jeff quadratic root 1

Percentage Accurate: 71.8% → 89.7%

Time: 17.5s

Alternatives: 8

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: 71.8% 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, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}\\ \mathbf{if}\;b \leq -5.8 \cdot 10^{+168}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0 - \left(b + b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{+137}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5 \cdot \left(b + t\_0\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{t\_0 - b}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* b b) (* (* a c) -4.0)))))
   (if (<= b -5.8e+168)
     (if (>= b 0.0) (/ (- 0.0 (+ b b)) (* 2.0 a)) (- 0.0 (/ c b)))
     (if (<= b 1e+137)
       (if (>= b 0.0) (/ (* -0.5 (+ b t_0)) a) (/ (* 2.0 c) (- t_0 b)))
       (- 0.0 (/ b a))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) + ((a * c) * -4.0)));
	double tmp_1;
	if (b <= -5.8e+168) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (0.0 - (b + b)) / (2.0 * a);
		} else {
			tmp_2 = 0.0 - (c / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1e+137) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-0.5 * (b + t_0)) / a;
		} else {
			tmp_3 = (2.0 * c) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = 0.0 - (b / a);
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = sqrt(((b * b) + ((a * c) * (-4.0d0))))
    if (b <= (-5.8d+168)) then
        if (b >= 0.0d0) then
            tmp_2 = (0.0d0 - (b + b)) / (2.0d0 * a)
        else
            tmp_2 = 0.0d0 - (c / b)
        end if
        tmp_1 = tmp_2
    else if (b <= 1d+137) then
        if (b >= 0.0d0) then
            tmp_3 = ((-0.5d0) * (b + t_0)) / a
        else
            tmp_3 = (2.0d0 * c) / (t_0 - b)
        end if
        tmp_1 = tmp_3
    else
        tmp_1 = 0.0d0 - (b / a)
    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) + ((a * c) * -4.0)));
	double tmp_1;
	if (b <= -5.8e+168) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (0.0 - (b + b)) / (2.0 * a);
		} else {
			tmp_2 = 0.0 - (c / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1e+137) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-0.5 * (b + t_0)) / a;
		} else {
			tmp_3 = (2.0 * c) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = 0.0 - (b / a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) + ((a * c) * -4.0)))
	tmp_1 = 0
	if b <= -5.8e+168:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (0.0 - (b + b)) / (2.0 * a)
		else:
			tmp_2 = 0.0 - (c / b)
		tmp_1 = tmp_2
	elif b <= 1e+137:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (-0.5 * (b + t_0)) / a
		else:
			tmp_3 = (2.0 * c) / (t_0 - b)
		tmp_1 = tmp_3
	else:
		tmp_1 = 0.0 - (b / a)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) + Float64(Float64(a * c) * -4.0)))
	tmp_1 = 0.0
	if (b <= -5.8e+168)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(0.0 - Float64(b + b)) / Float64(2.0 * a));
		else
			tmp_2 = Float64(0.0 - Float64(c / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1e+137)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-0.5 * Float64(b + t_0)) / a);
		else
			tmp_3 = Float64(Float64(2.0 * c) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	else
		tmp_1 = Float64(0.0 - Float64(b / a));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) + ((a * c) * -4.0)));
	tmp_2 = 0.0;
	if (b <= -5.8e+168)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (0.0 - (b + b)) / (2.0 * a);
		else
			tmp_3 = 0.0 - (c / b);
		end
		tmp_2 = tmp_3;
	elseif (b <= 1e+137)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (-0.5 * (b + t_0)) / a;
		else
			tmp_4 = (2.0 * c) / (t_0 - b);
		end
		tmp_2 = tmp_4;
	else
		tmp_2 = 0.0 - (b / a);
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -5.8e+168], If[GreaterEqual[b, 0.0], N[(N[(0.0 - N[(b + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1e+137], If[GreaterEqual[b, 0.0], N[(N[(-0.5 * N[(b + t$95$0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.8e168

   1. Initial program 30.4%

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

   3. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right)\right)\\ \end{array} \]
   4. Step-by-step derivation

   5. Simplified 30.4%

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

   6. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{neg}\left(\frac{c}{b}\right)\\ \end{array} \]

      2. neg-sub0 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(0, \left(\frac{c}{b}\right)\right)\\ \end{array} \]

      4. /-lowering-/.f64 100.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, b\right)\right)\\ \end{array} \]

   8. Simplified 100.0%

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

   if -5.8e168 < b < 1e137

   1. Initial program 89.9%

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

   4. Applied egg-rr 89.8%

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

   5. Taylor expanded in b around 0

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

      1. >=-lowering->=.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\color{blue}{\frac{-1}{2}} \cdot \frac{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

