jeff quadratic root 2

Percentage Accurate: 71.7% → 90.0%

Time: 15.7s

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: 71.7% 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: 90.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\\ t_1 := \frac{2 \cdot c}{\left(-b\right) - t\_0}\\ \mathbf{if}\;b \leq -1.55 \cdot 10^{+159}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{+111}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* 4.0 a)))))
        (t_1 (/ (* 2.0 c) (- (- b) t_0))))
   (if (<= b -1.55e+159)
     (if (>= b 0.0) t_1 (fma -1.0 (/ b a) (/ c b)))
     (if (<= b 9.5e+111)
       (if (>= b 0.0) t_1 (/ (- t_0 b) (* 2.0 a)))
       (if (>= b 0.0) (/ (- c) b) 0.0)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (4.0 * a))));
	double t_1 = (2.0 * c) / (-b - t_0);
	double tmp_1;
	if (b <= -1.55e+159) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = fma(-1.0, (b / a), (c / b));
		}
		tmp_1 = tmp_2;
	} else if (b <= 9.5e+111) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = (t_0 - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -c / b;
	} else {
		tmp_1 = 0.0;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a))))
	t_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0))
	tmp_1 = 0.0
	if (b <= -1.55e+159)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = fma(-1.0, Float64(b / a), Float64(c / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 9.5e+111)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-c) / b);
	else
		tmp_1 = 0.0;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.55e+159], If[GreaterEqual[b, 0.0], t$95$1, N[(-1.0 * N[(b / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 9.5e+111], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(t$95$0 - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[((-c) / b), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.5499999999999999e159

   1. Initial program 33.3%

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

      1. add-sqr-sqrt 25.9%

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

      2. pow2 25.9%

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

   4. Applied egg-rr 25.9%

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

   5. Taylor expanded in b around -inf 98.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(b \cdot \left(-2 \cdot \frac{c \cdot {\left(\sqrt{0.5}\right)}^{2}}{{b}^{2}} + 2 \cdot \frac{{\left(\sqrt{0.5}\right)}^{2}}{a}\right)\right)\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-*r* 98.6%

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

      2. neg-mul-1 98.6%

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

      3. fma-define 98.6%

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

      4. associate-/l* 98.6%

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

      5. unpow2 98.6%

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

      6. rem-square-sqrt 98.6%

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

      7. associate-*r/ 98.6%

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

      8. unpow2 98.6%

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

      9. rem-square-sqrt 99.7%

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

      10. metadata-eval 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in c around 0 100.0%

