jeff quadratic root 2

Percentage Accurate: 72.1% → 90.5%

Time: 28.1s

Alternatives: 6

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: 72.1% 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: 90.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\\ t_1 := \frac{2 \cdot c}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \mathbf{if}\;b \leq -2 \cdot 10^{+117}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \mathsf{fma}\left(a, \frac{c}{b \cdot -0.5}, b\right)}{a \cdot -2}\\ \end{array}\\ \mathbf{elif}\;b \leq 5.3 \cdot 10^{+94}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - t\_0}{a \cdot -2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|c \cdot \frac{a}{b \cdot -0.5}\right|}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma c (* a -4.0) (* b b))))
        (t_1 (/ (* 2.0 c) (* 2.0 (fma a (/ c b) (- b))))))
   (if (<= b -2e+117)
     (if (>= b 0.0) t_1 (/ (+ b (fma a (/ c (* b -0.5)) b)) (* a -2.0)))
     (if (<= b 5.3e+94)
       (if (>= b 0.0) (* c (/ -2.0 (+ b t_0))) (/ (- b t_0) (* a -2.0)))
       (if (>= b 0.0) t_1 (/ (fabs (* c (/ a (* b -0.5)))) (* 2.0 a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma(c, (a * -4.0), (b * b)));
	double t_1 = (2.0 * c) / (2.0 * fma(a, (c / b), -b));
	double tmp_1;
	if (b <= -2e+117) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = (b + fma(a, (c / (b * -0.5)), b)) / (a * -2.0);
		}
		tmp_1 = tmp_2;
	} else if (b <= 5.3e+94) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = c * (-2.0 / (b + t_0));
		} else {
			tmp_3 = (b - t_0) / (a * -2.0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = t_1;
	} else {
		tmp_1 = fabs((c * (a / (b * -0.5)))) / (2.0 * a);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(c, Float64(a * -4.0), Float64(b * b)))
	t_1 = Float64(Float64(2.0 * c) / Float64(2.0 * fma(a, Float64(c / b), Float64(-b))))
	tmp_1 = 0.0
	if (b <= -2e+117)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(Float64(b + fma(a, Float64(c / Float64(b * -0.5)), b)) / Float64(a * -2.0));
		end
		tmp_1 = tmp_2;
	elseif (b <= 5.3e+94)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(c * Float64(-2.0 / Float64(b + t_0)));
		else
			tmp_3 = Float64(Float64(b - t_0) / Float64(a * -2.0));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = t_1;
	else
		tmp_1 = Float64(abs(Float64(c * Float64(a / Float64(b * -0.5)))) / Float64(2.0 * a));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * c), $MachinePrecision] / N[(2.0 * N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2e+117], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(b + N[(a * N[(c / N[(b * -0.5), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 5.3e+94], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[(b + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - t$95$0), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$1, N[(N[Abs[N[(c * N[(a / N[(b * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.0000000000000001e117