      2. associate-*r/ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\frac{\frac{-1}{2} \cdot \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)}{\color{blue}{a}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)\right), \color{blue}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)\right), a\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(b, \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}\right)\right)\right), a\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(-4 \cdot \left(a \cdot c\right) + {b}^{2}\right)\right)\right)\right), a\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

      7. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left({b}^{2} + -4 \cdot \left(a \cdot c\right)\right)\right)\right)\right), a\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({b}^{2}\right), \left(-4 \cdot \left(a \cdot c\right)\right)\right)\right)\right)\right), a\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

      9. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(-4 \cdot \left(a \cdot c\right)\right)\right)\right)\right)\right), a\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

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

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(-4 \cdot \left(a \cdot c\right)\right)\right)\right)\right)\right), a\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

      11. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right), a\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

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

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\left(a \cdot c\right), -4\right)\right)\right)\right)\right), a\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

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

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, c\right), -4\right)\right)\right)\right)\right), a\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

   7. Simplified 89.9%

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

   if 1e137 < b

   1. Initial program 46.8%

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

   3. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right)\right)\\ \end{array} \]
   4. Step-by-step derivation

   5. Simplified 98.2%

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

   6. Taylor expanded in c around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{neg}\left(\frac{b}{a}\right)\\ \end{array} \]

      2. neg-sub0 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(0, \left(\frac{b}{a}\right)\right)\\ \end{array} \]

      4. /-lowering-/.f64 98.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, a\right)\right)\\ \end{array} \]

   8. Simplified 98.2%

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

   9. Taylor expanded in b around 0

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

       1. neg-mul-1 N/A

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

       2. if-same N/A

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

       3. neg-mul-1 N/A

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

       4. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\frac{b}{a}} \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right) \]

       6. /-lowering-/.f64 98.2%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   11. Simplified 98.2%

       \[\leadsto \color{blue}{0 - \frac{b}{a}} \]

   12. Taylor expanded in b around 0

       \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   13. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

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

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

          \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(-1 \cdot a\right)}\right) \]

       5. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(b, \left(\mathsf{neg}\left(a\right)\right)\right) \]

       6. neg-lowering-neg.f64 98.2%

          \[\leadsto \mathsf{/.f64}\left(b, \mathsf{neg.f64}\left(a\right)\right) \]

   14. Simplified 98.2%

       \[\leadsto \color{blue}{\frac{b}{-a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{+168}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0 - \left(b + b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{+137}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5 \cdot \left(b + \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4} - b}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\\ \mathbf{if}\;b \leq -5.8 \cdot 10^{+168}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0 - \left(b + b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{+139}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{\frac{a}{b + t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{t\_0 - b}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* b b) (* a (* c -4.0))))))
   (if (<= b -5.8e+168)
     (if (>= b 0.0) (/ (- 0.0 (+ b b)) (* 2.0 a)) (- 0.0 (/ c b)))
     (if (<= b 7.5e+139)
       (if (>= b 0.0) (/ -0.5 (/ a (+ b t_0))) (/ (* 2.0 c) (- t_0 b)))
       (- 0.0 (/ b a))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) + (a * (c * -4.0))));
	double tmp_1;
	if (b <= -5.8e+168) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (0.0 - (b + b)) / (2.0 * a);
		} else {
			tmp_2 = 0.0 - (c / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 7.5e+139) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -0.5 / (a / (b + t_0));
		} else {
			tmp_3 = (2.0 * c) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = 0.0 - (b / a);
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = sqrt(((b * b) + (a * (c * (-4.0d0)))))
    if (b <= (-5.8d+168)) then
        if (b >= 0.0d0) then
            tmp_2 = (0.0d0 - (b + b)) / (2.0d0 * a)
        else
            tmp_2 = 0.0d0 - (c / b)
        end if
        tmp_1 = tmp_2
    else if (b <= 7.5d+139) then
        if (b >= 0.0d0) then
            tmp_3 = (-0.5d0) / (a / (b + t_0))
        else
            tmp_3 = (2.0d0 * c) / (t_0 - b)
        end if
        tmp_1 = tmp_3
    else
        tmp_1 = 0.0d0 - (b / a)
    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) + (a * (c * -4.0))));
	double tmp_1;
	if (b <= -5.8e+168) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (0.0 - (b + b)) / (2.0 * a);
		} else {
			tmp_2 = 0.0 - (c / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 7.5e+139) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -0.5 / (a / (b + t_0));
		} else {
			tmp_3 = (2.0 * c) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = 0.0 - (b / a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) + (a * (c * -4.0))))
	tmp_1 = 0
	if b <= -5.8e+168:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (0.0 - (b + b)) / (2.0 * a)
		else:
			tmp_2 = 0.0 - (c / b)
		tmp_1 = tmp_2
	elif b <= 7.5e+139:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = -0.5 / (a / (b + t_0))
		else:
			tmp_3 = (2.0 * c) / (t_0 - b)
		tmp_1 = tmp_3
	else:
		tmp_1 = 0.0 - (b / a)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0))))
	tmp_1 = 0.0
	if (b <= -5.8e+168)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(0.0 - Float64(b + b)) / Float64(2.0 * a));
		else
			tmp_2 = Float64(0.0 - Float64(c / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 7.5e+139)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(-0.5 / Float64(a / Float64(b + t_0)));
		else
			tmp_3 = Float64(Float64(2.0 * c) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	else
		tmp_1 = Float64(0.0 - Float64(b / a));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) + (a * (c * -4.0))));
	tmp_2 = 0.0;
	if (b <= -5.8e+168)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (0.0 - (b + b)) / (2.0 * a);
		else
			tmp_3 = 0.0 - (c / b);
		end
		tmp_2 = tmp_3;
	elseif (b <= 7.5e+139)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = -0.5 / (a / (b + t_0));
		else
			tmp_4 = (2.0 * c) / (t_0 - b);
		end
		tmp_2 = tmp_4;
	else
		tmp_2 = 0.0 - (b / a);
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -5.8e+168], If[GreaterEqual[b, 0.0], N[(N[(0.0 - N[(b + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 7.5e+139], If[GreaterEqual[b, 0.0], N[(-0.5 / N[(a / N[(b + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.8e168

