jeff quadratic root 2

Percentage Accurate: 72.9% → 90.5%

Time: 21.2s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

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: 72.9% accurate, 1.0× speedup

Math

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

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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ \mathbf{if}\;b \leq -2.4 \cdot 10^{+73}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{-c}{b}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-2 \cdot \frac{b}{a} + \frac{\frac{c \cdot 2}{{\left(\sqrt[3]{b}\right)}^{2}}}{\sqrt[3]{b}}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{+65}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \mathsf{fma}\left(c \cdot \frac{a}{b}, -2, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0))))))
   (if (<= b -2.4e+73)
     (if (>= b 0.0)
       (fma 1.0 (/ (- c) b) 0.0)
       (*
        0.5
        (+ (* -2.0 (/ b a)) (/ (/ (* c 2.0) (pow (cbrt b) 2.0)) (cbrt b)))))
     (if (<= b 7.2e+65)
       (if (>= b 0.0) (/ (* c 2.0) (- (- b) t_0)) (/ (- t_0 b) (* a 2.0)))
       (if (>= b 0.0)
         (* -2.0 (/ c (+ b (fma (* c (/ a b)) -2.0 b))))
         (/ (- b) a))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -2.4e+73) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = fma(1.0, (-c / b), 0.0);
		} else {
			tmp_2 = 0.5 * ((-2.0 * (b / a)) + (((c * 2.0) / pow(cbrt(b), 2.0)) / cbrt(b)));
		}
		tmp_1 = tmp_2;
	} else if (b <= 7.2e+65) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c * 2.0) / (-b - t_0);
		} else {
			tmp_3 = (t_0 - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -2.0 * (c / (b + fma((c * (a / b)), -2.0, b)));
	} else {
		tmp_1 = -b / a;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	tmp_1 = 0.0
	if (b <= -2.4e+73)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = fma(1.0, Float64(Float64(-c) / b), 0.0);
		else
			tmp_2 = Float64(0.5 * Float64(Float64(-2.0 * Float64(b / a)) + Float64(Float64(Float64(c * 2.0) / (cbrt(b) ^ 2.0)) / cbrt(b))));
		end
		tmp_1 = tmp_2;
	elseif (b <= 7.2e+65)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(c * 2.0) / Float64(Float64(-b) - t_0));
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(a * 2.0));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(-2.0 * Float64(c / Float64(b + fma(Float64(c * Float64(a / b)), -2.0, b))));
	else
		tmp_1 = Float64(Float64(-b) / 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]}, If[LessEqual[b, -2.4e+73], If[GreaterEqual[b, 0.0], N[(1.0 * N[((-c) / b), $MachinePrecision] + 0.0), $MachinePrecision], N[(0.5 * N[(N[(-2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c * 2.0), $MachinePrecision] / N[Power[N[Power[b, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 7.2e+65], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(c / N[(b + N[(N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision] * -2.0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.40000000000000002e73

   1. Initial program 67.7%

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

   3. Simplified 67.7%

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

   5. Taylor expanded in c around 0 67.7%

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

      1. add0 67.7%

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

      2. *-un-lft-identity 67.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{1 \cdot \left(c \cdot \frac{-2}{b + b}\right)} + 0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

      3. fma-define 67.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\mathsf{fma}\left(1, c \cdot \frac{-2}{b + b}, 0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

      4. associate-*r/ 67.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \color{blue}{\frac{c \cdot -2}{b + b}}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

      5. count-2 67.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{c \cdot -2}{\color{blue}{2 \cdot b}}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

      6. *-commutative 67.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{c \cdot -2}{\color{blue}{b \cdot 2}}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

      7. times-frac 67.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \color{blue}{\frac{c}{b} \cdot \frac{-2}{2}}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

      8. metadata-eval 67.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{c}{b} \cdot \color{blue}{-1}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

   7. Applied egg-rr 67.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\mathsf{fma}\left(1, \frac{c}{b} \cdot -1, 0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

   8. Taylor expanded in b around -inf 96.3%

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

      1. associate-*r/ 96.3%

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

      2. add-cube-cbrt 96.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{c}{b} \cdot -1, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-2 \cdot \frac{b}{a} + \color{blue}{\frac{2 \cdot c}{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \sqrt[3]{b}}}\right)\\ \end{array} \]

      3. associate-/r* 96.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{c}{b} \cdot -1, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \color{blue}{\left(-2 \cdot \frac{b}{a} + \frac{\frac{2 \cdot c}{\sqrt[3]{b} \cdot \sqrt[3]{b}}}{\sqrt[3]{b}}\right)}\\ \end{array} \]