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

      1. fma-define 100.0%

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

   10. Simplified 100.0%

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

   if -1.5499999999999999e159 < b < 9.50000000000000019e111

   1. Initial program 83.2%

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

   if 9.50000000000000019e111 < b

   1. Initial program 61.3%

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

   3. Simplified 61.3%

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

   5. Taylor expanded in c around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in c around 0 100.0%

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

   9. Taylor expanded in b around 0 100.0%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{+159}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{+111}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\\ t_1 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(b \cdot \frac{0.5}{a}\right)\\ \end{array}\\ \mathbf{if}\;b \leq -1.55 \cdot 10^{+159}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \left(c \cdot \frac{-0.5}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{+107}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* 4.0 a)))))
        (t_1
         (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_0)) (* -2.0 (* b (/ 0.5 a))))))
   (if (<= b -1.55e+159)
     t_1
     (if (<= b -1e-310)
       (if (>= b 0.0) (* 2.0 (* c (/ -0.5 b))) (/ (- t_0 b) (* 2.0 a)))
       (if (<= b 2.3e+107) t_1 (if (>= b 0.0) (/ (- c) b) 0.0))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (4.0 * a))));
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (-b - t_0);
	} else {
		tmp = -2.0 * (b * (0.5 / a));
	}
	double t_1 = tmp;
	double tmp_1;
	if (b <= -1.55e+159) {
		tmp_1 = t_1;
	} else if (b <= -1e-310) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 2.0 * (c * (-0.5 / b));
		} else {
			tmp_2 = (t_0 - b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 2.3e+107) {
		tmp_1 = t_1;
	} else if (b >= 0.0) {
		tmp_1 = -c / b;
	} else {
		tmp_1 = 0.0;
	}
	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) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = sqrt(((b * b) - (c * (4.0d0 * a))))
    if (b >= 0.0d0) then
        tmp = (2.0d0 * c) / (-b - t_0)
    else
        tmp = (-2.0d0) * (b * (0.5d0 / a))
    end if
    t_1 = tmp
    if (b <= (-1.55d+159)) then
        tmp_1 = t_1
    else if (b <= (-1d-310)) then
        if (b >= 0.0d0) then
            tmp_2 = 2.0d0 * (c * ((-0.5d0) / b))
        else
            tmp_2 = (t_0 - b) / (2.0d0 * a)
        end if
        tmp_1 = tmp_2
    else if (b <= 2.3d+107) then
        tmp_1 = t_1
    else if (b >= 0.0d0) then
        tmp_1 = -c / b
    else
        tmp_1 = 0.0d0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (c * (4.0 * a))));
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (-b - t_0);
	} else {
		tmp = -2.0 * (b * (0.5 / a));
	}
	double t_1 = tmp;
	double tmp_1;
	if (b <= -1.55e+159) {
		tmp_1 = t_1;
	} else if (b <= -1e-310) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 2.0 * (c * (-0.5 / b));
		} else {
			tmp_2 = (t_0 - b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 2.3e+107) {
		tmp_1 = t_1;
	} else if (b >= 0.0) {
		tmp_1 = -c / b;
	} else {
		tmp_1 = 0.0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (c * (4.0 * a))))
	tmp = 0
	if b >= 0.0:
		tmp = (2.0 * c) / (-b - t_0)
	else:
		tmp = -2.0 * (b * (0.5 / a))
	t_1 = tmp
	tmp_1 = 0
	if b <= -1.55e+159:
		tmp_1 = t_1
	elif b <= -1e-310:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = 2.0 * (c * (-0.5 / b))
		else:
			tmp_2 = (t_0 - b) / (2.0 * a)
		tmp_1 = tmp_2
	elif b <= 2.3e+107:
		tmp_1 = t_1
	elif b >= 0.0:
		tmp_1 = -c / b
	else:
		tmp_1 = 0.0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a))))
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0));
	else
		tmp = Float64(-2.0 * Float64(b * Float64(0.5 / a)));
	end
	t_1 = tmp
	tmp_1 = 0.0
	if (b <= -1.55e+159)
		tmp_1 = t_1;
	elseif (b <= -1e-310)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(2.0 * Float64(c * Float64(-0.5 / b)));
		else
			tmp_2 = Float64(Float64(t_0 - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 2.3e+107)
		tmp_1 = t_1;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-c) / b);
	else
		tmp_1 = 0.0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = sqrt(((b * b) - (c * (4.0 * a))));
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (2.0 * c) / (-b - t_0);
	else
		tmp = -2.0 * (b * (0.5 / a));
	end
	t_1 = tmp;
	tmp_2 = 0.0;
	if (b <= -1.55e+159)
		tmp_2 = t_1;
	elseif (b <= -1e-310)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = 2.0 * (c * (-0.5 / b));
		else
			tmp_3 = (t_0 - b) / (2.0 * a);
		end
		tmp_2 = tmp_3;
	elseif (b <= 2.3e+107)
		tmp_2 = t_1;
	elseif (b >= 0.0)
		tmp_2 = -c / b;
	else
		tmp_2 = 0.0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(b * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]}, If[LessEqual[b, -1.55e+159], t$95$1, If[LessEqual[b, -1e-310], If[GreaterEqual[b, 0.0], N[(2.0 * N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 2.3e+107], t$95$1, If[GreaterEqual[b, 0.0], N[((-c) / b), $MachinePrecision], 0.0]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.5499999999999999e159 or -9.999999999999969e-311 < b < 2.3e107