   1. Initial program 30.4%

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

   3. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right)\right)\\ \end{array} \]
   4. Step-by-step derivation

   5. Simplified 30.4%

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

   6. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{neg}\left(\frac{c}{b}\right)\\ \end{array} \]

      2. neg-sub0 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(0, \left(\frac{c}{b}\right)\right)\\ \end{array} \]

      4. /-lowering-/.f64 100.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, b\right)\right)\\ \end{array} \]

   8. Simplified 100.0%

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

   if -5.8e168 < b < 7.49999999999999992e139

   1. Initial program 89.9%

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

   4. Applied egg-rr 89.8%

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

   6. Applied egg-rr 89.8%

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

   if 7.49999999999999992e139 < b

   1. Initial program 46.8%

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

   3. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right)\right)\\ \end{array} \]
   4. Step-by-step derivation

   5. Simplified 98.2%

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

   6. Taylor expanded in c around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{neg}\left(\frac{b}{a}\right)\\ \end{array} \]

      2. neg-sub0 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(0, \left(\frac{b}{a}\right)\right)\\ \end{array} \]

      4. /-lowering-/.f64 98.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, a\right)\right)\\ \end{array} \]

   8. Simplified 98.2%

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

   9. Taylor expanded in b around 0

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

       1. neg-mul-1 N/A

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

       2. if-same N/A

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

       3. neg-mul-1 N/A

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

       4. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\frac{b}{a}} \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right) \]

       6. /-lowering-/.f64 98.2%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   11. Simplified 98.2%

       \[\leadsto \color{blue}{0 - \frac{b}{a}} \]

   12. Taylor expanded in b around 0

       \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   13. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

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

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

          \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(-1 \cdot a\right)}\right) \]

       5. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(b, \left(\mathsf{neg}\left(a\right)\right)\right) \]

       6. neg-lowering-neg.f64 98.2%

          \[\leadsto \mathsf{/.f64}\left(b, \mathsf{neg.f64}\left(a\right)\right) \]

