jeff quadratic root 2

Percentage Accurate: 73.1% → 91.2%

Time: 18.8s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

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 = (2.0 * c) / (-b - t_0);
	} else {
		tmp = (-b + t_0) / (2.0 * a);
	}
	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 = (2.0d0 * c) / (-b - t_0)
    else
        tmp = (-b + t_0) / (2.0d0 * a)
    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 = (2.0 * c) / (-b - t_0);
	} else {
		tmp = (-b + t_0) / (2.0 * a);
	}
	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 = (2.0 * c) / (-b - t_0)
	else:
		tmp = (-b + t_0) / (2.0 * a)
	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(2.0 * c) / Float64(Float64(-b) - t_0));
	else
		tmp = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
	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 = (2.0 * c) / (-b - t_0);
	else
		tmp = (-b + t_0) / (2.0 * a);
	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[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Initial Program: 73.1% accurate, 1.0× speedup

Math

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

FPCore

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

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 = (2.0 * c) / (-b - t_0);
	} else {
		tmp = (-b + t_0) / (2.0 * a);
	}
	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 = (2.0d0 * c) / (-b - t_0)
    else
        tmp = (-b + t_0) / (2.0d0 * a)
    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 = (2.0 * c) / (-b - t_0);
	} else {
		tmp = (-b + t_0) / (2.0 * a);
	}
	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 = (2.0 * c) / (-b - t_0)
	else:
		tmp = (-b + t_0) / (2.0 * a)
	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(2.0 * c) / Float64(Float64(-b) - t_0));
	else
		tmp = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
	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 = (2.0 * c) / (-b - t_0);
	else
		tmp = (-b + t_0) / (2.0 * a);
	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[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 91.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ t_1 := \frac{t\_0 - b}{2 \cdot a}\\ \mathbf{if}\;b \leq -2 \cdot 10^{+147}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{+84}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot \mathsf{fma}\left(2, a \cdot \frac{c}{b \cdot b}, -2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0))))) (t_1 (/ (- t_0 b) (* 2.0 a))))
   (if (<= b -2e+147)
     (if (>= b 0.0) (/ b a) (/ b (- a)))
     (if (<= b 1.5e+84)
       (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_0)) t_1)
       (if (>= b 0.0)
         (/ (* 2.0 c) (* b (fma 2.0 (* a (/ c (* b b))) -2.0)))
         t_1)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double t_1 = (t_0 - b) / (2.0 * a);
	double tmp_1;
	if (b <= -2e+147) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b / a;
		} else {
			tmp_2 = b / -a;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.5e+84) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_3 = t_1;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (b * fma(2.0, (a * (c / (b * b))), -2.0));
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	t_1 = Float64(Float64(t_0 - b) / Float64(2.0 * a))
	tmp_1 = 0.0
	if (b <= -2e+147)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(b / a);
		else
			tmp_2 = Float64(b / Float64(-a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.5e+84)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0));
		else
			tmp_3 = t_1;
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(b * fma(2.0, Float64(a * Float64(c / Float64(b * b))), -2.0)));
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2e+147], If[GreaterEqual[b, 0.0], N[(b / a), $MachinePrecision], N[(b / (-a)), $MachinePrecision]], If[LessEqual[b, 1.5e+84], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(b * N[(2.0 * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2e147