   1. Initial program 66.6%

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

      1. add-sqr-sqrt 64.5%

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

      2. pow2 64.5%

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

   4. Applied egg-rr 64.5%

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

   5. Taylor expanded in b around -inf 84.7%

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

      1. associate-/l* 84.7%

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

      2. unpow2 84.7%

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

      3. rem-square-sqrt 85.0%

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

   7. Simplified 85.0%

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

   if -1.5499999999999999e159 < b < -9.999999999999969e-311

   1. Initial program 86.7%

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

      1. add-sqr-sqrt 86.7%

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

      2. pow2 86.7%

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

      3. *-commutative 86.7%

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

   4. Applied egg-rr 86.7%

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

   5. Taylor expanded in c around 0 86.7%

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

      1. associate-*r/ 86.7%

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

      2. associate-*r* 86.7%

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

      3. unpow2 86.7%

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

      4. rem-square-sqrt 86.7%

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

      5. rem-square-sqrt 86.7%

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

      6. unpow2 86.7%

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

      7. *-commutative 86.7%

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

      8. *-commutative 86.7%

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

      9. associate-*r/ 86.7%

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

      10. associate-/l* 86.7%

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

      11. unpow2 86.7%

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

      12. rem-square-sqrt 86.7%

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

   7. Simplified 86.7%

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

   if 2.3e107 < b

   1. Initial program 61.3%

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

   3. Simplified 61.3%

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

   5. Taylor expanded in c around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in c around 0 100.0%

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

   9. Taylor expanded in b around 0 100.0%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{+159}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(b \cdot \frac{0.5}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \left(c \cdot \frac{-0.5}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{+107}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(b \cdot \frac{0.5}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\\ t_1 := \frac{2 \cdot c}{\left(-b\right) - t\_0}\\ \mathbf{if}\;b \leq -1.55 \cdot 10^{+159}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(b \cdot \frac{0.5}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{+110}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* 4.0 a)))))
        (t_1 (/ (* 2.0 c) (- (- b) t_0))))
   (if (<= b -1.55e+159)
     (if (>= b 0.0) t_1 (* -2.0 (* b (/ 0.5 a))))
     (if (<= b 7.5e+110)
       (if (>= b 0.0) t_1 (/ (- t_0 b) (* 2.0 a)))
       (if (>= b 0.0) (/ (- c) b) 0.0)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (4.0 * a))));
	double t_1 = (2.0 * c) / (-b - t_0);
	double tmp_1;
	if (b <= -1.55e+159) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = -2.0 * (b * (0.5 / a));
		}
		tmp_1 = tmp_2;
	} else if (b <= 7.5e+110) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = (t_0 - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -c / b;
	} else {
		tmp_1 = 0.0;
	}
	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) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = sqrt(((b * b) - (c * (4.0d0 * a))))
    t_1 = (2.0d0 * c) / (-b - t_0)
    if (b <= (-1.55d+159)) then
        if (b >= 0.0d0) then
            tmp_2 = t_1
        else
            tmp_2 = (-2.0d0) * (b * (0.5d0 / a))
        end if
        tmp_1 = tmp_2
    else if (b <= 7.5d+110) then
        if (b >= 0.0d0) then
            tmp_3 = t_1
        else
            tmp_3 = (t_0 - b) / (2.0d0 * a)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = -c / b
    else
        tmp_1 = 0.0d0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (c * (4.0 * a))));
	double t_1 = (2.0 * c) / (-b - t_0);
	double tmp_1;
	if (b <= -1.55e+159) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = -2.0 * (b * (0.5 / a));
		}
		tmp_1 = tmp_2;
	} else if (b <= 7.5e+110) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = (t_0 - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -c / b;
	} else {
		tmp_1 = 0.0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (c * (4.0 * a))))
	t_1 = (2.0 * c) / (-b - t_0)
	tmp_1 = 0
	if b <= -1.55e+159:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_1
		else:
			tmp_2 = -2.0 * (b * (0.5 / a))
		tmp_1 = tmp_2
	elif b <= 7.5e+110:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = t_1
		else:
			tmp_3 = (t_0 - b) / (2.0 * a)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = -c / b
	else:
		tmp_1 = 0.0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a))))
	t_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0))
	tmp_1 = 0.0
	if (b <= -1.55e+159)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(-2.0 * Float64(b * Float64(0.5 / a)));
		end
		tmp_1 = tmp_2;
	elseif (b <= 7.5e+110)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-c) / b);
	else
		tmp_1 = 0.0;
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) - (c * (4.0 * a))));
	t_1 = (2.0 * c) / (-b - t_0);
	tmp_2 = 0.0;
	if (b <= -1.55e+159)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = -2.0 * (b * (0.5 / a));
		end
		tmp_2 = tmp_3;
	elseif (b <= 7.5e+110)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = t_1;
		else
			tmp_4 = (t_0 - b) / (2.0 * a);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = -c / b;
	else
		tmp_2 = 0.0;
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.55e+159], If[GreaterEqual[b, 0.0], t$95$1, N[(-2.0 * N[(b * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 7.5e+110], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(t$95$0 - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[((-c) / b), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.5499999999999999e159