   14. Simplified 98.2%

       \[\leadsto \color{blue}{\frac{b}{-a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{+168}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0 - \left(b + b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{+139}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{\frac{a}{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+76}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0 - \left(b + b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{+139}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(b + \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.2e+76)
   (if (>= b 0.0) (/ (- 0.0 (+ b b)) (* 2.0 a)) (- 0.0 (/ c b)))
   (if (<= b 4.5e+139)
     (if (>= b 0.0)
       (* (+ b (sqrt (+ (* b b) (* (* a c) -4.0)))) (/ -0.5 a))
       (* c (/ 2.0 (- (sqrt (+ (* b b) (* a (* c -4.0)))) b))))
     (- 0.0 (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1.2e+76) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (0.0 - (b + b)) / (2.0 * a);
		} else {
			tmp_2 = 0.0 - (c / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 4.5e+139) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (b + sqrt(((b * b) + ((a * c) * -4.0)))) * (-0.5 / a);
		} else {
			tmp_3 = c * (2.0 / (sqrt(((b * b) + (a * (c * -4.0)))) - b));
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = 0.0 - (b / a);
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    if (b <= (-1.2d+76)) then
        if (b >= 0.0d0) then
            tmp_2 = (0.0d0 - (b + b)) / (2.0d0 * a)
        else
            tmp_2 = 0.0d0 - (c / b)
        end if
        tmp_1 = tmp_2
    else if (b <= 4.5d+139) then
        if (b >= 0.0d0) then
            tmp_3 = (b + sqrt(((b * b) + ((a * c) * (-4.0d0))))) * ((-0.5d0) / a)
        else
            tmp_3 = c * (2.0d0 / (sqrt(((b * b) + (a * (c * (-4.0d0))))) - b))
        end if
        tmp_1 = tmp_3
    else
        tmp_1 = 0.0d0 - (b / a)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1.2e+76) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (0.0 - (b + b)) / (2.0 * a);
		} else {
			tmp_2 = 0.0 - (c / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 4.5e+139) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (b + Math.sqrt(((b * b) + ((a * c) * -4.0)))) * (-0.5 / a);
		} else {
			tmp_3 = c * (2.0 / (Math.sqrt(((b * b) + (a * (c * -4.0)))) - b));
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = 0.0 - (b / a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if b <= -1.2e+76:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (0.0 - (b + b)) / (2.0 * a)
		else:
			tmp_2 = 0.0 - (c / b)
		tmp_1 = tmp_2
	elif b <= 4.5e+139:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (b + math.sqrt(((b * b) + ((a * c) * -4.0)))) * (-0.5 / a)
		else:
			tmp_3 = c * (2.0 / (math.sqrt(((b * b) + (a * (c * -4.0)))) - b))
		tmp_1 = tmp_3
	else:
		tmp_1 = 0.0 - (b / a)
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -1.2e+76)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(0.0 - Float64(b + b)) / Float64(2.0 * a));
		else
			tmp_2 = Float64(0.0 - Float64(c / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 4.5e+139)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(b + sqrt(Float64(Float64(b * b) + Float64(Float64(a * c) * -4.0)))) * Float64(-0.5 / a));
		else
			tmp_3 = Float64(c * Float64(2.0 / Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0)))) - b)));
		end
		tmp_1 = tmp_3;
	else
		tmp_1 = Float64(0.0 - Float64(b / a));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	tmp_2 = 0.0;
	if (b <= -1.2e+76)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (0.0 - (b + b)) / (2.0 * a);
		else
			tmp_3 = 0.0 - (c / b);
		end
		tmp_2 = tmp_3;
	elseif (b <= 4.5e+139)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (b + sqrt(((b * b) + ((a * c) * -4.0)))) * (-0.5 / a);
		else
			tmp_4 = c * (2.0 / (sqrt(((b * b) + (a * (c * -4.0)))) - b));
		end
		tmp_2 = tmp_4;
	else
		tmp_2 = 0.0 - (b / a);
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.2e+76], If[GreaterEqual[b, 0.0], N[(N[(0.0 - N[(b + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 4.5e+139], If[GreaterEqual[b, 0.0], N[(N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(2.0 / N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.2e76

   1. Initial program 61.1%

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

   3. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right)\right)\\ \end{array} \]
   4. Step-by-step derivation

   5. Simplified 61.1%

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

   6. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{neg}\left(\frac{c}{b}\right)\\ \end{array} \]

      2. neg-sub0 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(0, \left(\frac{c}{b}\right)\right)\\ \end{array} \]

      4. /-lowering-/.f64 100.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, b\right)\right)\\ \end{array} \]

   8. Simplified 100.0%

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

   if -1.2e76 < b < 4.4999999999999999e139

   1. Initial program 88.1%

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

   4. Applied egg-rr 88.0%

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

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, c\right), -4\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      2. associate-/l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, c\right), -4\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, c\right), -4\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(c, \left(\frac{2}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)\right)\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, c\right), -4\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(2, \left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)\right)\right)\\ \end{array} \]

      5. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, c\right), -4\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(2, \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right)\right)\right)\\ \end{array} \]

      6. unsub-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, c\right), -4\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(2, \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)\right)\right)\\ \end{array} \]

      7. --lowering--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, c\right), -4\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right)\right)\right)\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, c\right), -4\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right)\right)\right)\\ \end{array} \]

      9. associate-*l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, c\right), -4\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right)\right), b\right)\right)\right)\\ \end{array} \]

      10. cancel-sign-sub-inv N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, c\right), -4\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)\right)\right), b\right)\right)\right)\\ \end{array} \]

      11. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, c\right), -4\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right)\right), b\right)\right)\right)\\ \end{array} \]

      12. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, c\right), -4\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(a \cdot c\right) \cdot -4\right)\right), b\right)\right)\right)\\ \end{array} \]

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

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, c\right), -4\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right), b\right)\right)\right)\\ \end{array} \]

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

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, c\right), -4\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right), b\right)\right)\right)\\ \end{array} \]

      15. associate-*l* N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, c\right), -4\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right)\\ \end{array} \]

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

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, c\right), -4\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot -4\right)\right)\right)\right), b\right)\right)\right)\\ \end{array} \]

      17. *-lowering-*.f64 87.9%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, c\right), -4\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right)\right)\right)\\ \end{array} \]

