jeff quadratic root 1

Percentage Accurate: 71.6% → 90.4%

Time: 15.1s

Alternatives: 4

Speedup: 0.9×

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: 71.6% 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: 90.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+85}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(2 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{+117}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}{\left(-a\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b - \left(-b\right)} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+85)
   (if (>= b 0.0) (* (/ (- 0.5) a) (* 2.0 b)) (/ c (- b)))
   (if (<= b 5.4e+117)
     (if (>= b 0.0)
       (/ (+ (sqrt (fma (* c a) -4.0 (* b b))) b) (* (- a) 2.0))
       (/ (* c 2.0) (- (sqrt (fma (* c -4.0) a (* b b))) b)))
     (if (>= b 0.0) (- (/ c b) (/ b a)) (* (/ -2.0 (- b (- b))) c)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -5e+85) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-0.5 / a) * (2.0 * b);
		} else {
			tmp_2 = c / -b;
		}
		tmp_1 = tmp_2;
	} else if (b <= 5.4e+117) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (sqrt(fma((c * a), -4.0, (b * b))) + b) / (-a * 2.0);
		} else {
			tmp_3 = (c * 2.0) / (sqrt(fma((c * -4.0), a, (b * b))) - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = (-2.0 / (b - -b)) * c;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -5e+85)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(Float64(-0.5) / a) * Float64(2.0 * b));
		else
			tmp_2 = Float64(c / Float64(-b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 5.4e+117)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(sqrt(fma(Float64(c * a), -4.0, Float64(b * b))) + b) / Float64(Float64(-a) * 2.0));
		else
			tmp_3 = Float64(Float64(c * 2.0) / Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp_1 = Float64(Float64(-2.0 / Float64(b - Float64(-b))) * c);
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+85], If[GreaterEqual[b, 0.0], N[(N[((-0.5) / a), $MachinePrecision] * N[(2.0 * b), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]], If[LessEqual[b, 5.4e+117], If[GreaterEqual[b, 0.0], N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] / N[((-a) * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 / N[(b - (-b)), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.0000000000000001e85

   1. Initial program 55.2%

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

   4. Applied rewrites 55.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 55.2

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-neg-out N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

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

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

   6. Applied rewrites 55.2%

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

   7. Taylor expanded in c around 0

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

      1. lower-*.f64 55.2

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

   9. Applied rewrites 55.2%

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

   10. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

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

       4. lower-/.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f64 95.2

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

   12. Applied rewrites 95.2%

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

   if -5.0000000000000001e85 < b < 5.4000000000000005e117

   1. Initial program 90.3%

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

   4. Applied rewrites 90.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 90.2

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

   6. Applied rewrites 90.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-neg.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. frac-2neg N/A

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

      11. lower-/.f64 90.3

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

   8. Applied rewrites 90.3%

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

   if 5.4000000000000005e117 < b

   1. Initial program 52.4%

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

   4. Applied rewrites 52.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 52.4

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

   6. Applied rewrites 52.4%

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

   7. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 52.4

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

   9. Applied rewrites 52.4%

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

   10. Taylor expanded in c around 0

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

       1. +-commutative N/A

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

       2. mul-1-neg N/A

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

       3. unsub-neg N/A

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

       4. lower--.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-/.f64 98.5

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

   12. Applied rewrites 98.5%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+85}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(2 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{+117}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}{\left(-a\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b - \left(-b\right)} \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\\ \mathbf{if}\;b \leq -5 \cdot 10^{+85}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(2 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{+117}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{t\_0 + b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b - \left(-b\right)} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma -4.0 (* c a) (* b b)))))
   (if (<= b -5e+85)
     (if (>= b 0.0) (* (/ (- 0.5) a) (* 2.0 b)) (/ c (- b)))
     (if (<= b 5.4e+117)
       (if (>= b 0.0) (* -0.5 (/ (+ t_0 b) a)) (/ (* c 2.0) (- t_0 b)))
       (if (>= b 0.0) (- (/ c b) (/ b a)) (* (/ -2.0 (- b (- b))) c))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma(-4.0, (c * a), (b * b)));
	double tmp_1;
	if (b <= -5e+85) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-0.5 / a) * (2.0 * b);
		} else {
			tmp_2 = c / -b;
		}
		tmp_1 = tmp_2;
	} else if (b <= 5.4e+117) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -0.5 * ((t_0 + b) / a);
		} else {
			tmp_3 = (c * 2.0) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = (-2.0 / (b - -b)) * c;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(-4.0, Float64(c * a), Float64(b * b)))
	tmp_1 = 0.0
	if (b <= -5e+85)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(Float64(-0.5) / a) * Float64(2.0 * b));
		else
			tmp_2 = Float64(c / Float64(-b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 5.4e+117)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(-0.5 * Float64(Float64(t_0 + b) / a));
		else
			tmp_3 = Float64(Float64(c * 2.0) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp_1 = Float64(Float64(-2.0 / Float64(b - Float64(-b))) * c);
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -5e+85], If[GreaterEqual[b, 0.0], N[(N[((-0.5) / a), $MachinePrecision] * N[(2.0 * b), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]], If[LessEqual[b, 5.4e+117], If[GreaterEqual[b, 0.0], N[(-0.5 * N[(N[(t$95$0 + b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 / N[(b - (-b)), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.0000000000000001e85

