jeff quadratic root 2

Percentage Accurate: 72.6% → 90.7%

Time: 23.6s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (-b - t_0);
	} else {
		tmp = (-b + t_0) / (2.0 * a);
	}
	return tmp;
}

Fortran

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

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (-b - t_0);
	} else {
		tmp = (-b + t_0) / (2.0 * a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - ((4.0 * a) * c)))
	tmp = 0
	if b >= 0.0:
		tmp = (2.0 * c) / (-b - t_0)
	else:
		tmp = (-b + t_0) / (2.0 * a)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0));
	else
		tmp = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (2.0 * c) / (-b - t_0);
	else
		tmp = (-b + t_0) / (2.0 * a);
	end
	tmp_2 = tmp;
end

Wolfram

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

Initial Program: 72.6% accurate, 1.0× speedup

Math

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

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: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)}\\ \mathbf{if}\;b \leq -5 \cdot 10^{+106}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right) \cdot -0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{+115}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - t_0}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma b b (* (* c a) -4.0)))))
   (if (<= b -5e+106)
     (if (>= b 0.0)
       (* c (/ -2.0 (+ b b)))
       (* (+ (* -2.0 (/ c b)) (* 2.0 (/ b a))) -0.5))
     (if (<= b 1.4e+115)
       (if (>= b 0.0) (/ 2.0 (/ (- (- b) t_0) c)) (/ (- t_0 b) (* 2.0 a)))
       (if (>= b 0.0)
         (/ (* c 2.0) (fma 2.0 (/ c (/ b a)) (* b -2.0)))
         (/ (+ b (sqrt (- (* b b) (* c (* a 4.0))))) (* 2.0 a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma(b, b, ((c * a) * -4.0)));
	double tmp_1;
	if (b <= -5e+106) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / (b + b));
		} else {
			tmp_2 = ((-2.0 * (c / b)) + (2.0 * (b / a))) * -0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.4e+115) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = 2.0 / ((-b - t_0) / c);
		} else {
			tmp_3 = (t_0 - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c * 2.0) / fma(2.0, (c / (b / a)), (b * -2.0));
	} else {
		tmp_1 = (b + sqrt(((b * b) - (c * (a * 4.0))))) / (2.0 * a);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(b, b, Float64(Float64(c * a) * -4.0)))
	tmp_1 = 0.0
	if (b <= -5e+106)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * Float64(-2.0 / Float64(b + b)));
		else
			tmp_2 = Float64(Float64(Float64(-2.0 * Float64(c / b)) + Float64(2.0 * Float64(b / a))) * -0.5);
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.4e+115)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(2.0 / Float64(Float64(Float64(-b) - t_0) / c));
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c * 2.0) / fma(2.0, Float64(c / Float64(b / a)), Float64(b * -2.0)));
	else
		tmp_1 = Float64(Float64(b + sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))) / Float64(2.0 * a));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(b * b + N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -5e+106], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], If[LessEqual[b, 1.4e+115], If[GreaterEqual[b, 0.0], N[(2.0 / N[(N[((-b) - t$95$0), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[(2.0 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(b * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.9999999999999998e106

   1. Initial program 47.0%

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

   3. Simplified 47.0%

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

   4. Taylor expanded in b around inf 47.0%

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

   5. Taylor expanded in b around -inf 94.7%

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

   if -4.9999999999999998e106 < b < 1.4e115

   1. Initial program 87.9%

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

   3. Simplified 88.4%

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

   if 1.4e115 < b

   1. Initial program 50.0%

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

      1. add-sqr-sqrt 50.1%

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

      2. pow2 50.1%

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

      3. pow1/2 50.1%

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

      4. sqrt-pow1 50.1%

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

      5. *-commutative 50.1%

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

      6. *-commutative 50.1%

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

      7. metadata-eval 50.1%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - {\left({\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   3. Applied egg-rr 50.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{{\left({\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{0.25}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Taylor expanded in b around inf 92.9%