   1. Initial program 46.2%

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

   3. Taylor expanded in a around 0 46.2%

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

      1. distribute-lft-out-- 46.2%

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

      2. associate-/l* 46.2%

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

      3. fma-neg 46.2%

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

   5. Simplified 46.2%

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

   6. Taylor expanded in a around 0 2.3%

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

      1. associate-*r/ 2.3%

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

      2. associate-*r* 2.3%

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

   8. Simplified 2.3%

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

      1. add-sqr-sqrt 1.5%

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

      2. sqrt-unprod 2.5%

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

      3. pow2 2.5%

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

      4. associate-*l* 2.5%

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

      5. *-un-lft-identity 2.5%

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

      6. times-frac 2.5%

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

      7. metadata-eval 2.5%

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

   10. Applied egg-rr 2.5%

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

       1. unpow2 2.5%

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

       2. rem-sqrt-square 2.5%

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

       3. *-commutative 2.5%

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

       4. associate-*l/ 2.5%

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

       5. associate-*l* 2.5%

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

       6. associate-*r/ 2.4%

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

       7. associate-*r/ 2.4%

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

   12. Simplified 2.4%

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

   14. Applied egg-rr 97.9%

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

       1. associate-*r/ 98.3%

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

       2. *-rgt-identity 98.3%

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

       3. *-commutative 98.3%

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

   16. Simplified 98.3%

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

   if -2.0000000000000001e117 < b < 5.30000000000000003e94

   1. Initial program 89.1%

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

   3. Simplified 89.2%

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

   if 5.30000000000000003e94 < b

   1. Initial program 54.5%

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

   3. Taylor expanded in a around 0 96.4%

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

      1. distribute-lft-out-- 96.4%

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

      2. associate-/l* 96.7%

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

      3. fma-neg 96.7%

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

   5. Simplified 96.7%

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

   6. Taylor expanded in a around 0 96.7%

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

      1. associate-*r/ 96.7%

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

      2. associate-*r* 96.7%

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

   8. Simplified 96.7%

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

      1. add-sqr-sqrt 96.7%

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

      2. sqrt-unprod 96.7%

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

      3. pow2 96.7%

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

      4. associate-*l* 96.7%

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

      5. *-un-lft-identity 96.7%

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

      6. times-frac 96.7%

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

      7. metadata-eval 96.7%

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

   10. Applied egg-rr 96.7%

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

       1. unpow2 96.7%

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

       2. rem-sqrt-square 96.7%

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

       3. *-commutative 96.7%

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

       4. associate-*l/ 96.7%

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

       5. associate-*l* 96.7%

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

       6. associate-*r/ 96.7%

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

       7. associate-*r/ 96.7%

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

   12. Simplified 96.7%

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

   13. Taylor expanded in b around 0 96.7%

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

       1. *-commutative 96.7%

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

       2. *-commutative 96.7%

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

       3. metadata-eval 96.7%

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

       4. times-frac 96.7%

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

       5. *-rgt-identity 96.7%

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

       6. associate-/l* 96.7%

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

   15. Simplified 96.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|c \cdot \frac{a}{b \cdot -0.5}\right|}{2 \cdot a}\\ \end{array} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+117}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \mathsf{fma}\left(a, \frac{c}{b \cdot -0.5}, b\right)}{a \cdot -2}\\ \end{array}\\ \mathbf{elif}\;b \leq 5.3 \cdot 10^{+94}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a \cdot -2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|c \cdot \frac{a}{b \cdot -0.5}\right|}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ t_1 := \frac{2 \cdot c}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \mathbf{if}\;b \leq -2 \cdot 10^{+117}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \mathsf{fma}\left(a, \frac{c}{b \cdot -0.5}, b\right)}{a \cdot -2}\\ \end{array}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+95}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|c \cdot \frac{a}{b \cdot -0.5}\right|}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0)))))
        (t_1 (/ (* 2.0 c) (* 2.0 (fma a (/ c b) (- b))))))
   (if (<= b -2e+117)
     (if (>= b 0.0) t_1 (/ (+ b (fma a (/ c (* b -0.5)) b)) (* a -2.0)))
     (if (<= b 4e+95)
       (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_0)) (/ (- t_0 b) (* 2.0 a)))
       (if (>= b 0.0) t_1 (/ (fabs (* c (/ a (* b -0.5)))) (* 2.0 a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double t_1 = (2.0 * c) / (2.0 * fma(a, (c / b), -b));
	double tmp_1;
	if (b <= -2e+117) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = (b + fma(a, (c / (b * -0.5)), b)) / (a * -2.0);
		}
		tmp_1 = tmp_2;
	} else if (b <= 4e+95) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_3 = (t_0 - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = t_1;
	} else {
		tmp_1 = fabs((c * (a / (b * -0.5)))) / (2.0 * a);
	}
	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(2.0 * c) / Float64(2.0 * fma(a, Float64(c / b), Float64(-b))))
	tmp_1 = 0.0
	if (b <= -2e+117)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(Float64(b + fma(a, Float64(c / Float64(b * -0.5)), b)) / Float64(a * -2.0));
		end
		tmp_1 = tmp_2;
	elseif (b <= 4e+95)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0));
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = t_1;
	else
		tmp_1 = Float64(abs(Float64(c * Float64(a / Float64(b * -0.5)))) / Float64(2.0 * a));
	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[(2.0 * c), $MachinePrecision] / N[(2.0 * N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2e+117], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(b + N[(a * N[(c / N[(b * -0.5), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 4e+95], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$1, N[(N[Abs[N[(c * N[(a / N[(b * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.0000000000000001e117

