jeff quadratic root 1

Percentage Accurate: 71.9% → 90.8%

Time: 17.7s

Alternatives: 13

Speedup: 1.1×

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: 71.9% 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: 90.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ t_1 := \frac{\left(-b\right) - t\_0}{a \cdot 2}\\ \mathbf{if}\;b \leq -1 \cdot 10^{+139}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left|b\right| \cdot \sqrt{\mathsf{fma}\left(\frac{c}{b \cdot b}, a \cdot -4, 1\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{+90}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b + b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* (* 4.0 a) c))))
        (t_1 (/ (- (- b) t_0) (* a 2.0))))
   (if (<= b -1e+139)
     (if (>= b 0.0)
       t_1
       (/
        (* c 2.0)
        (- (* (fabs b) (sqrt (fma (/ c (* b b)) (* a -4.0) 1.0))) b)))
     (if (<= b 1.55e+90)
       (if (>= b 0.0) t_1 (/ (* c 2.0) (- t_0 b)))
       (if (>= b 0.0) (/ (- (- b) b) (* a 2.0)) (- (/ c (+ b b))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	double t_1 = (-b - t_0) / (a * 2.0);
	double tmp_1;
	if (b <= -1e+139) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = (c * 2.0) / ((fabs(b) * sqrt(fma((c / (b * b)), (a * -4.0), 1.0))) - b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.55e+90) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = (c * 2.0) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (-b - b) / (a * 2.0);
	} else {
		tmp_1 = -(c / (b + b));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	t_1 = Float64(Float64(Float64(-b) - t_0) / Float64(a * 2.0))
	tmp_1 = 0.0
	if (b <= -1e+139)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(Float64(c * 2.0) / Float64(Float64(abs(b) * sqrt(fma(Float64(c / Float64(b * b)), Float64(a * -4.0), 1.0))) - b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.55e+90)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = Float64(Float64(c * 2.0) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(-b) - b) / Float64(a * 2.0));
	else
		tmp_1 = Float64(-Float64(c / Float64(b + b)));
	end
	return tmp_1
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]}, Block[{t$95$1 = N[(N[((-b) - t$95$0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1e+139], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(c * 2.0), $MachinePrecision] / N[(N[(N[Abs[b], $MachinePrecision] * N[Sqrt[N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(a * -4.0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.55e+90], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(c * 2.0), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[((-b) - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], (-N[(c / N[(b + b), $MachinePrecision]), $MachinePrecision])]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.00000000000000003e139

   1. Initial program 45.2%

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

   3. Taylor expanded in b around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 45.4

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

   5. Simplified 45.4%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

      4. sqrt-prod N/A

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

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

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

      6. rem-sqrt-square N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

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

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

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

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

      15. *-lowering-*.f64 97.8

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

   7. Applied egg-rr 97.8%

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

   if -1.00000000000000003e139 < b < 1.54999999999999994e90

   1. Initial program 87.9%

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

   if 1.54999999999999994e90 < b

   1. Initial program 60.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 60.3

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

   5. Simplified 60.3%

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

   6. Taylor expanded in b around inf

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

   8. Simplified 98.4%

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

      1. clear-num N/A

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

      2. sub-neg N/A

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

      3. associate-/r* N/A

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

      4. sub-neg N/A

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

      5. flip-+ N/A

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

      6. +-inverses N/A

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

      7. associate-/l/ N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. flip-+ N/A

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

      11. sub-neg N/A

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

      12. clear-num N/A

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

      13. frac-2neg N/A

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

      14. sub-neg N/A

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

      15. flip-+ N/A

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

      16. +-inverses N/A

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

   10. Applied egg-rr 98.4%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+139}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left|b\right| \cdot \sqrt{\mathsf{fma}\left(\frac{c}{b \cdot b}, a \cdot -4, 1\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{+90}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b + b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\\ t_1 := \left(-b\right) - b\\ t_2 := \frac{t\_1}{a \cdot 2}\\ \mathbf{if}\;b \leq -2 \cdot 10^{+153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{b + b}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;b \cdot \left(\frac{c}{b \cdot b} + \frac{-1}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{+90}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\left(-b\right) - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t\_1}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b + b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma a (* c -4.0) (* b b))))
        (t_1 (- (- b) b))
        (t_2 (/ t_1 (* a 2.0))))
   (if (<= b -2e+153)
     (if (>= b 0.0) t_2 (/ (* c -2.0) (+ b b)))
     (if (<= b -5e-311)
       (if (>= b 0.0)
         (* b (+ (/ c (* b b)) (/ -1.0 a)))
         (/ (* c 2.0) (- t_0 b)))
       (if (<= b 1.55e+90)
         (if (>= b 0.0) (* (/ 0.5 a) (- (- b) t_0)) (/ (* c 2.0) t_1))
         (if (>= b 0.0) t_2 (- (/ c (+ b b)))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma(a, (c * -4.0), (b * b)));
	double t_1 = -b - b;
	double t_2 = t_1 / (a * 2.0);
	double tmp_1;
	if (b <= -2e+153) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_2;
		} else {
			tmp_2 = (c * -2.0) / (b + b);
		}
		tmp_1 = tmp_2;
	} else if (b <= -5e-311) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = b * ((c / (b * b)) + (-1.0 / a));
		} else {
			tmp_3 = (c * 2.0) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.55e+90) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (0.5 / a) * (-b - t_0);
		} else {
			tmp_4 = (c * 2.0) / t_1;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = t_2;
	} else {
		tmp_1 = -(c / (b + b));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(a, Float64(c * -4.0), Float64(b * b)))
	t_1 = Float64(Float64(-b) - b)
	t_2 = Float64(t_1 / Float64(a * 2.0))
	tmp_1 = 0.0
	if (b <= -2e+153)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_2;
		else
			tmp_2 = Float64(Float64(c * -2.0) / Float64(b + b));
		end
		tmp_1 = tmp_2;
	elseif (b <= -5e-311)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(b * Float64(Float64(c / Float64(b * b)) + Float64(-1.0 / a)));
		else
			tmp_3 = Float64(Float64(c * 2.0) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	elseif (b <= 1.55e+90)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(0.5 / a) * Float64(Float64(-b) - t_0));
		else
			tmp_4 = Float64(Float64(c * 2.0) / t_1);
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = t_2;
	else
		tmp_1 = Float64(-Float64(c / Float64(b + b)));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-b) - b), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2e+153], If[GreaterEqual[b, 0.0], t$95$2, N[(N[(c * -2.0), $MachinePrecision] / N[(b + b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -5e-311], If[GreaterEqual[b, 0.0], N[(b * N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.55e+90], If[GreaterEqual[b, 0.0], N[(N[(0.5 / a), $MachinePrecision] * N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / t$95$1), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$2, (-N[(c / N[(b + b), $MachinePrecision]), $MachinePrecision])]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2e153