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

      1. fma-def 92.9%

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

      2. associate-*r* 92.9%

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

      3. metadata-eval 92.9%

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

      4. associate-/l* 96.5%

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

      5. *-commutative 96.5%

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

   6. Simplified 96.5%

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

      1. expm1-log1p-u 96.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)\right)\\ \end{array} \]

      2. expm1-udef 96.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)} - 1\\ \end{array} \]

      3. add-sqr-sqrt 96.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\sqrt{-b} \cdot \sqrt{-b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)} - 1\\ \end{array} \]

      4. sqrt-unprod 96.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\sqrt{\left(-b\right) \cdot \left(-b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)} - 1\\ \end{array} \]

      5. sqr-neg 96.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\sqrt{b \cdot b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)} - 1\\ \end{array} \]

      6. sqrt-prod 96.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\sqrt{b} \cdot \sqrt{b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)} - 1\\ \end{array} \]

      7. add-sqr-sqrt 96.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)} - 1\\ \end{array} \]

      8. *-commutative 96.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{b + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}{2 \cdot a}\right)} - 1\\ \end{array} \]

      9. *-commutative 96.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{2 \cdot a}\right)} - 1\\ \end{array} \]

      10. *-commutative 96.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot 2}\right)} - 1\\ \end{array} \]

   8. Applied egg-rr 96.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot 2}\right)} - 1\\ \end{array} \]
   9. Step-by-step derivation

      1. expm1-def 96.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot 2}\right)\right)\\ \end{array} \]

      2. expm1-log1p 96.5%

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

   10. Simplified 96.5%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+106}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right) \cdot -0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{+115}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)}}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{2 \cdot a}\\ \end{array} \]

Alternative 2: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -2 \cdot \frac{c}{b}\\ t_1 := c \cdot \left(a \cdot -4\right)\\ t_2 := c \cdot \frac{-2}{b + b}\\ \mathbf{if}\;b \leq -7.4 \cdot 10^{+106}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + 2 \cdot \frac{b}{a}\right) \cdot -0.5\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b - \sqrt{b \cdot b + t_1}}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-107}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{t_1}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \mathsf{fma}\left(2, \frac{b}{a}, t_0\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* -2.0 (/ c b)))
        (t_1 (* c (* a -4.0)))
        (t_2 (* c (/ -2.0 (+ b b)))))
   (if (<= b -7.4e+106)
     (if (>= b 0.0) t_2 (* (+ t_0 (* 2.0 (/ b a))) -0.5))
     (if (<= b -2e-310)
       (if (>= b 0.0) t_2 (* -0.5 (/ (- b (sqrt (+ (* b b) t_1))) a)))
       (if (<= b 9.2e-107)
         (if (>= b 0.0)
           (/ (* c -2.0) (+ b (sqrt t_1)))
           (* -0.5 (fma 2.0 (/ b a) t_0)))
         (if (>= b 0.0)
           (/ (* c 2.0) (fma 2.0 (/ c (/ b a)) (* b -2.0)))
           (/ (+ b (sqrt (- (* b b) (* c (* a 4.0))))) (* 2.0 a))))))))