   1. Initial program 46.2%

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

   3. Taylor expanded in a around 0 46.2%

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

      1. distribute-lft-out-- 46.2%

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

      2. associate-/l* 46.2%

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

      3. fma-neg 46.2%

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

   5. Simplified 46.2%

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

   6. Taylor expanded in a around 0 2.3%

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

      1. associate-*r/ 2.3%

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

      2. associate-*r* 2.3%

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

   8. Simplified 2.3%

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

      1. add-sqr-sqrt 1.5%

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

      2. sqrt-unprod 2.5%

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

      3. pow2 2.5%

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

      4. associate-*l* 2.5%

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

      5. *-un-lft-identity 2.5%

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

      6. times-frac 2.5%

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

      7. metadata-eval 2.5%

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

   10. Applied egg-rr 2.5%

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

       1. unpow2 2.5%

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

       2. rem-sqrt-square 2.5%

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

       3. *-commutative 2.5%

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

       4. associate-*l/ 2.5%

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

       5. associate-*l* 2.5%

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

       6. associate-*r/ 2.4%

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

       7. associate-*r/ 2.4%

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

   12. Simplified 2.4%

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

   14. Applied egg-rr 97.9%

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

       1. associate-*r/ 98.3%

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

       2. *-rgt-identity 98.3%

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

       3. *-commutative 98.3%

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

   16. Simplified 98.3%

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

   if -2.0000000000000001e117 < b < 4.00000000000000008e95

   1. Initial program 89.1%

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

   if 4.00000000000000008e95 < b

   1. Initial program 54.5%

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

   3. Taylor expanded in a around 0 96.4%

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

      1. distribute-lft-out-- 96.4%

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

      2. associate-/l* 96.7%

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

      3. fma-neg 96.7%

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

   5. Simplified 96.7%

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

   6. Taylor expanded in a around 0 96.7%

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

      1. associate-*r/ 96.7%

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

      2. associate-*r* 96.7%

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

   8. Simplified 96.7%

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

      1. add-sqr-sqrt 96.7%

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

      2. sqrt-unprod 96.7%

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

      3. pow2 96.7%

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

      4. associate-*l* 96.7%

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

      5. *-un-lft-identity 96.7%

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

      6. times-frac 96.7%

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

      7. metadata-eval 96.7%

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

   10. Applied egg-rr 96.7%

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

       1. unpow2 96.7%

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

       2. rem-sqrt-square 96.7%

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

       3. *-commutative 96.7%

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

       4. associate-*l/ 96.7%

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

       5. associate-*l* 96.7%

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

       6. associate-*r/ 96.7%

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

       7. associate-*r/ 96.7%

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

   12. Simplified 96.7%

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

   13. Taylor expanded in b around 0 96.7%

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

       1. *-commutative 96.7%

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

       2. *-commutative 96.7%

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

       3. metadata-eval 96.7%

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

       4. times-frac 96.7%

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

       5. *-rgt-identity 96.7%

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

       6. associate-/l* 96.7%

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

   15. Simplified 96.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|c \cdot \frac{a}{b \cdot -0.5}\right|}{2 \cdot a}\\ \end{array} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+117}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \mathsf{fma}\left(a, \frac{c}{b \cdot -0.5}, b\right)}{a \cdot -2}\\ \end{array}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+95}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|c \cdot \frac{a}{b \cdot -0.5}\right|}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* 2.0 c) (* 2.0 (fma a (/ c b) (- b))))))
   (if (<= b -2e+117)
     (if (>= b 0.0) t_0 (/ (+ b (fma a (/ c (* b -0.5)) b)) (* a -2.0)))
     (if (>= b 0.0)
       t_0
       (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a))))))