      4. pow2 96.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{c}{b} \cdot -1, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-2 \cdot \color{blue}{\frac{b}{a}} + \frac{\frac{2 \cdot c}{{\left(\sqrt[3]{b}\right)}^{2}}}{\sqrt[3]{b}}\right)\\ \end{array} \]

   10. Applied egg-rr 96.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{c}{b} \cdot -1, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \color{blue}{\left(-2 \cdot \frac{b}{a} + \frac{\frac{2 \cdot c}{{\left(\sqrt[3]{b}\right)}^{2}}}{\sqrt[3]{b}}\right)}\\ \end{array} \]

   if -2.40000000000000002e73 < b < 7.19999999999999957e65

   1. Initial program 88.7%

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

   if 7.19999999999999957e65 < b

   1. Initial program 56.2%

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

   3. Simplified 56.2%

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

   5. Taylor expanded in b around -inf 56.2%

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

      1. neg-mul-1 56.2%

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

      2. distribute-neg-frac 56.2%

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

   7. Simplified 56.2%

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

   8. Taylor expanded in b around inf 88.0%

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

   10. Applied egg-rr 98.4%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{+73}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{-c}{b}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-2 \cdot \frac{b}{a} + \frac{\frac{c \cdot 2}{{\left(\sqrt[3]{b}\right)}^{2}}}{\sqrt[3]{b}}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{+65}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \mathsf{fma}\left(c \cdot \frac{a}{b}, -2, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ \mathbf{if}\;b \leq -2.5 \cdot 10^{+73}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{-c}{b}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-2 \cdot \frac{b}{a} + \frac{c}{b} \cdot 2\right)\\ \end{array}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{+65}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - t\_0}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \mathsf{fma}\left(c \cdot \frac{a}{b}, -2, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0))))))
   (if (<= b -2.5e+73)
     (if (>= b 0.0)
       (fma 1.0 (/ (- c) b) 0.0)
       (* 0.5 (+ (* -2.0 (/ b a)) (* (/ c b) 2.0))))
     (if (<= b 7.2e+65)
       (if (>= b 0.0) (/ 2.0 (/ (- (- b) t_0) c)) (/ (- t_0 b) (* a 2.0)))
       (if (>= b 0.0)
         (* -2.0 (/ c (+ b (fma (* c (/ a b)) -2.0 b))))
         (/ (- b) a))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -2.5e+73) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = fma(1.0, (-c / b), 0.0);
		} else {
			tmp_2 = 0.5 * ((-2.0 * (b / a)) + ((c / b) * 2.0));
		}
		tmp_1 = tmp_2;
	} else if (b <= 7.2e+65) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = 2.0 / ((-b - t_0) / c);
		} else {
			tmp_3 = (t_0 - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -2.0 * (c / (b + fma((c * (a / b)), -2.0, b)));
	} else {
		tmp_1 = -b / a;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	tmp_1 = 0.0
	if (b <= -2.5e+73)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = fma(1.0, Float64(Float64(-c) / b), 0.0);
		else
			tmp_2 = Float64(0.5 * Float64(Float64(-2.0 * Float64(b / a)) + Float64(Float64(c / b) * 2.0)));
		end
		tmp_1 = tmp_2;
	elseif (b <= 7.2e+65)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(2.0 / Float64(Float64(Float64(-b) - t_0) / c));
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(a * 2.0));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(-2.0 * Float64(c / Float64(b + fma(Float64(c * Float64(a / b)), -2.0, b))));
	else
		tmp_1 = Float64(Float64(-b) / 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]}, If[LessEqual[b, -2.5e+73], If[GreaterEqual[b, 0.0], N[(1.0 * N[((-c) / b), $MachinePrecision] + 0.0), $MachinePrecision], N[(0.5 * N[(N[(-2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 7.2e+65], If[GreaterEqual[b, 0.0], N[(2.0 / N[(N[((-b) - t$95$0), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(c / N[(b + N[(N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision] * -2.0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.49999999999999988e73

   1. Initial program 67.7%

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

   3. Simplified 67.7%

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

   5. Taylor expanded in c around 0 67.7%

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

      1. add0 67.7%

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

      2. *-un-lft-identity 67.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{1 \cdot \left(c \cdot \frac{-2}{b + b}\right)} + 0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

      3. fma-define 67.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\mathsf{fma}\left(1, c \cdot \frac{-2}{b + b}, 0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

      4. associate-*r/ 67.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \color{blue}{\frac{c \cdot -2}{b + b}}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