C

double code(double a, double b, double c) {
	double t_0 = -2.0 * (c / b);
	double t_1 = c * (a * -4.0);
	double t_2 = c * (-2.0 / (b + b));
	double tmp_1;
	if (b <= -7.4e+106) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_2;
		} else {
			tmp_2 = (t_0 + (2.0 * (b / a))) * -0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_2;
		} else {
			tmp_3 = -0.5 * ((b - sqrt(((b * b) + t_1))) / a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 9.2e-107) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c * -2.0) / (b + sqrt(t_1));
		} else {
			tmp_4 = -0.5 * fma(2.0, (b / a), t_0);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c * 2.0) / fma(2.0, (c / (b / a)), (b * -2.0));
	} else {
		tmp_1 = (b + sqrt(((b * b) - (c * (a * 4.0))))) / (2.0 * a);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(-2.0 * Float64(c / b))
	t_1 = Float64(c * Float64(a * -4.0))
	t_2 = Float64(c * Float64(-2.0 / Float64(b + b)))
	tmp_1 = 0.0
	if (b <= -7.4e+106)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_2;
		else
			tmp_2 = Float64(Float64(t_0 + Float64(2.0 * Float64(b / a))) * -0.5);
		end
		tmp_1 = tmp_2;
	elseif (b <= -2e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_2;
		else
			tmp_3 = Float64(-0.5 * Float64(Float64(b - sqrt(Float64(Float64(b * b) + t_1))) / a));
		end
		tmp_1 = tmp_3;
	elseif (b <= 9.2e-107)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(c * -2.0) / Float64(b + sqrt(t_1)));
		else
			tmp_4 = Float64(-0.5 * fma(2.0, Float64(b / a), t_0));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c * 2.0) / fma(2.0, Float64(c / Float64(b / a)), Float64(b * -2.0)));
	else
		tmp_1 = Float64(Float64(b + sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))) / Float64(2.0 * a));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-2.0 * N[(c / b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c * N[(-2.0 / N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -7.4e+106], If[GreaterEqual[b, 0.0], t$95$2, N[(N[(t$95$0 + N[(2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], If[LessEqual[b, -2e-310], If[GreaterEqual[b, 0.0], t$95$2, N[(-0.5 * N[(N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 9.2e-107], If[GreaterEqual[b, 0.0], N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(2.0 * N[(b / a), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[(2.0 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(b * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -7.3999999999999999e106

   1. Initial program 47.0%

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

   3. Simplified 47.0%

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

   4. Taylor expanded in b around inf 47.0%

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

   5. Taylor expanded in b around -inf 94.7%

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

   if -7.3999999999999999e106 < b < -1.999999999999994e-310

   1. Initial program 86.1%

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

   3. Simplified 86.1%

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

   4. Taylor expanded in b around inf 86.1%

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

      1. fma-udef 86.1%

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

   6. Applied egg-rr 86.1%

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

   if -1.999999999999994e-310 < b < 9.20000000000000014e-107

   1. Initial program 77.9%

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

   3. Simplified 77.6%

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

   4. Taylor expanded in b around -inf 77.6%

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

      1. +-commutative 77.6%

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

      2. fma-def 77.6%

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

   6. Simplified 77.6%

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

   7. Taylor expanded in b around 0 70.7%

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

      1. *-commutative 70.7%

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

      2. associate-*r* 70.7%

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

      3. *-commutative 70.7%

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

   9. Simplified 70.7%

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

       1. associate-*r/ 70.9%

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

       2. *-commutative 70.9%

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

   11. Applied egg-rr 70.9%

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

   if 9.20000000000000014e-107 < b

   1. Initial program 71.9%

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

      1. add-sqr-sqrt 71.6%

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

      2. pow2 71.6%

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

      3. pow1/2 71.6%

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

      4. sqrt-pow1 71.6%

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

      5. *-commutative 71.6%

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

      6. *-commutative 71.6%

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

      7. metadata-eval 71.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - {\left({\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   3. Applied egg-rr 71.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{{\left({\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{0.25}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Taylor expanded in b around inf 85.4%

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

      1. fma-def 85.4%

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

      2. associate-*r* 85.4%

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

      3. metadata-eval 85.4%

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

      4. associate-/l* 87.3%

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

      5. *-commutative 87.3%

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

   6. Simplified 87.3%

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

      1. expm1-log1p-u 87.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)\right)\\ \end{array} \]

      2. expm1-udef 87.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)} - 1\\ \end{array} \]

      3. add-sqr-sqrt 87.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\sqrt{-b} \cdot \sqrt{-b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)} - 1\\ \end{array} \]

      4. sqrt-unprod 87.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\sqrt{\left(-b\right) \cdot \left(-b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)} - 1\\ \end{array} \]