   1. Initial program 37.1%

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

   4. Applied rewrites 37.1%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 37.1

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

   7. Applied rewrites 37.1%

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

   8. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 97.3

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

   10. Applied rewrites 97.3%

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

   if -2e147 < b < 1.49999999999999998e84

   1. Initial program 92.0%

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

   if 1.49999999999999998e84 < b

   1. Initial program 65.0%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 92.5

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

   5. Applied rewrites 92.5%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+147}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{+84}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot \mathsf{fma}\left(2, a \cdot \frac{c}{b \cdot b}, -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{b \cdot b}\\ \mathbf{if}\;b \leq -2 \cdot 10^{+147}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;b \cdot \left(\frac{1}{a} - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{+84}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(t\_0 + \frac{-1}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ c (* b b))))
   (if (<= b -2e+147)
     (if (>= b 0.0) (/ b a) (/ b (- a)))
     (if (<= b -5e-310)
       (if (>= b 0.0)
         (* b (- (/ 1.0 a) t_0))
         (/ (- (sqrt (fma c (* a -4.0) (* b b))) b) (* 2.0 a)))
       (if (<= b 1.5e+84)
         (if (>= b 0.0)
           (* c (/ -2.0 (+ b (sqrt (fma b b (* a (* c -4.0)))))))
           (* b (+ t_0 (/ -1.0 a))))
         (if (>= b 0.0) (/ c (- b)) (/ (- (- b) b) (* 2.0 a))))))))

C

double code(double a, double b, double c) {
	double t_0 = c / (b * b);
	double tmp_1;
	if (b <= -2e+147) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b / a;
		} else {
			tmp_2 = b / -a;
		}
		tmp_1 = tmp_2;
	} else if (b <= -5e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = b * ((1.0 / a) - t_0);
		} else {
			tmp_3 = (sqrt(fma(c, (a * -4.0), (b * b))) - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.5e+84) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = c * (-2.0 / (b + sqrt(fma(b, b, (a * (c * -4.0))))));
		} else {
			tmp_4 = b * (t_0 + (-1.0 / a));
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = c / -b;
	} else {
		tmp_1 = (-b - b) / (2.0 * a);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(c / Float64(b * b))
	tmp_1 = 0.0
	if (b <= -2e+147)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(b / a);
		else
			tmp_2 = Float64(b / Float64(-a));
		end
		tmp_1 = tmp_2;
	elseif (b <= -5e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(b * Float64(Float64(1.0 / a) - t_0));
		else
			tmp_3 = Float64(Float64(sqrt(fma(c, Float64(a * -4.0), Float64(b * b))) - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (b <= 1.5e+84)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(c * Float64(-2.0 / Float64(b + sqrt(fma(b, b, Float64(a * Float64(c * -4.0)))))));
		else
			tmp_4 = Float64(b * Float64(t_0 + Float64(-1.0 / a)));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(c / Float64(-b));
	else
		tmp_1 = Float64(Float64(Float64(-b) - b) / Float64(2.0 * a));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2e+147], If[GreaterEqual[b, 0.0], N[(b / a), $MachinePrecision], N[(b / (-a)), $MachinePrecision]], If[LessEqual[b, -5e-310], If[GreaterEqual[b, 0.0], N[(b * N[(N[(1.0 / a), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.5e+84], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[(b + N[Sqrt[N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(t$95$0 + N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(c / (-b)), $MachinePrecision], N[(N[((-b) - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2e147

   1. Initial program 43.7%

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

   4. Applied rewrites 43.7%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 43.7

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

   7. Applied rewrites 43.7%

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

   8. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 98.3

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

   10. Applied rewrites 98.3%

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

   if -2e147 < b < -4.999999999999985e-310

   1. Initial program 88.4%

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

   4. Applied rewrites 88.4%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 88.4

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

   7. Applied rewrites 88.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 88.4

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lift-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-fma.f64 88.4

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

   9. Applied rewrites 88.4%

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

   if -4.999999999999985e-310 < b < 1.49999999999999998e84

   1. Initial program 86.4%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 86.4

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

   5. Applied rewrites 86.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

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

      9. lift-neg.f64 N/A

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

      10. distribute-neg-out N/A

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

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

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

      16. lift-*.f64 N/A

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

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

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

      18. metadata-eval N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

   7. Applied rewrites 86.2%

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

   if 1.49999999999999998e84 < b

   1. Initial program 57.4%

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

   4. Applied rewrites 30.9%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 2.8

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

   7. Applied rewrites 2.8%

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

   8. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 2.8

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

   10. Applied rewrites 2.8%

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

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 95.5%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+147}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;b \cdot \left(\frac{1}{a} - \frac{c}{b \cdot b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{+84}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\frac{c}{b \cdot b} + \frac{-1}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024228 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))