   1. Initial program 42.2%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 97.7

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

   5. Simplified 97.7%

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

   6. Taylor expanded in b around inf

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

   8. Simplified 97.7%

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

      1. frac-2neg N/A

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

      2. flip-+ N/A

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

      3. +-inverses N/A

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

      4. distribute-neg-frac2 N/A

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

      5. metadata-eval N/A

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

      6. +-inverses N/A

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

      7. flip-+ N/A

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

      8. sub-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

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

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. flip-+ N/A

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

      16. +-inverses N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac2 N/A

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

   10. Applied egg-rr 97.7%

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

   if -2e153 < b < -5.00000000000023e-311

   1. Initial program 83.2%

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

   3. Taylor expanded in b around inf

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

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

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

      2. sub-neg N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 83.2

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

   5. Simplified 83.2%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. associate-*l* N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-commutative N/A

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

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

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

      15. *-lowering-*.f64 83.2

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

   7. Applied egg-rr 83.2%

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

   if -5.00000000000023e-311 < b < 1.54999999999999994e90

   1. Initial program 92.4%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 92.4

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

   5. Simplified 92.4%

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

      1. clear-num N/A

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

      2. metadata-eval N/A

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

      3. associate-/r/ N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

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

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

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

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

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

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

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

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

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

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

      12. associate-*l* N/A

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

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

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. +-commutative N/A

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

      19. accelerator-lowering-fma.f64 N/A

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

      20. *-commutative N/A

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

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

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

      22. *-lowering-*.f64 92.3

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

   7. Applied egg-rr 92.3%

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

   if 1.54999999999999994e90 < b

   1. Initial program 60.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 60.3

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

   5. Simplified 60.3%

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

   6. Taylor expanded in b around inf

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

   8. Simplified 98.4%

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

      1. clear-num N/A

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

      2. sub-neg N/A

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

      3. associate-/r* N/A

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

      4. sub-neg N/A

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

      5. flip-+ N/A

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

      6. +-inverses N/A

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

      7. associate-/l/ N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. flip-+ N/A

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

      11. sub-neg N/A

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

      12. clear-num N/A

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

      13. frac-2neg N/A

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

      14. sub-neg N/A

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

      15. flip-+ N/A

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

      16. +-inverses N/A

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

   10. Applied egg-rr 98.4%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{b + b}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;b \cdot \left(\frac{c}{b \cdot b} + \frac{-1}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{+90}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b + b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(-b\right) - b}{a \cdot 2}\\ t_1 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;b \leq -3.3 \cdot 10^{+153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{b + b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{+90}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t\_1 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b + b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (- b) b) (* a 2.0)))
        (t_1 (sqrt (- (* b b) (* (* 4.0 a) c)))))
   (if (<= b -3.3e+153)
     (if (>= b 0.0) t_0 (/ (* c -2.0) (+ b b)))
     (if (<= b 1.55e+90)
       (if (>= b 0.0) (/ (- (- b) t_1) (* a 2.0)) (/ (* c 2.0) (- t_1 b)))
       (if (>= b 0.0) t_0 (- (/ c (+ b b))))))))