      5. sqr-neg 87.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\sqrt{b \cdot b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)} - 1\\ \end{array} \]

      6. sqrt-prod 87.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\sqrt{b} \cdot \sqrt{b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)} - 1\\ \end{array} \]

      7. add-sqr-sqrt 87.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)} - 1\\ \end{array} \]

      8. *-commutative 87.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{b + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}{2 \cdot a}\right)} - 1\\ \end{array} \]

      9. *-commutative 87.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{2 \cdot a}\right)} - 1\\ \end{array} \]

      10. *-commutative 87.3%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot 2}\right)} - 1\\ \end{array} \]

   8. Applied egg-rr 87.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot 2}\right)} - 1\\ \end{array} \]
   9. Step-by-step derivation

      1. expm1-def 87.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot 2}\right)\right)\\ \end{array} \]

      2. expm1-log1p 87.3%

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

   10. Simplified 87.3%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.4 \cdot 10^{+106}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right) \cdot -0.5\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-107}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \mathsf{fma}\left(2, \frac{b}{a}, -2 \cdot \frac{c}{b}\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{2 \cdot a}\\ \end{array} \]

Alternative 3: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ \mathbf{if}\;b \leq -8.2 \cdot 10^{+107}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right) \cdot -0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+114}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + t_0}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0))))))
   (if (<= b -8.2e+107)
     (if (>= b 0.0)
       (* c (/ -2.0 (+ b b)))
       (* (+ (* -2.0 (/ c b)) (* 2.0 (/ b a))) -0.5))
     (if (<= b 4e+114)
       (if (>= b 0.0) (/ (* c 2.0) (- (- b) t_0)) (/ (- t_0 b) (* 2.0 a)))
       (if (>= b 0.0)
         (/ (* c 2.0) (fma 2.0 (/ c (/ b a)) (* b -2.0)))
         (/ (+ b t_0) (* 2.0 a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -8.2e+107) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / (b + b));
		} else {
			tmp_2 = ((-2.0 * (c / b)) + (2.0 * (b / a))) * -0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= 4e+114) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c * 2.0) / (-b - t_0);
		} else {
			tmp_3 = (t_0 - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c * 2.0) / fma(2.0, (c / (b / a)), (b * -2.0));
	} else {
		tmp_1 = (b + t_0) / (2.0 * a);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	tmp_1 = 0.0
	if (b <= -8.2e+107)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * Float64(-2.0 / Float64(b + b)));
		else
			tmp_2 = Float64(Float64(Float64(-2.0 * Float64(c / b)) + Float64(2.0 * Float64(b / a))) * -0.5);
		end
		tmp_1 = tmp_2;
	elseif (b <= 4e+114)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(c * 2.0) / Float64(Float64(-b) - t_0));
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c * 2.0) / fma(2.0, Float64(c / Float64(b / a)), Float64(b * -2.0)));
	else
		tmp_1 = Float64(Float64(b + t_0) / Float64(2.0 * a));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -8.2e+107], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], If[LessEqual[b, 4e+114], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[(2.0 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(b * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.1999999999999998e107

   1. Initial program 47.0%

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

   3. Simplified 47.0%

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

   4. Taylor expanded in b around inf 47.0%

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

   5. Taylor expanded in b around -inf 94.7%

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

   if -8.1999999999999998e107 < b < 4e114

   1. Initial program 87.9%

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

   if 4e114 < b

   1. Initial program 50.0%

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

      1. add-sqr-sqrt 50.1%

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

      2. pow2 50.1%

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

      3. pow1/2 50.1%

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

      4. sqrt-pow1 50.1%

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

      5. *-commutative 50.1%

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

      6. *-commutative 50.1%

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

      7. metadata-eval 50.1%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - {\left({\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   3. Applied egg-rr 50.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{{\left({\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{0.25}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Taylor expanded in b around inf 92.9%