   1. Initial program 33.3%

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

      1. add-sqr-sqrt 25.9%

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

      2. pow2 25.9%

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

   4. Applied egg-rr 25.9%

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

   5. Taylor expanded in b around -inf 98.7%

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

      1. associate-/l* 98.6%

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

      2. unpow2 98.6%

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

      3. rem-square-sqrt 99.7%

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

   7. Simplified 99.7%

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

   if -1.5499999999999999e159 < b < 7.5e110

   1. Initial program 83.2%

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

   if 7.5e110 < b

   1. Initial program 61.3%

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

   3. Simplified 61.3%

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

   5. Taylor expanded in c around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in c around 0 100.0%

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

   9. Taylor expanded in b around 0 100.0%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{+159}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(b \cdot \frac{0.5}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{+110}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{+159}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \left(c \cdot \frac{-0.5}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.55e+159)
   (if (>= b 0.0) (* 2.0 (* c (/ -0.5 b))) (/ (* b -2.0) (* 2.0 a)))
   (if (>= b 0.0)
     (/ (* 2.0 c) (* 2.0 (fma a (/ c b) (- b))))
     (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* 2.0 a)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1.55e+159) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 2.0 * (c * (-0.5 / b));
		} else {
			tmp_2 = (b * -2.0) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (2.0 * fma(a, (c / b), -b));
	} else {
		tmp_1 = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (2.0 * a);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -1.55e+159)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(2.0 * Float64(c * Float64(-0.5 / b)));
		else
			tmp_2 = Float64(Float64(b * -2.0) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(2.0 * fma(a, Float64(c / b), Float64(-b))));
	else
		tmp_1 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(2.0 * a));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.55e+159], If[GreaterEqual[b, 0.0], N[(2.0 * N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * -2.0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], 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[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.5499999999999999e159

   1. Initial program 33.3%

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

      1. add-sqr-sqrt 33.3%

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

      2. pow2 33.3%

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

      3. *-commutative 33.3%

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

   4. Applied egg-rr 33.3%

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

   5. Taylor expanded in c around 0 33.3%

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

      1. associate-*r/ 33.3%

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

      2. associate-*r* 33.3%

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

      3. unpow2 33.3%

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

      4. rem-square-sqrt 33.3%

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

      5. rem-square-sqrt 33.3%

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

      6. unpow2 33.3%

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

      7. *-commutative 33.3%

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

      8. *-commutative 33.3%

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

      9. associate-*r/ 33.3%

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

      10. associate-/l* 33.3%

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

      11. unpow2 33.3%

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

      12. rem-square-sqrt 33.3%

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

   7. Simplified 33.3%

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

   8. Taylor expanded in b around -inf 96.8%

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

      1. *-commutative 96.8%

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

   10. Simplified 96.8%

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

   if -1.5499999999999999e159 < b

   1. Initial program 77.2%

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

   3. Taylor expanded in a around 0 73.1%

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

      1. distribute-lft-out-- 73.1%

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

      2. associate-/l* 74.8%

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

      3. fmm-def 74.8%

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

   5. Simplified 74.8%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{+159}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \left(c \cdot \frac{-0.5}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(c \cdot \frac{-0.5}{b}\right)\\ \mathbf{if}\;b \leq -1.55 \cdot 10^{+159}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* 2.0 (* c (/ -0.5 b)))))
   (if (<= b -1.55e+159)
     (if (>= b 0.0) t_0 (/ (* b -2.0) (* 2.0 a)))
     (if (>= b 0.0)
       t_0
       (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* 2.0 a))))))