C

double code(double a, double b, double c) {
	double t_0 = (-b - b) / (a * 2.0);
	double t_1 = sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp_1;
	if (b <= -3.3e+153) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = (c * -2.0) / (b + b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.55e+90) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_1) / (a * 2.0);
		} else {
			tmp_3 = (c * 2.0) / (t_1 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = -(c / (b + b));
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = (-b - b) / (a * 2.0d0)
    t_1 = sqrt(((b * b) - ((4.0d0 * a) * c)))
    if (b <= (-3.3d+153)) then
        if (b >= 0.0d0) then
            tmp_2 = t_0
        else
            tmp_2 = (c * (-2.0d0)) / (b + b)
        end if
        tmp_1 = tmp_2
    else if (b <= 1.55d+90) then
        if (b >= 0.0d0) then
            tmp_3 = (-b - t_1) / (a * 2.0d0)
        else
            tmp_3 = (c * 2.0d0) / (t_1 - b)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = t_0
    else
        tmp_1 = -(c / (b + b))
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (-b - b) / (a * 2.0);
	double t_1 = Math.sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp_1;
	if (b <= -3.3e+153) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = (c * -2.0) / (b + b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.55e+90) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_1) / (a * 2.0);
		} else {
			tmp_3 = (c * 2.0) / (t_1 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = -(c / (b + b));
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (-b - b) / (a * 2.0)
	t_1 = math.sqrt(((b * b) - ((4.0 * a) * c)))
	tmp_1 = 0
	if b <= -3.3e+153:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_0
		else:
			tmp_2 = (c * -2.0) / (b + b)
		tmp_1 = tmp_2
	elif b <= 1.55e+90:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (-b - t_1) / (a * 2.0)
		else:
			tmp_3 = (c * 2.0) / (t_1 - b)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = t_0
	else:
		tmp_1 = -(c / (b + b))
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(-b) - b) / Float64(a * 2.0))
	t_1 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	tmp_1 = 0.0
	if (b <= -3.3e+153)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(Float64(c * -2.0) / Float64(b + b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.55e+90)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(-b) - t_1) / Float64(a * 2.0));
		else
			tmp_3 = Float64(Float64(c * 2.0) / Float64(t_1 - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = t_0;
	else
		tmp_1 = Float64(-Float64(c / Float64(b + b)));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = (-b - b) / (a * 2.0);
	t_1 = sqrt(((b * b) - ((4.0 * a) * c)));
	tmp_2 = 0.0;
	if (b <= -3.3e+153)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_0;
		else
			tmp_3 = (c * -2.0) / (b + b);
		end
		tmp_2 = tmp_3;
	elseif (b <= 1.55e+90)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (-b - t_1) / (a * 2.0);
		else
			tmp_4 = (c * 2.0) / (t_1 - b);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = t_0;
	else
		tmp_2 = -(c / (b + b));
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[((-b) - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -3.3e+153], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(c * -2.0), $MachinePrecision] / N[(b + b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.55e+90], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$1), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(t$95$1 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$0, (-N[(c / N[(b + b), $MachinePrecision]), $MachinePrecision])]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.29999999999999994e153

   1. Initial program 42.2%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 97.7

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

   5. Simplified 97.7%

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

   6. Taylor expanded in b around inf

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

   8. Simplified 97.7%

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

      1. frac-2neg N/A

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

      2. flip-+ N/A

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

      3. +-inverses N/A

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

      4. distribute-neg-frac2 N/A

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

      5. metadata-eval N/A

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

      6. +-inverses N/A

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

      7. flip-+ N/A

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

      8. sub-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

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

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. flip-+ N/A

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

      16. +-inverses N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac2 N/A

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

   10. Applied egg-rr 97.7%

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

   if -3.29999999999999994e153 < b < 1.54999999999999994e90

   1. Initial program 87.6%

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

   if 1.54999999999999994e90 < b

   1. Initial program 60.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 60.3

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

   5. Simplified 60.3%

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

   6. Taylor expanded in b around inf

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

   8. Simplified 98.4%

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

      1. clear-num N/A

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

      2. sub-neg N/A

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

      3. associate-/r* N/A

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

      4. sub-neg N/A

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

      5. flip-+ N/A

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

      6. +-inverses N/A

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

      7. associate-/l/ N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. flip-+ N/A

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

      11. sub-neg N/A

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

      12. clear-num N/A

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

      13. frac-2neg N/A

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

      14. sub-neg N/A

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

      15. flip-+ N/A

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

      16. +-inverses N/A

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

   10. Applied egg-rr 98.4%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{+153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{b + b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{+90}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b + b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-b\right) - b\\ t_1 := \frac{c \cdot 2}{t\_0}\\ \mathbf{if}\;b \leq -2 \cdot 10^{+153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{t\_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{b + b}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;b \cdot \left(\frac{c}{b \cdot b} + \frac{-1}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-44}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\left(-b\right) - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (- b) b)) (t_1 (/ (* c 2.0) t_0)))
   (if (<= b -2e+153)
     (if (>= b 0.0) (/ t_0 (* a 2.0)) (/ (* c -2.0) (+ b b)))
     (if (<= b -5e-311)
       (if (>= b 0.0)
         (* b (+ (/ c (* b b)) (/ -1.0 a)))
         (/ (* c 2.0) (- (sqrt (fma a (* c -4.0) (* b b))) b)))
       (if (<= b 2.6e-44)
         (if (>= b 0.0) (* (/ 0.5 a) (- (- b) (sqrt (* a (* c -4.0))))) t_1)
         (if (>= b 0.0) (- (/ c b) (/ b a)) t_1))))))