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

      1. fma-def 92.9%

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

      2. associate-*r* 92.9%

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

      3. metadata-eval 92.9%

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

      4. associate-/l* 96.5%

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

      5. *-commutative 96.5%

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

   6. Simplified 96.5%

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

      1. expm1-log1p-u 96.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)\right)\\ \end{array} \]

      2. expm1-udef 96.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)} - 1\\ \end{array} \]

      3. add-sqr-sqrt 96.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\sqrt{-b} \cdot \sqrt{-b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)} - 1\\ \end{array} \]

      4. sqrt-unprod 96.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\sqrt{\left(-b\right) \cdot \left(-b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)} - 1\\ \end{array} \]

      5. sqr-neg 96.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\sqrt{b \cdot b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)} - 1\\ \end{array} \]

      6. sqrt-prod 96.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\sqrt{b} \cdot \sqrt{b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)} - 1\\ \end{array} \]

      7. add-sqr-sqrt 96.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)} - 1\\ \end{array} \]

      8. *-commutative 96.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{b + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}{2 \cdot a}\right)} - 1\\ \end{array} \]

      9. *-commutative 96.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{2 \cdot a}\right)} - 1\\ \end{array} \]

      10. *-commutative 96.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot 2}\right)} - 1\\ \end{array} \]

   8. Applied egg-rr 96.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot 2}\right)} - 1\\ \end{array} \]
   9. Step-by-step derivation

      1. expm1-def 96.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot 2}\right)\right)\\ \end{array} \]

      2. expm1-log1p 96.5%

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

   10. Simplified 96.5%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{+107}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right) \cdot -0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+114}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{2 \cdot a}\\ \end{array} \]

Alternative 4: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -2 \cdot \frac{c}{b}\\ t_1 := c \cdot \left(a \cdot -4\right)\\ t_2 := c \cdot \frac{-2}{b + b}\\ \mathbf{if}\;b \leq -7.8 \cdot 10^{+108}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + 2 \cdot \frac{b}{a}\right) \cdot -0.5\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b - \sqrt{b \cdot b + t_1}}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-107}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{t_1}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \mathsf{fma}\left(2, \frac{b}{a}, t_0\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + b}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* -2.0 (/ c b)))
        (t_1 (* c (* a -4.0)))
        (t_2 (* c (/ -2.0 (+ b b)))))
   (if (<= b -7.8e+108)
     (if (>= b 0.0) t_2 (* (+ t_0 (* 2.0 (/ b a))) -0.5))
     (if (<= b -2e-310)
       (if (>= b 0.0) t_2 (* -0.5 (/ (- b (sqrt (+ (* b b) t_1))) a)))
       (if (<= b 9.2e-107)
         (if (>= b 0.0)
           (/ (* c -2.0) (+ b (sqrt t_1)))
           (* -0.5 (fma 2.0 (/ b a) t_0)))
         (if (>= b 0.0) (/ (* c -2.0) (+ b b)) (* -0.5 (/ (+ b b) a))))))))

