (b + b))
	tmp_1 = 0
	if b <= 7.5e-56:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (-0.5 / a) * (b + math.sqrt((c * (a * -4.0))))
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = (c / b) - (b / a)
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(-2.0 / Float64(b + b)))
	tmp_1 = 0.0
	if (b <= 7.5e-56)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-0.5 / a) * Float64(b + sqrt(Float64(c * Float64(a * -4.0)))));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = c * (-2.0 / (b + b));
	tmp_2 = 0.0;
	if (b <= 7.5e-56)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (-0.5 / a) * (b + sqrt((c * (a * -4.0))));
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = (c / b) - (b / a);
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(-2.0 / N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 7.5e-56], If[GreaterEqual[b, 0.0], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < 7.50000000000000041e-56

   1. Initial program 71.9%

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

   3. Simplified 71.8%

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

   4. Taylor expanded in b around -inf 71.6%

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

      1. fma-udef 58.7%

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

      2. *-commutative 58.7%

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

      3. associate-*l* 58.7%

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

      4. *-commutative 58.7%

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

   6. Applied egg-rr 71.6%

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

   7. Taylor expanded in b around 0 69.0%

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

      1. *-commutative 69.0%

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

      2. associate-*r* 69.0%

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

   9. Simplified 69.0%

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

   if 7.50000000000000041e-56 < b

   1. Initial program 59.8%

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

   3. Simplified 59.7%

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

   4. Taylor expanded in b around -inf 59.7%

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

   5. Taylor expanded in a around 0 91.7%

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

      1. mul-1-neg 91.7%

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

      2. unsub-neg 91.7%

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

   7. Simplified 91.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c}{b} - \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-2}{b - -1 \cdot b}\\ \end{array} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.5 \cdot 10^{-56}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \end{array} \]

Alternative 8: 67.7% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.1%

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

3. Simplified 68.0%

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

4. Taylor expanded in b around inf 67.6%

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

   1. +-commutative 67.6%

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

   2. *-commutative 67.6%

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

   3. fma-def 67.6%

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

   4. associate-/l* 69.1%

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

6. Simplified 69.1%

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

7. Taylor expanded in b around -inf 67.2%

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

   1. fma-def 67.2%

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

   2. associate-/l* 69.2%

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

   3. mul-1-neg 69.2%

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

9. Simplified 69.2%

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

    1. associate-*r/ 69.3%

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

11. Applied egg-rr 69.3%

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

    1. associate-*r/ 69.2%

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

    2. fma-udef 69.2%

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

    3. associate-/l* 67.2%

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

    4. neg-mul-1 67.2%

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

    5. +-commutative 67.2%

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

    6. associate--r+ 67.2%

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

    7. cancel-sign-sub-inv 67.2%

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

    8. metadata-eval 67.2%

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

    9. *-lft-identity 67.2%

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

    10. *-commutative 67.2%

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

    11. associate-/l* 69.2%

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

    12. associate-/r/ 69.2%

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

    13. associate-*l* 69.2%

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

13. Simplified 69.2%

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

    1. associate-*l/ 69.3%

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

15. Applied egg-rr 69.3%

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

    1. fma-udef 69.3%

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

    2. *-commutative 69.3%

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

    3. count-2 69.3%

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

    4. distribute-rgt-in 69.3%

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

    5. count-2 69.3%

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

    6. *-commutative 69.3%

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

    7. associate-*l* 69.3%

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

    8. metadata-eval 69.3%

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

    9. *-commutative 69.3%

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

    10. neg-mul-1 69.3%

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

    11. associate-/l* 67.8%

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

    12. *-commutative 67.8%

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

    13. associate-*l* 67.8%

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

    14. associate-/l* 69.3%

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

    15. associate-/r/ 69.3%

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

    16. metadata-eval 69.3%

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

17. Simplified 69.3%

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

18. Final simplification 69.3%

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

Alternative 9: 67.5% accurate, 13.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.1%

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

   1. associate-*l* 68.1%

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

   2. *-commutative 68.1%

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

   3. associate-/l* 67.9%

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

   4. associate-*l* 67.9%

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

3. Simplified 67.9%

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

4. Taylor expanded in b around inf 68.9%

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

   1. *-commutative 68.9%

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

6. Simplified 68.9%

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

7. Taylor expanded in b around -inf 68.6%

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

   1. associate-*r/ 67.6%

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

   2. *-commutative 67.6%

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

9. Simplified 68.6%

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

    1. associate-/r/ 68.9%

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

    2. *-commutative 68.9%

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

11. Applied egg-rr 68.9%

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

12. Final simplification 68.9%

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

Alternative 10: 67.6% accurate, 13.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.1%

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

3. Simplified 68.0%

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

4. Taylor expanded in b around -inf 67.9%

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

5. Taylor expanded in a around 0 69.1%

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

   1. mul-1-neg 69.1%

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

   2. unsub-neg 69.1%

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

7. Simplified 69.1%

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

8. Final simplification 69.1%

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

Reproduce

herbie shell --seed 2023171 
(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)))))))