C

double code(double a, double b, double c) {
	double t_0 = -b - b;
	double t_1 = (c * 2.0) / t_0;
	double tmp_1;
	if (b <= -2e+153) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0 / (a * 2.0);
		} else {
			tmp_2 = (c * -2.0) / (b + b);
		}
		tmp_1 = tmp_2;
	} else if (b <= -5e-311) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = b * ((c / (b * b)) + (-1.0 / a));
		} else {
			tmp_3 = (c * 2.0) / (sqrt(fma(a, (c * -4.0), (b * b))) - b);
		}
		tmp_1 = tmp_3;
	} else if (b <= 2.6e-44) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (0.5 / a) * (-b - sqrt((a * (c * -4.0))));
		} else {
			tmp_4 = t_1;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-b) - b)
	t_1 = Float64(Float64(c * 2.0) / t_0)
	tmp_1 = 0.0
	if (b <= -2e+153)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(t_0 / Float64(a * 2.0));
		else
			tmp_2 = Float64(Float64(c * -2.0) / Float64(b + b));
		end
		tmp_1 = tmp_2;
	elseif (b <= -5e-311)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(b * Float64(Float64(c / Float64(b * b)) + Float64(-1.0 / a)));
		else
			tmp_3 = Float64(Float64(c * 2.0) / Float64(sqrt(fma(a, Float64(c * -4.0), Float64(b * b))) - b));
		end
		tmp_1 = tmp_3;
	elseif (b <= 2.6e-44)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(0.5 / a) * Float64(Float64(-b) - sqrt(Float64(a * Float64(c * -4.0)))));
		else
			tmp_4 = t_1;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) - b), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * 2.0), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[b, -2e+153], If[GreaterEqual[b, 0.0], N[(t$95$0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -2.0), $MachinePrecision] / N[(b + b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -5e-311], If[GreaterEqual[b, 0.0], N[(b * N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 2.6e-44], If[GreaterEqual[b, 0.0], N[(N[(0.5 / a), $MachinePrecision] * N[((-b) - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1], If[GreaterEqual[b, 0.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2e153

   1. Initial program 42.2%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 97.7

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

   5. Simplified 97.7%

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

   6. Taylor expanded in b around inf

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

   8. Simplified 97.7%

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

      1. frac-2neg N/A

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

      2. flip-+ N/A

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

      3. +-inverses N/A

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

      4. distribute-neg-frac2 N/A

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

      5. metadata-eval N/A

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

      6. +-inverses N/A

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

      7. flip-+ N/A

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

      8. sub-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

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

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. flip-+ N/A

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

      16. +-inverses N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac2 N/A

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

   10. Applied egg-rr 97.7%

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

   if -2e153 < b < -5.00000000000023e-311

   1. Initial program 83.2%

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

   3. Taylor expanded in b around inf

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

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

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

      2. sub-neg N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 83.2

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

   5. Simplified 83.2%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. associate-*l* N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-commutative N/A

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

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

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

      15. *-lowering-*.f64 83.2

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

   7. Applied egg-rr 83.2%

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

   if -5.00000000000023e-311 < b < 2.5999999999999998e-44

   1. Initial program 89.6%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 89.6

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

   5. Simplified 89.6%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

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

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

      3. *-lowering-*.f64 75.8

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

   8. Simplified 75.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

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

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

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

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

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

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 75.9

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

   10. Applied egg-rr 75.9%

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

   if 2.5999999999999998e-44 < 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

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 73.7

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

   5. Simplified 73.7%

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

   6. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. /-lowering-/.f64 89.7

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

   8. Simplified 89.7%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{b + b}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;b \cdot \left(\frac{c}{b \cdot b} + \frac{-1}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-44}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\left(-b\right) - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-b\right) - b\\ t_1 := \frac{c \cdot 2}{t\_0}\\ \mathbf{if}\;b \leq -4.2 \cdot 10^{-83}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{t\_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{b + b}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;b \cdot \left(\frac{c}{b \cdot b} + \frac{-1}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 8.8 \cdot 10^{-48}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\left(-b\right) - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (- b) b)) (t_1 (/ (* c 2.0) t_0)))
   (if (<= b -4.2e-83)
     (if (>= b 0.0) (/ t_0 (* a 2.0)) (/ (* c -2.0) (+ b b)))
     (if (<= b -5e-311)
       (if (>= b 0.0)
         (* b (+ (/ c (* b b)) (/ -1.0 a)))
         (/ (* c 2.0) (- (sqrt (* -4.0 (* a c))) b)))
       (if (<= b 8.8e-48)
         (if (>= b 0.0) (* (/ 0.5 a) (- (- b) (sqrt (* a (* c -4.0))))) t_1)
         (if (>= b 0.0) (- (/ c b) (/ b a)) t_1))))))