      5. count-2 67.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{c \cdot -2}{\color{blue}{2 \cdot b}}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

      6. *-commutative 67.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{c \cdot -2}{\color{blue}{b \cdot 2}}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

      7. times-frac 67.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \color{blue}{\frac{c}{b} \cdot \frac{-2}{2}}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

      8. metadata-eval 67.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{c}{b} \cdot \color{blue}{-1}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

   7. Applied egg-rr 67.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\mathsf{fma}\left(1, \frac{c}{b} \cdot -1, 0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

   8. Taylor expanded in b around -inf 96.3%

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

   if -2.49999999999999988e73 < b < 7.19999999999999957e65

   1. Initial program 88.7%

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

   3. Simplified 88.7%

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

   if 7.19999999999999957e65 < b

   1. Initial program 56.2%

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

   3. Simplified 56.2%

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

   5. Taylor expanded in b around -inf 56.2%

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

      1. neg-mul-1 56.2%

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

      2. distribute-neg-frac 56.2%

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

   7. Simplified 56.2%

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

   8. Taylor expanded in b around inf 88.0%

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

   10. Applied egg-rr 98.4%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{+73}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{-c}{b}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-2 \cdot \frac{b}{a} + \frac{c}{b} \cdot 2\right)\\ \end{array}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{+65}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \mathsf{fma}\left(c \cdot \frac{a}{b}, -2, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ \mathbf{if}\;b \leq -5.5 \cdot 10^{+74}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{-c}{b}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-2 \cdot \frac{b}{a} + \frac{c}{b} \cdot 2\right)\\ \end{array}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{+65}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \mathsf{fma}\left(c \cdot \frac{a}{b}, -2, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0))))))
   (if (<= b -5.5e+74)
     (if (>= b 0.0)
       (fma 1.0 (/ (- c) b) 0.0)
       (* 0.5 (+ (* -2.0 (/ b a)) (* (/ c b) 2.0))))
     (if (<= b 7.2e+65)
       (if (>= b 0.0) (/ (* c 2.0) (- (- b) t_0)) (/ (- t_0 b) (* a 2.0)))
       (if (>= b 0.0)
         (* -2.0 (/ c (+ b (fma (* c (/ a b)) -2.0 b))))
         (/ (- b) a))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -5.5e+74) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = fma(1.0, (-c / b), 0.0);
		} else {
			tmp_2 = 0.5 * ((-2.0 * (b / a)) + ((c / b) * 2.0));
		}
		tmp_1 = tmp_2;
	} else if (b <= 7.2e+65) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c * 2.0) / (-b - t_0);
		} else {
			tmp_3 = (t_0 - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -2.0 * (c / (b + fma((c * (a / b)), -2.0, b)));
	} else {
		tmp_1 = -b / a;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	tmp_1 = 0.0
	if (b <= -5.5e+74)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = fma(1.0, Float64(Float64(-c) / b), 0.0);
		else
			tmp_2 = Float64(0.5 * Float64(Float64(-2.0 * Float64(b / a)) + Float64(Float64(c / b) * 2.0)));
		end
		tmp_1 = tmp_2;
	elseif (b <= 7.2e+65)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(c * 2.0) / Float64(Float64(-b) - t_0));
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(a * 2.0));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(-2.0 * Float64(c / Float64(b + fma(Float64(c * Float64(a / b)), -2.0, b))));
	else
		tmp_1 = Float64(Float64(-b) / 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]}, If[LessEqual[b, -5.5e+74], If[GreaterEqual[b, 0.0], N[(1.0 * N[((-c) / b), $MachinePrecision] + 0.0), $MachinePrecision], N[(0.5 * N[(N[(-2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 7.2e+65], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(c / N[(b + N[(N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision] * -2.0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.5000000000000003e74

   1. Initial program 67.7%

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

   3. Simplified 67.7%

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

   5. Taylor expanded in c around 0 67.7%

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

      1. add0 67.7%

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

      2. *-un-lft-identity 67.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{1 \cdot \left(c \cdot \frac{-2}{b + b}\right)} + 0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

      3. fma-define 67.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\mathsf{fma}\left(1, c \cdot \frac{-2}{b + b}, 0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

      4. associate-*r/ 67.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \color{blue}{\frac{c \cdot -2}{b + b}}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

      5. count-2 67.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{c \cdot -2}{\color{blue}{2 \cdot b}}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

      6. *-commutative 67.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{c \cdot -2}{\color{blue}{b \cdot 2}}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

      7. times-frac 67.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \color{blue}{\frac{c}{b} \cdot \frac{-2}{2}}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