C

double code(double a, double b, double c) {
	double t_0 = 2.0 * (c * (-0.5 / b));
	double tmp_1;
	if (b <= -1.55e+159) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = (b * -2.0) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (2.0 * a);
	}
	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 = 2.0d0 * (c * ((-0.5d0) / b))
    if (b <= (-1.55d+159)) then
        if (b >= 0.0d0) then
            tmp_2 = t_0
        else
            tmp_2 = (b * (-2.0d0)) / (2.0d0 * a)
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = t_0
    else
        tmp_1 = (sqrt(((b * b) - (c * (4.0d0 * a)))) - b) / (2.0d0 * a)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = 2.0 * (c * (-0.5 / b));
	double tmp_1;
	if (b <= -1.55e+159) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = (b * -2.0) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = (Math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (2.0 * a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = 2.0 * (c * (-0.5 / b))
	tmp_1 = 0
	if b <= -1.55e+159:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_0
		else:
			tmp_2 = (b * -2.0) / (2.0 * a)
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = t_0
	else:
		tmp_1 = (math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (2.0 * a)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(2.0 * Float64(c * Float64(-0.5 / b)))
	tmp_1 = 0.0
	if (b <= -1.55e+159)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(Float64(b * -2.0) / Float64(2.0 * a));
		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(4.0 * a)))) - b) / Float64(2.0 * a));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = 2.0 * (c * (-0.5 / b));
	tmp_2 = 0.0;
	if (b <= -1.55e+159)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_0;
		else
			tmp_3 = (b * -2.0) / (2.0 * a);
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = t_0;
	else
		tmp_2 = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (2.0 * a);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(2.0 * N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.55e+159], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(b * -2.0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.5499999999999999e159

   1. Initial program 33.3%

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

      1. add-sqr-sqrt 33.3%

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

      2. pow2 33.3%

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

      3. *-commutative 33.3%

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

   4. Applied egg-rr 33.3%

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

   5. Taylor expanded in c around 0 33.3%

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

      1. associate-*r/ 33.3%

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

      2. associate-*r* 33.3%

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

      3. unpow2 33.3%

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

      4. rem-square-sqrt 33.3%

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

      5. rem-square-sqrt 33.3%

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

      6. unpow2 33.3%

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

      7. *-commutative 33.3%

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

      8. *-commutative 33.3%

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

      9. associate-*r/ 33.3%

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

      10. associate-/l* 33.3%

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

      11. unpow2 33.3%

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

      12. rem-square-sqrt 33.3%

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

   7. Simplified 33.3%

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

   8. Taylor expanded in b around -inf 96.8%

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

      1. *-commutative 96.8%

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

   10. Simplified 96.8%

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

   if -1.5499999999999999e159 < b

   1. Initial program 77.2%

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

      1. add-sqr-sqrt 60.4%

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

      2. pow2 60.4%

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

      3. *-commutative 60.4%

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

   4. Applied egg-rr 60.4%

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

   5. Taylor expanded in c around 0 74.3%

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

      1. associate-*r/ 74.3%

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

      2. associate-*r* 74.3%

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

      3. unpow2 74.3%

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

      4. rem-square-sqrt 74.8%

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

      5. rem-square-sqrt 33.0%

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

      6. unpow2 33.0%

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

      7. *-commutative 33.0%

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

      8. *-commutative 33.0%

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

      9. associate-*r/ 33.0%

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

      10. associate-/l* 33.0%

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

      11. unpow2 33.0%

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

      12. rem-square-sqrt 74.7%

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

   7. Simplified 74.7%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{+159}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \left(c \cdot \frac{-0.5}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;2 \cdot \left(c \cdot \frac{-0.5}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.65 \cdot 10^{-90}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \left(c \cdot \frac{-0.5}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \sqrt{-4 \cdot \left(c \cdot a\right)}}{a \cdot -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.65e-90)
   (if (>= b 0.0) (* 2.0 (* c (/ -0.5 b))) (/ (* b -2.0) (* 2.0 a)))
   (if (>= b 0.0) (/ (- c) b) (/ (- b (sqrt (* -4.0 (* c a)))) (* a -2.0)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -2.65e-90) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 2.0 * (c * (-0.5 / b));
		} else {
			tmp_2 = (b * -2.0) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = -c / b;
	} else {
		tmp_1 = (b - sqrt((-4.0 * (c * a)))) / (a * -2.0);
	}
	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) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    if (b <= (-2.65d-90)) then
        if (b >= 0.0d0) then
            tmp_2 = 2.0d0 * (c * ((-0.5d0) / b))
        else
            tmp_2 = (b * (-2.0d0)) / (2.0d0 * a)
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = -c / b
    else
        tmp_1 = (b - sqrt(((-4.0d0) * (c * a)))) / (a * (-2.0d0))
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -2.65e-90) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 2.0 * (c * (-0.5 / b));
		} else {
			tmp_2 = (b * -2.0) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = -c / b;
	} else {
		tmp_1 = (b - Math.sqrt((-4.0 * (c * a)))) / (a * -2.0);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if b <= -2.65e-90:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = 2.0 * (c * (-0.5 / b))
		else:
			tmp_2 = (b * -2.0) / (2.0 * a)
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = -c / b
	else:
		tmp_1 = (b - math.sqrt((-4.0 * (c * a)))) / (a * -2.0)
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -2.65e-90)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(2.0 * Float64(c * Float64(-0.5 / b)));
		else
			tmp_2 = Float64(Float64(b * -2.0) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-c) / b);
	else
		tmp_1 = Float64(Float64(b - sqrt(Float64(-4.0 * Float64(c * a)))) / Float64(a * -2.0));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp_2 = 0.0;
	if (b <= -2.65e-90)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = 2.0 * (c * (-0.5 / b));
		else
			tmp_3 = (b * -2.0) / (2.0 * a);
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = -c / b;
	else
		tmp_2 = (b - sqrt((-4.0 * (c * a)))) / (a * -2.0);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.65e-90], If[GreaterEqual[b, 0.0], N[(2.0 * N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * -2.0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[((-c) / b), $MachinePrecision], N[(N[(b - N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.6500000000000002e-90