C

double code(double a, double b, double c) {
	double t_0 = -b - b;
	double t_1 = (c * 2.0) / t_0;
	double tmp_1;
	if (b <= -4.2e-83) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0 / (a * 2.0);
		} else {
			tmp_2 = (c * -2.0) / (b + b);
		}
		tmp_1 = tmp_2;
	} else if (b <= -5e-311) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = b * ((c / (b * b)) + (-1.0 / a));
		} else {
			tmp_3 = (c * 2.0) / (sqrt((-4.0 * (a * c))) - b);
		}
		tmp_1 = tmp_3;
	} else if (b <= 8.8e-48) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (0.5 / a) * (-b - sqrt((a * (c * -4.0))));
		} else {
			tmp_4 = t_1;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    real(8) :: tmp_4
    t_0 = -b - b
    t_1 = (c * 2.0d0) / t_0
    if (b <= (-4.2d-83)) then
        if (b >= 0.0d0) then
            tmp_2 = t_0 / (a * 2.0d0)
        else
            tmp_2 = (c * (-2.0d0)) / (b + b)
        end if
        tmp_1 = tmp_2
    else if (b <= (-5d-311)) then
        if (b >= 0.0d0) then
            tmp_3 = b * ((c / (b * b)) + ((-1.0d0) / a))
        else
            tmp_3 = (c * 2.0d0) / (sqrt(((-4.0d0) * (a * c))) - b)
        end if
        tmp_1 = tmp_3
    else if (b <= 8.8d-48) then
        if (b >= 0.0d0) then
            tmp_4 = (0.5d0 / a) * (-b - sqrt((a * (c * (-4.0d0)))))
        else
            tmp_4 = t_1
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = (c / b) - (b / a)
    else
        tmp_1 = t_1
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -b - b;
	double t_1 = (c * 2.0) / t_0;
	double tmp_1;
	if (b <= -4.2e-83) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0 / (a * 2.0);
		} else {
			tmp_2 = (c * -2.0) / (b + b);
		}
		tmp_1 = tmp_2;
	} else if (b <= -5e-311) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = b * ((c / (b * b)) + (-1.0 / a));
		} else {
			tmp_3 = (c * 2.0) / (Math.sqrt((-4.0 * (a * c))) - b);
		}
		tmp_1 = tmp_3;
	} else if (b <= 8.8e-48) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (0.5 / a) * (-b - Math.sqrt((a * (c * -4.0))));
		} else {
			tmp_4 = t_1;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = -b - b
	t_1 = (c * 2.0) / t_0
	tmp_1 = 0
	if b <= -4.2e-83:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_0 / (a * 2.0)
		else:
			tmp_2 = (c * -2.0) / (b + b)
		tmp_1 = tmp_2
	elif b <= -5e-311:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = b * ((c / (b * b)) + (-1.0 / a))
		else:
			tmp_3 = (c * 2.0) / (math.sqrt((-4.0 * (a * c))) - b)
		tmp_1 = tmp_3
	elif b <= 8.8e-48:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = (0.5 / a) * (-b - math.sqrt((a * (c * -4.0))))
		else:
			tmp_4 = t_1
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = (c / b) - (b / a)
	else:
		tmp_1 = t_1
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-b) - b)
	t_1 = Float64(Float64(c * 2.0) / t_0)
	tmp_1 = 0.0
	if (b <= -4.2e-83)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(t_0 / Float64(a * 2.0));
		else
			tmp_2 = Float64(Float64(c * -2.0) / Float64(b + b));
		end
		tmp_1 = tmp_2;
	elseif (b <= -5e-311)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(b * Float64(Float64(c / Float64(b * b)) + Float64(-1.0 / a)));
		else
			tmp_3 = Float64(Float64(c * 2.0) / Float64(sqrt(Float64(-4.0 * Float64(a * c))) - b));
		end
		tmp_1 = tmp_3;
	elseif (b <= 8.8e-48)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(0.5 / a) * Float64(Float64(-b) - sqrt(Float64(a * Float64(c * -4.0)))));
		else
			tmp_4 = t_1;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = -b - b;
	t_1 = (c * 2.0) / t_0;
	tmp_2 = 0.0;
	if (b <= -4.2e-83)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_0 / (a * 2.0);
		else
			tmp_3 = (c * -2.0) / (b + b);
		end
		tmp_2 = tmp_3;
	elseif (b <= -5e-311)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = b * ((c / (b * b)) + (-1.0 / a));
		else
			tmp_4 = (c * 2.0) / (sqrt((-4.0 * (a * c))) - b);
		end
		tmp_2 = tmp_4;
	elseif (b <= 8.8e-48)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = (0.5 / a) * (-b - sqrt((a * (c * -4.0))));
		else
			tmp_5 = t_1;
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = (c / b) - (b / a);
	else
		tmp_2 = t_1;
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) - b), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * 2.0), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[b, -4.2e-83], If[GreaterEqual[b, 0.0], N[(t$95$0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -2.0), $MachinePrecision] / N[(b + b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -5e-311], If[GreaterEqual[b, 0.0], N[(b * N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 8.8e-48], If[GreaterEqual[b, 0.0], N[(N[(0.5 / a), $MachinePrecision] * N[((-b) - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1], If[GreaterEqual[b, 0.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -4.1999999999999998e-83

   1. Initial program 71.5%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 87.1

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

   5. Simplified 87.1%

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

   6. Taylor expanded in b around inf

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

   8. Simplified 87.1%

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

      1. frac-2neg N/A

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

      2. flip-+ N/A

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

      3. +-inverses N/A

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

      4. distribute-neg-frac2 N/A

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

      5. metadata-eval N/A

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

      6. +-inverses N/A

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

      7. flip-+ N/A

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

      8. sub-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

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

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. flip-+ N/A

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

      16. +-inverses N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac2 N/A

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

   10. Applied egg-rr 87.1%

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

   if -4.1999999999999998e-83 < b < -5.00000000000023e-311

   1. Initial program 66.6%

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

   3. Taylor expanded in b around inf

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

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

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

      2. sub-neg N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 66.6

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

   5. Simplified 66.6%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

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

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

      3. *-lowering-*.f64 64.1

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

   8. Simplified 64.1%

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

   if -5.00000000000023e-311 < b < 8.8000000000000005e-48

   1. Initial program 89.6%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 89.6

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

   5. Simplified 89.6%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

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

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

      3. *-lowering-*.f64 75.8

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

   8. Simplified 75.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

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

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

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

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

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

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 75.9

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

   10. Applied egg-rr 75.9%

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

   if 8.8000000000000005e-48 < 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