C

double code(double a, double b, double c) {
	double t_0 = -2.0 * (c / b);
	double t_1 = c * (a * -4.0);
	double t_2 = c * (-2.0 / (b + b));
	double tmp_1;
	if (b <= -7.8e+108) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_2;
		} else {
			tmp_2 = (t_0 + (2.0 * (b / a))) * -0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_2;
		} else {
			tmp_3 = -0.5 * ((b - sqrt(((b * b) + t_1))) / a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 9.2e-107) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c * -2.0) / (b + sqrt(t_1));
		} else {
			tmp_4 = -0.5 * fma(2.0, (b / a), t_0);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c * -2.0) / (b + b);
	} else {
		tmp_1 = -0.5 * ((b + b) / a);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(-2.0 * Float64(c / b))
	t_1 = Float64(c * Float64(a * -4.0))
	t_2 = Float64(c * Float64(-2.0 / Float64(b + b)))
	tmp_1 = 0.0
	if (b <= -7.8e+108)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_2;
		else
			tmp_2 = Float64(Float64(t_0 + Float64(2.0 * Float64(b / a))) * -0.5);
		end
		tmp_1 = tmp_2;
	elseif (b <= -2e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_2;
		else
			tmp_3 = Float64(-0.5 * Float64(Float64(b - sqrt(Float64(Float64(b * b) + t_1))) / a));
		end
		tmp_1 = tmp_3;
	elseif (b <= 9.2e-107)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(c * -2.0) / Float64(b + sqrt(t_1)));
		else
			tmp_4 = Float64(-0.5 * fma(2.0, Float64(b / a), t_0));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c * -2.0) / Float64(b + b));
	else
		tmp_1 = Float64(-0.5 * Float64(Float64(b + b) / a));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-2.0 * N[(c / b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c * N[(-2.0 / N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -7.8e+108], If[GreaterEqual[b, 0.0], t$95$2, N[(N[(t$95$0 + N[(2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], If[LessEqual[b, -2e-310], If[GreaterEqual[b, 0.0], t$95$2, N[(-0.5 * N[(N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 9.2e-107], If[GreaterEqual[b, 0.0], N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(2.0 * N[(b / a), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c * -2.0), $MachinePrecision] / N[(b + b), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -7.79999999999999969e108

   1. Initial program 47.0%

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

   3. Simplified 47.0%

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

   4. Taylor expanded in b around inf 47.0%

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

   5. Taylor expanded in b around -inf 94.7%

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

   if -7.79999999999999969e108 < b < -1.999999999999994e-310

   1. Initial program 86.1%

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

   3. Simplified 86.1%

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

   4. Taylor expanded in b around inf 86.1%

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

      1. fma-udef 86.1%

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

   6. Applied egg-rr 86.1%

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

   if -1.999999999999994e-310 < b < 9.20000000000000014e-107

   1. Initial program 77.9%

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

   3. Simplified 77.6%

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

   4. Taylor expanded in b around -inf 77.6%

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

      1. +-commutative 77.6%

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

      2. fma-def 77.6%

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

   6. Simplified 77.6%

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

   7. Taylor expanded in b around 0 70.7%

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

      1. *-commutative 70.7%

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

      2. associate-*r* 70.7%

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

      3. *-commutative 70.7%

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

   9. Simplified 70.7%

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

       1. associate-*r/ 70.9%

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

       2. *-commutative 70.9%

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

   11. Applied egg-rr 70.9%

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

   if 9.20000000000000014e-107 < b

   1. Initial program 71.9%

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

   3. Simplified 71.7%

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

   4. Taylor expanded in b around inf 86.7%

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

   5. Taylor expanded in b around -inf 86.7%

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

      1. count-2 86.7%

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

   7. Simplified 86.7%

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

      1. associate-*r/ 87.0%

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

   9. Applied egg-rr 87.0%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.8 \cdot 10^{+108}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right) \cdot -0.5\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-107}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \mathsf{fma}\left(2, \frac{b}{a}, -2 \cdot \frac{c}{b}\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + b}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \end{array} \]

Alternative 5: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -2 \cdot \frac{c}{b}\\ t_1 := c \cdot \frac{-2}{b + b}\\ \mathbf{if}\;b \leq -3.4 \cdot 10^{-99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + 2 \cdot \frac{b}{a}\right) \cdot -0.5\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b - \sqrt{\left(c \cdot a\right) \cdot -4}}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{-106}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \mathsf{fma}\left(2, \frac{b}{a}, t_0\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + b}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* -2.0 (/ c b))) (t_1 (* c (/ -2.0 (+ b b)))))
   (if (<= b -3.4e-99)
     (if (>= b 0.0) t_1 (* (+ t_0 (* 2.0 (/ b a))) -0.5))
     (if (<= b -2e-310)
       (if (>= b 0.0) t_1 (* -0.5 (/ (- b (sqrt (* (* c a) -4.0))) a)))
       (if (<= b 1e-106)
         (if (>= b 0.0)
           (* c (/ -2.0 (+ b (sqrt (* c (* a -4.0))))))
           (* -0.5 (fma 2.0 (/ b a) t_0)))
         (if (>= b 0.0) (/ (* c -2.0) (+ b b)) (* -0.5 (/ (+ b b) a))))))))