   1. Initial program 55.2%

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

   4. Applied rewrites 55.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 55.2

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-neg-out N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

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

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

   6. Applied rewrites 55.2%

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

   7. Taylor expanded in c around 0

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

      1. lower-*.f64 55.2

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

   9. Applied rewrites 55.2%

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

   10. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

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

       4. lower-/.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f64 95.2

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

   12. Applied rewrites 95.2%

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

   if -5.0000000000000001e85 < b < 5.4000000000000005e117

   1. Initial program 90.3%

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

   4. Applied rewrites 90.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 90.2

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

   6. Applied rewrites 90.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. lift-neg.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. frac-2neg N/A

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

      11. lower-/.f64 90.3

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

   8. Applied rewrites 90.3%

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

   9. Taylor expanded in c around 0

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

       1. lower->=.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-+.f64 N/A

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

       7. lower-sqrt.f64 N/A

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

       8. lower-fma.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. associate-*r/ N/A

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

   11. Applied rewrites 90.3%

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

   if 5.4000000000000005e117 < b

   1. Initial program 52.4%

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

   4. Applied rewrites 52.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 52.4

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

   6. Applied rewrites 52.4%

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

   7. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 52.4

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

   9. Applied rewrites 52.4%

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

   10. Taylor expanded in c around 0

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

       1. +-commutative N/A

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

       2. mul-1-neg N/A

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

       3. unsub-neg N/A

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

       4. lower--.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-/.f64 98.5

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

   12. Applied rewrites 98.5%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+85}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(2 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{+117}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b - \left(-b\right)} \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b >= 0.0d0) then
        tmp = (-0.5d0 / a) * (2.0d0 * b)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

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

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = (-0.5 / a) * (2.0 * b)
	else:
		tmp = c / -b
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (-0.5 / a) * (2.0 * b);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Initial program 73.6%

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

4. Applied rewrites 73.6%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. lift-/.f64 N/A

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

   8. lower-*.f64 73.5

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

   9. lift--.f64 N/A

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

   10. sub-neg N/A

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

   11. lift-neg.f64 N/A

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

   12. distribute-neg-out N/A

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

   13. lift--.f64 N/A

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

   14. lift-*.f64 N/A

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

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

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

6. Applied rewrites 73.5%

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

7. Taylor expanded in c around 0

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

   1. lower-*.f64 73.2

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

9. Applied rewrites 73.2%

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

10. Taylor expanded in b around -inf

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

    1. mul-1-neg N/A

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

    2. distribute-neg-frac2 N/A

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

    3. mul-1-neg N/A

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

    4. lower-/.f64 N/A

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

    5. mul-1-neg N/A

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

    6. lower-neg.f64 72.8

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

12. Applied rewrites 72.8%

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

13. Final simplification 72.8%

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

Alternative 4: 68.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.6%

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

4. Applied rewrites 73.6%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

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

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

   7. metadata-eval N/A

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

   8. associate-*r* N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-*r* N/A

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

   16. lower-fma.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 73.6

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

6. Applied rewrites 73.6%

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

7. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 73.1

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

9. Applied rewrites 73.1%

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

10. Taylor expanded in c around 0

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

    1. lower-*.f64 72.8

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

12. Applied rewrites 72.8%

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

13. Final simplification 72.8%

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

Reproduce

herbie shell --seed 2024272 
(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)))))))