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 73.7

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

   5. Simplified 73.7%

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

   6. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. /-lowering-/.f64 89.7

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

   8. Simplified 89.7%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-83}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{b + b}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;b \cdot \left(\frac{c}{b \cdot b} + \frac{-1}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 8.8 \cdot 10^{-48}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\left(-b\right) - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{if}\;b \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{b + b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{+90}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b + b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (- b) b) (* a 2.0))))
   (if (<= b -1e+154)
     (if (>= b 0.0) t_0 (/ (* c -2.0) (+ b b)))
     (if (<= b 1.55e+90)
       (if (>= b 0.0)
         (* (/ 0.5 a) (- (- b) (sqrt (fma a (* c -4.0) (* b b)))))
         (/ (* c 2.0) (- (sqrt (- (* b b) (* (* 4.0 a) c))) b)))
       (if (>= b 0.0) t_0 (- (/ c (+ b b))))))))

C

double code(double a, double b, double c) {
	double t_0 = (-b - b) / (a * 2.0);
	double tmp_1;
	if (b <= -1e+154) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = (c * -2.0) / (b + b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.55e+90) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (0.5 / a) * (-b - sqrt(fma(a, (c * -4.0), (b * b))));
		} else {
			tmp_3 = (c * 2.0) / (sqrt(((b * b) - ((4.0 * a) * c))) - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = -(c / (b + b));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(-b) - b) / Float64(a * 2.0))
	tmp_1 = 0.0
	if (b <= -1e+154)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(Float64(c * -2.0) / Float64(b + b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.55e+90)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(0.5 / a) * Float64(Float64(-b) - sqrt(fma(a, Float64(c * -4.0), Float64(b * b)))));
		else
			tmp_3 = Float64(Float64(c * 2.0) / Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = t_0;
	else
		tmp_1 = Float64(-Float64(c / Float64(b + b)));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[((-b) - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1e+154], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(c * -2.0), $MachinePrecision] / N[(b + b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.55e+90], If[GreaterEqual[b, 0.0], N[(N[(0.5 / a), $MachinePrecision] * N[((-b) - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$0, (-N[(c / N[(b + b), $MachinePrecision]), $MachinePrecision])]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.00000000000000004e154

   1. Initial program 42.2%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 97.7

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

   5. Simplified 97.7%

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

   6. Taylor expanded in b around inf

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

   8. Simplified 97.7%

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

      1. frac-2neg N/A

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

      2. flip-+ N/A

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

      3. +-inverses N/A

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

      4. distribute-neg-frac2 N/A

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

      5. metadata-eval N/A

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

      6. +-inverses N/A

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

      7. flip-+ N/A

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

      8. sub-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

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

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. flip-+ N/A

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

      16. +-inverses N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac2 N/A

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

   10. Applied egg-rr 97.7%

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

   if -1.00000000000000004e154 < b < 1.54999999999999994e90

   1. Initial program 87.6%

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

   4. Applied egg-rr 87.5%

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

   if 1.54999999999999994e90 < b

   1. Initial program 60.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 60.3

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

   5. Simplified 60.3%

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

   6. Taylor expanded in b around inf

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

   8. Simplified 98.4%

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

      1. clear-num N/A

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

      2. sub-neg N/A

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

      3. associate-/r* N/A

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

      4. sub-neg N/A

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

      5. flip-+ N/A

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

      6. +-inverses N/A

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

      7. associate-/l/ N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. flip-+ N/A

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

      11. sub-neg N/A

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

      12. clear-num N/A

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

      13. frac-2neg N/A

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

      14. sub-neg N/A

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

      15. flip-+ N/A

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

      16. +-inverses N/A

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

   10. Applied egg-rr 98.4%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{b + b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{+90}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b + b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{if}\;b \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{b + b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{+90}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b + b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (- b) b) (* a 2.0))))
   (if (<= b -1e+153)
     (if (>= b 0.0) t_0 (/ (* c -2.0) (+ b b)))
     (if (<= b 1.55e+90)
       (if (>= b 0.0)
         (/ (- (- b) (sqrt (fma (* c -4.0) a (* b b)))) (* a 2.0))
         (* c (/ 2.0 (- (sqrt (fma a (* c -4.0) (* b b))) b))))
       (if (>= b 0.0) t_0 (- (/ c (+ b b))))))))