C

double code(double a, double b, double c) {
	double t_0 = -2.0 * (c / b);
	double t_1 = c * (-2.0 / (b + b));
	double tmp_1;
	if (b <= -3.4e-99) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = (t_0 + (2.0 * (b / a))) * -0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = -0.5 * ((b - sqrt(((c * a) * -4.0))) / a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1e-106) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = c * (-2.0 / (b + sqrt((c * (a * -4.0)))));
		} else {
			tmp_4 = -0.5 * fma(2.0, (b / a), t_0);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c * -2.0) / (b + b);
	} else {
		tmp_1 = -0.5 * ((b + b) / a);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(-2.0 * Float64(c / b))
	t_1 = Float64(c * Float64(-2.0 / Float64(b + b)))
	tmp_1 = 0.0
	if (b <= -3.4e-99)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(Float64(t_0 + Float64(2.0 * Float64(b / a))) * -0.5);
		end
		tmp_1 = tmp_2;
	elseif (b <= -2e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = Float64(-0.5 * Float64(Float64(b - sqrt(Float64(Float64(c * a) * -4.0))) / a));
		end
		tmp_1 = tmp_3;
	elseif (b <= 1e-106)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(c * Float64(-2.0 / Float64(b + sqrt(Float64(c * Float64(a * -4.0))))));
		else
			tmp_4 = Float64(-0.5 * fma(2.0, Float64(b / a), t_0));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c * -2.0) / Float64(b + b));
	else
		tmp_1 = Float64(-0.5 * Float64(Float64(b + b) / a));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-2.0 * N[(c / b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(-2.0 / N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.4e-99], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(t$95$0 + N[(2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], If[LessEqual[b, -2e-310], If[GreaterEqual[b, 0.0], t$95$1, N[(-0.5 * N[(N[(b - N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1e-106], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(2.0 * N[(b / a), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c * -2.0), $MachinePrecision] / N[(b + b), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -3.40000000000000007e-99

   1. Initial program 67.2%

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

   3. Simplified 67.2%

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

   4. Taylor expanded in b around inf 67.2%

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

   5. Taylor expanded in b around -inf 83.0%

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

   if -3.40000000000000007e-99 < b < -1.999999999999994e-310

   1. Initial program 80.2%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in b around inf 80.2%

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

   5. Taylor expanded in b around 0 83.6%

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

   if -1.999999999999994e-310 < b < 9.99999999999999941e-107

   1. Initial program 77.9%

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

   3. Simplified 77.6%

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

   4. Taylor expanded in b around -inf 77.6%

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

      1. +-commutative 77.6%

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

      2. fma-def 77.6%

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

   6. Simplified 77.6%

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

   7. Taylor expanded in b around 0 70.7%

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

      1. *-commutative 70.7%

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

      2. associate-*r* 70.7%

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

      3. *-commutative 70.7%

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

   9. Simplified 70.7%

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

   if 9.99999999999999941e-107 < b

   1. Initial program 71.9%

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

   3. Simplified 71.7%

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

   4. Taylor expanded in b around inf 86.7%

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

   5. Taylor expanded in b around -inf 86.7%

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

      1. count-2 86.7%

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

   7. Simplified 86.7%

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

      1. associate-*r/ 87.0%

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

   9. Applied egg-rr 87.0%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{-99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right) \cdot -0.5\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b - \sqrt{\left(c \cdot a\right) \cdot -4}}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{-106}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \mathsf{fma}\left(2, \frac{b}{a}, -2 \cdot \frac{c}{b}\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + b}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \end{array} \]