C

double code(double a, double b, double c) {
	double t_0 = (2.0 * c) / (2.0 * fma(a, (c / b), -b));
	double tmp_1;
	if (b <= -2e+117) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = (b + fma(a, (c / (b * -0.5)), b)) / (a * -2.0);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(2.0 * c) / Float64(2.0 * fma(a, Float64(c / b), Float64(-b))))
	tmp_1 = 0.0
	if (b <= -2e+117)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(Float64(b + fma(a, Float64(c / Float64(b * -0.5)), b)) / Float64(a * -2.0));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = t_0;
	else
		tmp_1 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(2.0 * c), $MachinePrecision] / N[(2.0 * N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2e+117], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(b + N[(a * N[(c / N[(b * -0.5), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.0000000000000001e117

   1. Initial program 46.2%

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

   3. Taylor expanded in a around 0 46.2%

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

      1. distribute-lft-out-- 46.2%

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

      2. associate-/l* 46.2%

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

      3. fma-neg 46.2%

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

   5. Simplified 46.2%

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

   6. Taylor expanded in a around 0 2.3%

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

      1. associate-*r/ 2.3%

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

      2. associate-*r* 2.3%

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

   8. Simplified 2.3%

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

      1. add-sqr-sqrt 1.5%

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

      2. sqrt-unprod 2.5%

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

      3. pow2 2.5%

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

      4. associate-*l* 2.5%

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

      5. *-un-lft-identity 2.5%

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

      6. times-frac 2.5%

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

      7. metadata-eval 2.5%

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

   10. Applied egg-rr 2.5%

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

       1. unpow2 2.5%

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

       2. rem-sqrt-square 2.5%

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

       3. *-commutative 2.5%

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

       4. associate-*l/ 2.5%

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

       5. associate-*l* 2.5%

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

       6. associate-*r/ 2.4%

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

       7. associate-*r/ 2.4%

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

   12. Simplified 2.4%

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

   14. Applied egg-rr 97.9%

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

       1. associate-*r/ 98.3%

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

       2. *-rgt-identity 98.3%

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

       3. *-commutative 98.3%

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

   16. Simplified 98.3%

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

   if -2.0000000000000001e117 < b

   1. Initial program 79.8%

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

   3. Taylor expanded in a around 0 72.7%

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

      1. distribute-lft-out-- 72.7%

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

      2. associate-/l* 72.7%

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

      3. fma-neg 72.7%

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

   5. Simplified 72.7%

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

4. Final simplification 78.2%

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

Alternative 4: 67.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.6%

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

3. Taylor expanded in a around 0 67.0%

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

   1. distribute-lft-out-- 67.0%

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

   2. associate-/l* 67.0%

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

   3. fma-neg 67.0%

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

5. Simplified 67.0%

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

6. Taylor expanded in a around 0 32.4%

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

   1. associate-*r/ 32.4%

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

   2. associate-*r* 32.4%

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

8. Simplified 32.4%

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

   1. add-sqr-sqrt 32.0%

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

   2. sqrt-unprod 33.1%

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

   3. pow2 33.1%

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

   4. associate-*l* 33.1%

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

   5. *-un-lft-identity 33.1%

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

   6. times-frac 33.1%

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

   7. metadata-eval 33.1%

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

10. Applied egg-rr 33.1%

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

    1. unpow2 33.1%

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

    2. rem-sqrt-square 33.1%

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

    3. *-commutative 33.1%

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

    4. associate-*l/ 33.1%

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

    5. associate-*l* 33.1%

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

    6. associate-*r/ 33.1%

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

    7. associate-*r/ 33.1%

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

12. Simplified 33.1%

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

14. Applied egg-rr 65.7%

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

    1. associate-*r/ 65.9%

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

    2. *-rgt-identity 65.9%

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

    3. *-commutative 65.9%

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

16. Simplified 65.9%

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

17. Final simplification 65.9%

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

Alternative 5: 67.6% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.6%

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

3. Simplified 72.6%

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

5. Taylor expanded in b around -inf 71.4%

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

6. Taylor expanded in c around 0 65.8%

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

   1. associate-*r/ 65.8%

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

   2. mul-1-neg 65.8%

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

8. Simplified 65.8%

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

9. Final simplification 65.8%

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

Alternative 6: 67.5% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.6%

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

3. Simplified 72.6%

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

5. Taylor expanded in b around -inf 71.4%

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

6. Taylor expanded in c around 0 65.7%

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

7. Final simplification 65.7%

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

Reproduce

herbie shell --seed 2024101 
(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))))