C

double code(double a, double b, double c) {
	double t_0 = (-b - b) / (a * 2.0);
	double tmp_1;
	if (b <= -1e+153) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = (c * -2.0) / (b + b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.55e+90) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - sqrt(fma((c * -4.0), a, (b * b)))) / (a * 2.0);
		} else {
			tmp_3 = c * (2.0 / (sqrt(fma(a, (c * -4.0), (b * b))) - b));
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = -(c / (b + b));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(-b) - b) / Float64(a * 2.0))
	tmp_1 = 0.0
	if (b <= -1e+153)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(Float64(c * -2.0) / Float64(b + b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.55e+90)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(-b) - sqrt(fma(Float64(c * -4.0), a, Float64(b * b)))) / Float64(a * 2.0));
		else
			tmp_3 = Float64(c * Float64(2.0 / Float64(sqrt(fma(a, Float64(c * -4.0), Float64(b * b))) - b)));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = t_0;
	else
		tmp_1 = Float64(-Float64(c / Float64(b + b)));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[((-b) - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1e+153], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(c * -2.0), $MachinePrecision] / N[(b + b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.55e+90], If[GreaterEqual[b, 0.0], N[(N[((-b) - N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c * N[(2.0 / N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$0, (-N[(c / N[(b + b), $MachinePrecision]), $MachinePrecision])]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1e153

   1. Initial program 42.2%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 97.7

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

   5. Simplified 97.7%

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

   6. Taylor expanded in b around inf

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

   8. Simplified 97.7%

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

      1. frac-2neg N/A

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

      2. flip-+ N/A

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

      3. +-inverses N/A

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

      4. distribute-neg-frac2 N/A

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

      5. metadata-eval N/A

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

      6. +-inverses N/A

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

      7. flip-+ N/A

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

      8. sub-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

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

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. flip-+ N/A

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

      16. +-inverses N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac2 N/A

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

   10. Applied egg-rr 97.7%

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

   if -1e153 < b < 1.54999999999999994e90

   1. Initial program 87.6%

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

   4. Applied egg-rr 87.5%

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

      1. associate-*l* N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

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

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

      12. *-lowering-*.f64 87.5

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

   6. Applied egg-rr 87.5%

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

   if 1.54999999999999994e90 < b

   1. Initial program 60.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 60.3

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

   5. Simplified 60.3%

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

   6. Taylor expanded in b around inf

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

   8. Simplified 98.4%

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

      1. clear-num N/A

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

      2. sub-neg N/A

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

      3. associate-/r* N/A

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

      4. sub-neg N/A

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

      5. flip-+ N/A

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

      6. +-inverses N/A

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

      7. associate-/l/ N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. flip-+ N/A

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

      11. sub-neg N/A

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

      12. clear-num N/A

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

      13. frac-2neg N/A

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

      14. sub-neg N/A

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

      15. flip-+ N/A

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

      16. +-inverses N/A

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

   10. Applied egg-rr 98.4%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{b + b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{+90}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b + b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.22 \cdot 10^{-82}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{b + b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;b \cdot \left(\frac{c}{b \cdot b} + \frac{-1}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.22e-82)
   (if (>= b 0.0) (/ (- (- b) b) (* a 2.0)) (/ (* c -2.0) (+ b b)))
   (if (>= b 0.0)
     (* b (+ (/ c (* b b)) (/ -1.0 a)))
     (/ (* c 2.0) (- (sqrt (* -4.0 (* a c))) b)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1.22e-82) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-b - b) / (a * 2.0);
		} else {
			tmp_2 = (c * -2.0) / (b + b);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = b * ((c / (b * b)) + (-1.0 / a));
	} else {
		tmp_1 = (c * 2.0) / (sqrt((-4.0 * (a * c))) - 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) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    if (b <= (-1.22d-82)) then
        if (b >= 0.0d0) then
            tmp_2 = (-b - b) / (a * 2.0d0)
        else
            tmp_2 = (c * (-2.0d0)) / (b + b)
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = b * ((c / (b * b)) + ((-1.0d0) / a))
    else
        tmp_1 = (c * 2.0d0) / (sqrt(((-4.0d0) * (a * c))) - b)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1.22e-82) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-b - b) / (a * 2.0);
		} else {
			tmp_2 = (c * -2.0) / (b + b);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = b * ((c / (b * b)) + (-1.0 / a));
	} else {
		tmp_1 = (c * 2.0) / (Math.sqrt((-4.0 * (a * c))) - b);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if b <= -1.22e-82:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (-b - b) / (a * 2.0)
		else:
			tmp_2 = (c * -2.0) / (b + b)
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = b * ((c / (b * b)) + (-1.0 / a))
	else:
		tmp_1 = (c * 2.0) / (math.sqrt((-4.0 * (a * c))) - b)
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -1.22e-82)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(Float64(-b) - b) / Float64(a * 2.0));
		else
			tmp_2 = Float64(Float64(c * -2.0) / Float64(b + b));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(b * Float64(Float64(c / Float64(b * b)) + Float64(-1.0 / a)));
	else
		tmp_1 = Float64(Float64(c * 2.0) / Float64(sqrt(Float64(-4.0 * Float64(a * c))) - b));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp_2 = 0.0;
	if (b <= -1.22e-82)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (-b - b) / (a * 2.0);
		else
			tmp_3 = (c * -2.0) / (b + b);
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = b * ((c / (b * b)) + (-1.0 / a));
	else
		tmp_2 = (c * 2.0) / (sqrt((-4.0 * (a * c))) - b);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.22e-82], If[GreaterEqual[b, 0.0], N[(N[((-b) - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -2.0), $MachinePrecision] / N[(b + b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(b * N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.22000000000000001e-82