   1. Initial program 70.0%

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

      1. add-sqr-sqrt 70.0%

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

      2. pow2 70.0%

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

      3. *-commutative 70.0%

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

   4. Applied egg-rr 70.0%

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

   5. Taylor expanded in c around 0 70.0%

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

      1. associate-*r/ 70.0%

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

      2. associate-*r* 70.0%

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

      3. unpow2 70.0%

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

      4. rem-square-sqrt 70.0%

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

      5. rem-square-sqrt 70.0%

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

      6. unpow2 70.0%

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

      7. *-commutative 70.0%

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

      8. *-commutative 70.0%

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

      9. associate-*r/ 70.0%

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

      10. associate-/l* 70.0%

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

      11. unpow2 70.0%

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

      12. rem-square-sqrt 70.0%

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

   7. Simplified 70.0%

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

   8. Taylor expanded in b around -inf 84.0%

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

      1. *-commutative 84.0%

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

   10. Simplified 84.0%

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

   if -2.6500000000000002e-90 < b

   1. Initial program 73.0%

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

   3. Simplified 72.9%

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

   5. Taylor expanded in c around 0 69.9%

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

      1. associate-*r/ 69.9%

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

      2. mul-1-neg 69.9%

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

   7. Simplified 69.9%

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

   8. Taylor expanded in c around inf 68.5%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.65 \cdot 10^{-90}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \left(c \cdot \frac{-0.5}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \sqrt{-4 \cdot \left(c \cdot a\right)}}{a \cdot -2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.1% accurate, 10.1× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = 2.0 * (c * (-0.5 / b));
	} else {
		tmp = (b * -2.0) / (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) :: tmp
    if (b >= 0.0d0) then
        tmp = 2.0d0 * (c * ((-0.5d0) / b))
    else
        tmp = (b * (-2.0d0)) / (2.0d0 * 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 = 2.0 * (c * (-0.5 / b));
	} else {
		tmp = (b * -2.0) / (2.0 * a);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

   1. add-sqr-sqrt 57.2%

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

   2. pow2 57.2%

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

   3. *-commutative 57.2%

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

4. Applied egg-rr 57.2%

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

5. Taylor expanded in c around 0 69.5%

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

   1. associate-*r/ 69.5%

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

   2. associate-*r* 69.5%

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

   3. unpow2 69.5%

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

   4. rem-square-sqrt 69.9%

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

   5. rem-square-sqrt 33.0%

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

   6. unpow2 33.0%

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

   7. *-commutative 33.0%

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

   8. *-commutative 33.0%

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

   9. associate-*r/ 33.0%

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

   10. associate-/l* 33.0%

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

   11. unpow2 33.0%

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

   12. rem-square-sqrt 69.9%

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

7. Simplified 69.9%

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

8. Taylor expanded in b around -inf 66.4%

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

   1. *-commutative 66.4%

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

10. Simplified 66.4%

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

Alternative 8: 35.5% accurate, 13.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

3. Simplified 72.0%

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

5. Taylor expanded in c around 0 69.9%

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

   1. associate-*r/ 69.9%

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

   2. mul-1-neg 69.9%

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

7. Simplified 69.9%

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

8. Taylor expanded in c around 0 38.2%

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

9. Taylor expanded in b around 0 38.2%

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

Reproduce

herbie shell --seed 2024188 
(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))))