   6. Applied egg-rr 87.9%

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

   if 4.4999999999999999e139 < b

   1. Initial program 46.8%

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

   3. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right)\right)\\ \end{array} \]
   4. Step-by-step derivation

   5. Simplified 98.2%

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

   6. Taylor expanded in c around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{neg}\left(\frac{b}{a}\right)\\ \end{array} \]

      2. neg-sub0 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(0, \left(\frac{b}{a}\right)\right)\\ \end{array} \]

      4. /-lowering-/.f64 98.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, a\right)\right)\\ \end{array} \]

   8. Simplified 98.2%

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

   9. Taylor expanded in b around 0

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

       1. neg-mul-1 N/A

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

       2. if-same N/A

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

       3. neg-mul-1 N/A

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

       4. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\frac{b}{a}} \]

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

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right) \]

       6. /-lowering-/.f64 98.2%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   11. Simplified 98.2%

       \[\leadsto \color{blue}{0 - \frac{b}{a}} \]

   12. Taylor expanded in b around 0

       \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   13. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

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

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

          \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(-1 \cdot a\right)}\right) \]

       5. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(b, \left(\mathsf{neg}\left(a\right)\right)\right) \]

       6. neg-lowering-neg.f64 98.2%

          \[\leadsto \mathsf{/.f64}\left(b, \mathsf{neg.f64}\left(a\right)\right) \]

   14. Simplified 98.2%

       \[\leadsto \color{blue}{\frac{b}{-a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+76}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0 - \left(b + b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{+139}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(b + \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0 - \left(b + b\right)}{2 \cdot a}\\ \mathbf{if}\;b \leq -1.38 \cdot 10^{-62}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{a \cdot \left(c \cdot -4\right)} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- 0.0 (+ b b)) (* 2.0 a))))
   (if (<= b -1.38e-62)
     (if (>= b 0.0) t_0 (- 0.0 (/ c b)))
     (if (>= b 0.0) t_0 (/ (* 2.0 c) (- (sqrt (* a (* c -4.0))) b))))))

C

double code(double a, double b, double c) {
	double t_0 = (0.0 - (b + b)) / (2.0 * a);
	double tmp_1;
	if (b <= -1.38e-62) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = 0.0 - (c / b);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = (2.0 * c) / (sqrt((a * (c * -4.0))) - 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
    t_0 = (0.0d0 - (b + b)) / (2.0d0 * a)
    if (b <= (-1.38d-62)) then
        if (b >= 0.0d0) then
            tmp_2 = t_0
        else
            tmp_2 = 0.0d0 - (c / b)
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = t_0
    else
        tmp_1 = (2.0d0 * c) / (sqrt((a * (c * (-4.0d0)))) - b)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (0.0 - (b + b)) / (2.0 * a);
	double tmp_1;
	if (b <= -1.38e-62) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = 0.0 - (c / b);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = (2.0 * c) / (Math.sqrt((a * (c * -4.0))) - b);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (0.0 - (b + b)) / (2.0 * a)
	tmp_1 = 0
	if b <= -1.38e-62:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_0
		else:
			tmp_2 = 0.0 - (c / b)
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = t_0
	else:
		tmp_1 = (2.0 * c) / (math.sqrt((a * (c * -4.0))) - b)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(0.0 - Float64(b + b)) / Float64(2.0 * a))
	tmp_1 = 0.0
	if (b <= -1.38e-62)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(0.0 - Float64(c / b));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = t_0;
	else
		tmp_1 = Float64(Float64(2.0 * c) / Float64(sqrt(Float64(a * Float64(c * -4.0))) - b));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = (0.0 - (b + b)) / (2.0 * a);
	tmp_2 = 0.0;
	if (b <= -1.38e-62)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_0;
		else
			tmp_3 = 0.0 - (c / b);
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = t_0;
	else
		tmp_2 = (2.0 * c) / (sqrt((a * (c * -4.0))) - b);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(0.0 - N[(b + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.38e-62], If[GreaterEqual[b, 0.0], t$95$0, N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(2.0 * c), $MachinePrecision] / N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.38e-62

   1. Initial program 73.2%

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

   3. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right)\right)\\ \end{array} \]
   4. Step-by-step derivation

   5. Simplified 73.2%

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

   6. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{neg}\left(\frac{c}{b}\right)\\ \end{array} \]

      2. neg-sub0 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(0, \left(\frac{c}{b}\right)\right)\\ \end{array} \]

      4. /-lowering-/.f64 91.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, b\right)\right)\\ \end{array} \]

   8. Simplified 91.5%

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

   if -1.38e-62 < b

   1. Initial program 73.7%

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

   3. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right)\right)\\ \end{array} \]
   4. Step-by-step derivation

   5. Simplified 76.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{neg.f64}\left(b\right)}, \mathsf{sqrt.f64}\left(\left(-4 \cdot \left(a \cdot c\right)\right)\right)\right)\right)\\ \end{array} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a \cdot c\right), -4\right)\right)\right)\right)\\ \end{array} \]