   1. Initial program 71.5%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 87.1

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

   5. Simplified 87.1%

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

   6. Taylor expanded in b around inf

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

   8. Simplified 87.1%

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

      1. frac-2neg N/A

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

      2. flip-+ N/A

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

      3. +-inverses N/A

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

      4. distribute-neg-frac2 N/A

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

      5. metadata-eval N/A

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

      6. +-inverses N/A

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

      7. flip-+ N/A

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

      8. sub-neg N/A

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

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

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

      10. *-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

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

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. flip-+ N/A

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

      16. +-inverses N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac2 N/A

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

   10. Applied egg-rr 87.1%

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

   if -1.22000000000000001e-82 < b

   1. Initial program 76.6%

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

   3. Taylor expanded in b around inf

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

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

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

      2. sub-neg N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 64.6

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

   5. Simplified 64.6%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

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

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

      3. *-lowering-*.f64 64.0

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

   8. Simplified 64.0%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.22 \cdot 10^{-82}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{b + b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;b \cdot \left(\frac{c}{b \cdot b} + \frac{-1}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.8%

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

3. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 73.2

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

5. Simplified 73.2%

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

6. Taylor expanded in c around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. /-lowering-/.f64 65.9

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

8. Simplified 65.9%

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

9. Final simplification 65.9%

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

Alternative 10: 67.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.8%

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

3. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 73.2

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

5. Simplified 73.2%

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

6. Taylor expanded in b around inf

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

8. Simplified 65.6%

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

   1. frac-2neg N/A

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

   2. flip-+ N/A

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

   3. +-inverses N/A

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

   4. distribute-neg-frac2 N/A

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

   5. metadata-eval N/A

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

   6. +-inverses N/A

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

   7. flip-+ N/A

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

   8. sub-neg N/A

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

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

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

   10. *-commutative N/A

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

   11. distribute-rgt-neg-in N/A

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

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

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

   13. metadata-eval N/A

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

   14. sub-neg N/A

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

   15. flip-+ N/A

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

   16. +-inverses N/A

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

   17. metadata-eval N/A

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

   18. distribute-neg-frac2 N/A

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

10. Applied egg-rr 65.6%

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

11. Final simplification 65.6%

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

Alternative 11: 47.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.8%

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

3. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 73.2

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

5. Simplified 73.2%

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

6. Taylor expanded in b around inf

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

8. Simplified 65.6%

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

   1. clear-num N/A

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

   2. sub-neg N/A

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

   3. associate-/r* N/A

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

   4. sub-neg N/A

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

   5. flip-+ N/A

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

   6. +-inverses N/A

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

   7. associate-/l/ N/A

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

   8. metadata-eval N/A

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

   9. +-inverses N/A

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

   10. flip-+ N/A

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

   11. sub-neg N/A

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

   12. clear-num N/A

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

   13. frac-2neg N/A

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

   14. sub-neg N/A

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

   15. flip-+ N/A

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

   16. +-inverses N/A

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

10. Applied egg-rr 46.9%

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

11. Final simplification 46.9%

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

Alternative 12: 36.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.8%

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

3. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 73.2

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

5. Simplified 73.2%

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

6. Taylor expanded in b around inf

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

8. Simplified 65.6%

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

   1. frac-2neg N/A

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

   2. sub-neg N/A

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

   3. div-inv N/A

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

   4. sub-neg N/A

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

   5. flip-+ N/A

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

   6. +-inverses N/A

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

   7. distribute-neg-frac2 N/A

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

   8. metadata-eval N/A

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

   9. sqr-neg N/A

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

   10. sqr-neg N/A

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

   11. +-inverses N/A

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

   12. +-inverses N/A

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

   13. +-inverses N/A

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

   14. sqr-neg N/A

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

   15. sqr-neg N/A

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

10. Applied egg-rr 35.4%

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

11. Final simplification 35.4%

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

Alternative 13: 36.3% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.8%

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

3. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 73.2

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

5. Simplified 73.2%

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

6. Taylor expanded in b around inf

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

8. Simplified 65.6%

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

   1. frac-2neg N/A

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

   2. sub-neg N/A

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

   3. distribute-frac-neg N/A

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

   4. sub-neg N/A

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

   5. flip-+ N/A

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

   6. +-inverses N/A

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

   7. distribute-neg-frac2 N/A

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

   8. metadata-eval N/A

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

   9. +-inverses N/A

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

   10. flip-+ N/A

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

   11. sub-neg N/A

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

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

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

   13. clear-num N/A

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

   14. associate-/r* N/A

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

10. Applied egg-rr 35.4%

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

11. Taylor expanded in b around 0

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

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

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

    2. mul-1-neg N/A

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

    3. distribute-neg-frac2 N/A

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

    4. mul-1-neg N/A

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

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

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

    6. mul-1-neg N/A

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

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

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

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

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

    9. metadata-eval N/A

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

    10. *-commutative N/A

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

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

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

    12. *-commutative N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{\mathsf{neg}\left(a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\mathsf{neg}\left(a\right)}\\ \end{array} \]

    13. *-lowering-*.f64 35.4

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot b\right) \cdot -2\\ \end{array} \]

13. Simplified 35.4%

    \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot b\right) \cdot -2\\ } \end{array}} \]

14. Final simplification 35.4%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(b \cdot c\right)\\ \end{array} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024205 
(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)))))))