      8. metadata-eval 67.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{c}{b} \cdot \color{blue}{-1}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

   7. Applied egg-rr 67.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\mathsf{fma}\left(1, \frac{c}{b} \cdot -1, 0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

   8. Taylor expanded in b around -inf 96.3%

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

   if -5.5000000000000003e74 < b < 7.19999999999999957e65

   1. Initial program 88.7%

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

   if 7.19999999999999957e65 < b

   1. Initial program 56.2%

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

   3. Simplified 56.2%

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

   5. Taylor expanded in b around -inf 56.2%

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

      1. neg-mul-1 56.2%

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

      2. distribute-neg-frac 56.2%

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

   7. Simplified 56.2%

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

   8. Taylor expanded in b around inf 88.0%

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

   10. Applied egg-rr 98.4%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{+74}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{-c}{b}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-2 \cdot \frac{b}{a} + \frac{c}{b} \cdot 2\right)\\ \end{array}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{+65}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \mathsf{fma}\left(c \cdot \frac{a}{b}, -2, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-b}{a}\\ \mathbf{if}\;b \leq 7.2 \cdot 10^{+65}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{c}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \mathsf{fma}\left(c \cdot \frac{a}{b}, -2, b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- b) a)))
   (if (<= b 7.2e+65)
     (if (>= b 0.0)
       (/ 2.0 (/ (- (- b) (sqrt (- (* b b) (* c (* a 4.0))))) c))
       t_0)
     (if (>= b 0.0) (* -2.0 (/ c (+ b (fma (* c (/ a b)) -2.0 b)))) t_0))))

C

double code(double a, double b, double c) {
	double t_0 = -b / a;
	double tmp_1;
	if (b <= 7.2e+65) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 2.0 / ((-b - sqrt(((b * b) - (c * (a * 4.0))))) / c);
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = -2.0 * (c / (b + fma((c * (a / b)), -2.0, b)));
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-b) / a)
	tmp_1 = 0.0
	if (b <= 7.2e+65)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(2.0 / Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))) / c));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(-2.0 * Float64(c / Float64(b + fma(Float64(c * Float64(a / b)), -2.0, b))));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) / a), $MachinePrecision]}, If[LessEqual[b, 7.2e+65], If[GreaterEqual[b, 0.0], N[(2.0 / N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(c / N[(b + N[(N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision] * -2.0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < 7.19999999999999957e65

   1. Initial program 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in b around -inf 72.8%

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

      1. neg-mul-1 72.8%

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

      2. distribute-neg-frac 72.8%

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

   7. Simplified 72.8%

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

   if 7.19999999999999957e65 < b

   1. Initial program 56.2%

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

   3. Simplified 56.2%

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

   5. Taylor expanded in b around -inf 56.2%

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

      1. neg-mul-1 56.2%

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

      2. distribute-neg-frac 56.2%

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

   7. Simplified 56.2%

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

   8. Taylor expanded in b around inf 88.0%

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

   10. Applied egg-rr 98.4%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.2 \cdot 10^{+65}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \mathsf{fma}\left(c \cdot \frac{a}{b}, -2, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-b}{a}\\ \mathbf{if}\;b \leq 1.24 \cdot 10^{-88}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - \sqrt{-4 \cdot \left(c \cdot a\right)}}{c}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- b) a)))
   (if (<= b 1.24e-88)
     (if (>= b 0.0) (/ 2.0 (/ (- (- b) (sqrt (* -4.0 (* c a)))) c)) t_0)
     (if (>= b 0.0) (/ (- c) b) t_0))))

C

double code(double a, double b, double c) {
	double t_0 = -b / a;
	double tmp_1;
	if (b <= 1.24e-88) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 2.0 / ((-b - sqrt((-4.0 * (c * a)))) / c);
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = -c / b;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = -b / a
    if (b <= 1.24d-88) then
        if (b >= 0.0d0) then
            tmp_2 = 2.0d0 / ((-b - sqrt(((-4.0d0) * (c * a)))) / c)
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = -c / b
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -b / a;
	double tmp_1;
	if (b <= 1.24e-88) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 2.0 / ((-b - Math.sqrt((-4.0 * (c * a)))) / c);
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = -c / b;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = -b / a
	tmp_1 = 0
	if b <= 1.24e-88:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = 2.0 / ((-b - math.sqrt((-4.0 * (c * a)))) / c)
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = -c / b
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-b) / a)
	tmp_1 = 0.0
	if (b <= 1.24e-88)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(2.0 / Float64(Float64(Float64(-b) - sqrt(Float64(-4.0 * Float64(c * a)))) / c));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-c) / b);
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = -b / a;
	tmp_2 = 0.0;
	if (b <= 1.24e-88)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = 2.0 / ((-b - sqrt((-4.0 * (c * a)))) / c);
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = -c / b;
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) / a), $MachinePrecision]}, If[LessEqual[b, 1.24e-88], If[GreaterEqual[b, 0.0], N[(2.0 / N[(N[((-b) - N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[((-c) / b), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.23999999999999999e-88