      3. *-lowering-*.f64 72.9%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, c\right), -4\right)\right)\right)\right)\\ \end{array} \]

   8. Simplified 72.9%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(2 \cdot c\right), \left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(a \cdot c\right) \cdot -4}\right)\right)\\ \end{array} \]

      2. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\left(c \cdot 2\right), \left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(a \cdot c\right) \cdot -4}\right)\right)\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(a \cdot c\right) \cdot -4}\right)\right)\\ \end{array} \]

      4. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(\sqrt{\left(a \cdot c\right) \cdot -4} + \left(\mathsf{neg}\left(b\right)\right)\right)\right)\\ \end{array} \]

      5. unsub-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \left(\sqrt{\left(a \cdot c\right) \cdot -4} - b\right)\right)\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \mathsf{\_.f64}\left(\left(\sqrt{\left(a \cdot c\right) \cdot -4}\right), b\right)\right)\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(a \cdot c\right) \cdot -4\right)\right), b\right)\right)\\ \end{array} \]

      8. associate-*r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(a \cdot \left(c \cdot -4\right)\right)\right), b\right)\right)\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(a, \left(c \cdot -4\right)\right)\right), b\right)\right)\\ \end{array} \]

      10. *-lowering-*.f64 72.9%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, 2\right), \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right), b\right)\right)\\ \end{array} \]

   10. Applied egg-rr 72.9%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.38 \cdot 10^{-62}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0 - \left(b + b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{0 - \left(b + b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{a \cdot \left(c \cdot -4\right)} - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0 - \left(b + b\right)}{2 \cdot a}\\ \mathbf{if}\;b \leq -2.25 \cdot 10^{-61}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{a \cdot \left(c \cdot -4\right)} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- 0.0 (+ b b)) (* 2.0 a))))
   (if (<= b -2.25e-61)
     (if (>= b 0.0) t_0 (- 0.0 (/ c b)))
     (if (>= b 0.0) t_0 (* c (/ 2.0 (- (sqrt (* a (* c -4.0))) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = (0.0 - (b + b)) / (2.0 * a);
	double tmp_1;
	if (b <= -2.25e-61) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = 0.0 - (c / b);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = c * (2.0 / (sqrt((a * (c * -4.0))) - 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
    t_0 = (0.0d0 - (b + b)) / (2.0d0 * a)
    if (b <= (-2.25d-61)) then
        if (b >= 0.0d0) then
            tmp_2 = t_0
        else
            tmp_2 = 0.0d0 - (c / b)
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = t_0
    else
        tmp_1 = c * (2.0d0 / (sqrt((a * (c * (-4.0d0)))) - b))
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (0.0 - (b + b)) / (2.0 * a);
	double tmp_1;
	if (b <= -2.25e-61) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = 0.0 - (c / b);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = c * (2.0 / (Math.sqrt((a * (c * -4.0))) - b));
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (0.0 - (b + b)) / (2.0 * a)
	tmp_1 = 0
	if b <= -2.25e-61:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_0
		else:
			tmp_2 = 0.0 - (c / b)
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = t_0
	else:
		tmp_1 = c * (2.0 / (math.sqrt((a * (c * -4.0))) - b))
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(0.0 - Float64(b + b)) / Float64(2.0 * a))
	tmp_1 = 0.0
	if (b <= -2.25e-61)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(0.0 - Float64(c / b));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = t_0;
	else
		tmp_1 = Float64(c * Float64(2.0 / Float64(sqrt(Float64(a * Float64(c * -4.0))) - b)));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = (0.0 - (b + b)) / (2.0 * a);
	tmp_2 = 0.0;
	if (b <= -2.25e-61)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_0;
		else
			tmp_3 = 0.0 - (c / b);
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = t_0;
	else
		tmp_2 = c * (2.0 / (sqrt((a * (c * -4.0))) - b));
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(0.0 - N[(b + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.25e-61], If[GreaterEqual[b, 0.0], t$95$0, N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$0, N[(c * N[(2.0 / N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.25e-61

   1. Initial program 73.2%

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

   3. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right)\right)\\ \end{array} \]
   4. Step-by-step derivation

   5. Simplified 73.2%

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

   6. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{neg}\left(\frac{c}{b}\right)\\ \end{array} \]

      2. neg-sub0 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(0, \left(\frac{c}{b}\right)\right)\\ \end{array} \]

      4. /-lowering-/.f64 91.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, b\right)\right)\\ \end{array} \]

   8. Simplified 91.5%

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

   if -2.25e-61 < b

   1. Initial program 73.7%

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

   3. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right)\right)\\ \end{array} \]
   4. Step-by-step derivation