   1. Initial program 78.8%

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

   3. Simplified 78.8%

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

   5. Taylor expanded in b around -inf 69.3%

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

      1. neg-mul-1 69.3%

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

      2. distribute-neg-frac 69.3%

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

   7. Simplified 69.3%

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

   8. Taylor expanded in b around 0 69.2%

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

   if 1.23999999999999999e-88 < b

   1. Initial program 68.3%

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

   3. Simplified 68.2%

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

   5. Taylor expanded in b around -inf 68.2%

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

      1. neg-mul-1 68.2%

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

      2. distribute-neg-frac 68.2%

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

   7. Simplified 68.2%

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

   8. Taylor expanded in b around inf 83.0%

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

   9. Taylor expanded in b around inf 90.2%

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

       1. associate-*r/ 90.2%

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

       2. neg-mul-1 90.2%

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

   11. Simplified 90.2%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.24 \cdot 10^{-88}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - \sqrt{-4 \cdot \left(c \cdot a\right)}}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.3% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0)
   (fma 1.0 (/ (- c) b) 0.0)
   (* 0.5 (+ (* -2.0 (/ b a)) (* (/ c b) 2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = fma(1.0, (-c / b), 0.0);
	} else {
		tmp = 0.5 * ((-2.0 * (b / a)) + ((c / b) * 2.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = fma(1.0, Float64(Float64(-c) / b), 0.0);
	else
		tmp = Float64(0.5 * Float64(Float64(-2.0 * Float64(b / a)) + Float64(Float64(c / b) * 2.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(1.0 * N[((-c) / b), $MachinePrecision] + 0.0), $MachinePrecision], N[(0.5 * N[(N[(-2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 75.3%

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

3. Simplified 75.3%

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

5. Taylor expanded in c around 0 76.5%

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

   1. add0 76.5%

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

   2. *-un-lft-identity 76.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{1 \cdot \left(c \cdot \frac{-2}{b + b}\right)} + 0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

   3. fma-define 76.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\mathsf{fma}\left(1, c \cdot \frac{-2}{b + b}, 0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

   4. associate-*r/ 76.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \color{blue}{\frac{c \cdot -2}{b + b}}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

   5. count-2 76.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{c \cdot -2}{\color{blue}{2 \cdot b}}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

   6. *-commutative 76.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{c \cdot -2}{\color{blue}{b \cdot 2}}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

   7. times-frac 76.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \color{blue}{\frac{c}{b} \cdot \frac{-2}{2}}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

   8. metadata-eval 76.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{c}{b} \cdot \color{blue}{-1}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

7. Applied egg-rr 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\mathsf{fma}\left(1, \frac{c}{b} \cdot -1, 0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}\\ \end{array} \]

8. Taylor expanded in b around -inf 70.3%

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

9. Final simplification 70.3%

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

Alternative 7: 36.5% accurate, 13.4× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = b / a;
	} else {
		tmp = -b / 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) :: tmp
    if (b >= 0.0d0) then
        tmp = b / a
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

3. Simplified 75.3%

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

5. Taylor expanded in b around -inf 68.9%

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

   1. neg-mul-1 68.9%

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

   2. distribute-neg-frac 68.9%

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

7. Simplified 68.9%

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

8. Taylor expanded in b around inf 67.7%

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

9. Taylor expanded in b around 0 39.4%

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

10. Final simplification 39.4%

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

Alternative 8: 68.1% accurate, 13.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(Float64(-b) / a);
	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 / a;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Initial program 75.3%

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

3. Simplified 75.3%

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

5. Taylor expanded in b around -inf 68.9%

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

   1. neg-mul-1 68.9%

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

   2. distribute-neg-frac 68.9%

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

7. Simplified 68.9%

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

8. Taylor expanded in b around inf 67.7%

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

9. Taylor expanded in b around inf 70.2%

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

    1. associate-*r/ 70.2%

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

    2. neg-mul-1 70.2%

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

11. Simplified 70.2%

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

12. Final simplification 70.2%

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

Reproduce

herbie shell --seed 2024034 
(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))))