   5. Simplified 76.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{neg.f64}\left(b\right)}, \mathsf{sqrt.f64}\left(\left(-4 \cdot \left(a \cdot c\right)\right)\right)\right)\right)\\ \end{array} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a \cdot c\right), -4\right)\right)\right)\right)\\ \end{array} \]

      3. *-lowering-*.f64 72.9%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, c\right), -4\right)\right)\right)\right)\\ \end{array} \]

   8. Simplified 72.9%

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

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(a \cdot c\right) \cdot -4}}\\ \end{array} \]

      2. associate-/l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(a \cdot c\right) \cdot -4}}\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(c, \left(\frac{2}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(a \cdot c\right) \cdot -4}}\right)\right)\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(2, \left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(a \cdot c\right) \cdot -4}\right)\right)\right)\\ \end{array} \]

      5. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(2, \left(\sqrt{\left(a \cdot c\right) \cdot -4} + \left(\mathsf{neg}\left(b\right)\right)\right)\right)\right)\\ \end{array} \]

      6. unsub-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(2, \left(\sqrt{\left(a \cdot c\right) \cdot -4} - b\right)\right)\right)\\ \end{array} \]

      7. --lowering--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\left(\sqrt{\left(a \cdot c\right) \cdot -4}\right), b\right)\right)\right)\\ \end{array} \]

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

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(a \cdot c\right) \cdot -4\right)\right), b\right)\right)\right)\\ \end{array} \]

      9. associate-*r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(a \cdot \left(c \cdot -4\right)\right)\right), b\right)\right)\right)\\ \end{array} \]

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

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(a, \left(c \cdot -4\right)\right)\right), b\right)\right)\right)\\ \end{array} \]

      11. *-lowering-*.f64 72.9%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right), b\right)\right)\right)\\ \end{array} \]

   10. Applied egg-rr 72.9%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{-61}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0 - \left(b + b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{0 - \left(b + b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{a \cdot \left(c \cdot -4\right)} - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.6% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

3. Taylor expanded in b around inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right)\right)\\ \end{array} \]
4. Step-by-step derivation

5. Simplified 75.0%

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

6. Taylor expanded in b around -inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{neg}\left(\frac{c}{b}\right)\\ \end{array} \]

   2. neg-sub0 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{c}{b}\\ \end{array} \]

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

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(0, \left(\frac{c}{b}\right)\right)\\ \end{array} \]

   4. /-lowering-/.f64 70.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, b\right)\right)\\ \end{array} \]

8. Simplified 70.0%

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

9. Final simplification 70.0%

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

Alternative 7: 35.7% accurate, 24.2× speedup

Math

\[\begin{array}{l} \\ 0 - \frac{b}{a} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return 0.0 - (b / a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0 - (b / a)
end function

Java

public static double code(double a, double b, double c) {
	return 0.0 - (b / a);
}

Python

def code(a, b, c):
	return 0.0 - (b / a)

Julia

function code(a, b, c)
	return Float64(0.0 - Float64(b / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 - (b / a);
end

Wolfram

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

Derivation

1. Initial program 73.5%

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

3. Taylor expanded in b around inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right)\right)\\ \end{array} \]
4. Step-by-step derivation

5. Simplified 75.0%

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

6. Taylor expanded in c around 0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{neg}\left(\frac{b}{a}\right)\\ \end{array} \]

   2. neg-sub0 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]

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

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(0, \left(\frac{b}{a}\right)\right)\\ \end{array} \]

   4. /-lowering-/.f64 35.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, a\right)\right)\\ \end{array} \]

8. Simplified 35.4%

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

9. Taylor expanded in b around 0

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

    1. neg-mul-1 N/A

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

    2. if-same N/A

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

    3. neg-mul-1 N/A

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

    4. neg-sub0 N/A

       \[\leadsto 0 - \color{blue}{\frac{b}{a}} \]

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

       \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right) \]

    6. /-lowering-/.f64 35.4%

       \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

11. Simplified 35.4%

    \[\leadsto \color{blue}{0 - \frac{b}{a}} \]

12. Taylor expanded in b around 0

    \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
13. Step-by-step derivation

    1. mul-1-neg N/A

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

    2. distribute-neg-frac2 N/A

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

    3. mul-1-neg N/A

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

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

       \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(-1 \cdot a\right)}\right) \]

    5. mul-1-neg N/A

       \[\leadsto \mathsf{/.f64}\left(b, \left(\mathsf{neg}\left(a\right)\right)\right) \]

    6. neg-lowering-neg.f64 35.4%

       \[\leadsto \mathsf{/.f64}\left(b, \mathsf{neg.f64}\left(a\right)\right) \]

14. Simplified 35.4%

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

15. Final simplification 35.4%

    \[\leadsto 0 - \frac{b}{a} \]
16. Add Preprocessing

Alternative 8: 2.5% accurate, 40.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

3. Taylor expanded in b around inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), \color{blue}{b}\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, c\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, a\right), c\right)\right)\right)\right)\right)\\ \end{array} \]
4. Step-by-step derivation

5. Simplified 75.0%

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

6. Taylor expanded in c around 0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{neg}\left(\frac{b}{a}\right)\\ \end{array} \]

   2. neg-sub0 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]

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

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(0, \left(\frac{b}{a}\right)\right)\\ \end{array} \]

   4. /-lowering-/.f64 35.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, a\right)\right)\\ \end{array} \]

8. Simplified 35.4%

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

9. Taylor expanded in b around 0

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

    1. neg-mul-1 N/A

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

    2. if-same N/A

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

    3. neg-mul-1 N/A

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

    4. neg-sub0 N/A

       \[\leadsto 0 - \color{blue}{\frac{b}{a}} \]

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

       \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right) \]

    6. /-lowering-/.f64 35.4%

       \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

11. Simplified 35.4%

    \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
12. Step-by-step derivation

    1. flip3-- N/A

       \[\leadsto \frac{{0}^{3} - {\left(\frac{b}{a}\right)}^{3}}{\color{blue}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)}} \]

    2. metadata-eval N/A

       \[\leadsto \frac{0 - {\left(\frac{b}{a}\right)}^{3}}{\color{blue}{0} \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

    3. sub0-neg N/A

       \[\leadsto \frac{\mathsf{neg}\left({\left(\frac{b}{a}\right)}^{3}\right)}{\color{blue}{0 \cdot 0} + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

    4. cube-neg N/A

       \[\leadsto \frac{{\left(\mathsf{neg}\left(\frac{b}{a}\right)\right)}^{3}}{\color{blue}{0 \cdot 0} + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

    5. div-inv N/A

       \[\leadsto \frac{{\left(\mathsf{neg}\left(b \cdot \frac{1}{a}\right)\right)}^{3}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

    6. distribute-lft-neg-in N/A

       \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(b\right)\right) \cdot \frac{1}{a}\right)}^{3}}{\color{blue}{0} \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

    7. unpow-prod-down N/A

       \[\leadsto \frac{{\left(\mathsf{neg}\left(b\right)\right)}^{3} \cdot {\left(\frac{1}{a}\right)}^{3}}{\color{blue}{0 \cdot 0} + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

    8. metadata-eval N/A

       \[\leadsto \frac{{\left(\mathsf{neg}\left(b\right)\right)}^{\left(2 \cdot \frac{3}{2}\right)} \cdot {\left(\frac{1}{a}\right)}^{3}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

    9. metadata-eval N/A

       \[\leadsto \frac{{\left(\mathsf{neg}\left(b\right)\right)}^{\left(2 \cdot \left(\frac{1}{2} \cdot 3\right)\right)} \cdot {\left(\frac{1}{a}\right)}^{3}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

    10. pow-pow N/A

        \[\leadsto \frac{{\left({\left(\mathsf{neg}\left(b\right)\right)}^{2}\right)}^{\left(\frac{1}{2} \cdot 3\right)} \cdot {\left(\frac{1}{a}\right)}^{3}}{\color{blue}{0} \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

    11. pow2 N/A

        \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(b\right)\right) \cdot \left(\mathsf{neg}\left(b\right)\right)\right)}^{\left(\frac{1}{2} \cdot 3\right)} \cdot {\left(\frac{1}{a}\right)}^{3}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

    12. sqr-neg N/A

        \[\leadsto \frac{{\left(b \cdot b\right)}^{\left(\frac{1}{2} \cdot 3\right)} \cdot {\left(\frac{1}{a}\right)}^{3}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

    13. pow2 N/A

        \[\leadsto \frac{{\left({b}^{2}\right)}^{\left(\frac{1}{2} \cdot 3\right)} \cdot {\left(\frac{1}{a}\right)}^{3}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

    14. pow-pow N/A

        \[\leadsto \frac{{b}^{\left(2 \cdot \left(\frac{1}{2} \cdot 3\right)\right)} \cdot {\left(\frac{1}{a}\right)}^{3}}{\color{blue}{0} \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

    15. metadata-eval N/A

        \[\leadsto \frac{{b}^{\left(2 \cdot \frac{3}{2}\right)} \cdot {\left(\frac{1}{a}\right)}^{3}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

    16. metadata-eval N/A

        \[\leadsto \frac{{b}^{3} \cdot {\left(\frac{1}{a}\right)}^{3}}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

    17. unpow-prod-down N/A

        \[\leadsto \frac{{\left(b \cdot \frac{1}{a}\right)}^{3}}{\color{blue}{0 \cdot 0} + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

    18. div-inv N/A

        \[\leadsto \frac{{\left(\frac{b}{a}\right)}^{3}}{\color{blue}{0} \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

13. Applied egg-rr 2.5%

    \[\leadsto \color{blue}{\frac{b}{a}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(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)))))))