Alternative 6: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -2 \cdot \frac{c}{b}\\ t_1 := c \cdot \frac{-2}{b + b}\\ \mathbf{if}\;b \leq -3.4 \cdot 10^{-99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + 2 \cdot \frac{b}{a}\right) \cdot -0.5\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b - \sqrt{\left(c \cdot a\right) \cdot -4}}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{-106}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \mathsf{fma}\left(2, \frac{b}{a}, t_0\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + b}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* -2.0 (/ c b))) (t_1 (* c (/ -2.0 (+ b b)))))
   (if (<= b -3.4e-99)
     (if (>= b 0.0) t_1 (* (+ t_0 (* 2.0 (/ b a))) -0.5))
     (if (<= b -2e-310)
       (if (>= b 0.0) t_1 (* -0.5 (/ (- b (sqrt (* (* c a) -4.0))) a)))
       (if (<= b 1e-106)
         (if (>= b 0.0)
           (/ (* c -2.0) (+ b (sqrt (* c (* a -4.0)))))
           (* -0.5 (fma 2.0 (/ b a) t_0)))
         (if (>= b 0.0) (/ (* c -2.0) (+ b b)) (* -0.5 (/ (+ b b) a))))))))

C

double code(double a, double b, double c) {
	double t_0 = -2.0 * (c / b);
	double t_1 = c * (-2.0 / (b + b));
	double tmp_1;
	if (b <= -3.4e-99) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = (t_0 + (2.0 * (b / a))) * -0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = -0.5 * ((b - sqrt(((c * a) * -4.0))) / a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1e-106) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c * -2.0) / (b + sqrt((c * (a * -4.0))));
		} else {
			tmp_4 = -0.5 * fma(2.0, (b / a), t_0);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c * -2.0) / (b + b);
	} else {
		tmp_1 = -0.5 * ((b + b) / a);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(-2.0 * Float64(c / b))
	t_1 = Float64(c * Float64(-2.0 / Float64(b + b)))
	tmp_1 = 0.0
	if (b <= -3.4e-99)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(Float64(t_0 + Float64(2.0 * Float64(b / a))) * -0.5);
		end
		tmp_1 = tmp_2;
	elseif (b <= -2e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = Float64(-0.5 * Float64(Float64(b - sqrt(Float64(Float64(c * a) * -4.0))) / a));
		end
		tmp_1 = tmp_3;
	elseif (b <= 1e-106)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(c * -2.0) / Float64(b + sqrt(Float64(c * Float64(a * -4.0)))));
		else
			tmp_4 = Float64(-0.5 * fma(2.0, Float64(b / a), t_0));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c * -2.0) / Float64(b + b));
	else
		tmp_1 = Float64(-0.5 * Float64(Float64(b + b) / a));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-2.0 * N[(c / b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(-2.0 / N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.4e-99], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(t$95$0 + N[(2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], If[LessEqual[b, -2e-310], If[GreaterEqual[b, 0.0], t$95$1, N[(-0.5 * N[(N[(b - N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1e-106], If[GreaterEqual[b, 0.0], N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(2.0 * N[(b / a), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c * -2.0), $MachinePrecision] / N[(b + b), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -3.40000000000000007e-99

   1. Initial program 67.2%

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

   3. Simplified 67.2%

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

   4. Taylor expanded in b around inf 67.2%

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

   5. Taylor expanded in b around -inf 83.0%

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

   if -3.40000000000000007e-99 < b < -1.999999999999994e-310

   1. Initial program 80.2%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in b around inf 80.2%

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

   5. Taylor expanded in b around 0 83.6%

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

   if -1.999999999999994e-310 < b < 9.99999999999999941e-107

   1. Initial program 77.9%

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

   3. Simplified 77.6%

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

   4. Taylor expanded in b around -inf 77.6%

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

      1. +-commutative 77.6%

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

      2. fma-def 77.6%

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

   6. Simplified 77.6%

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

   7. Taylor expanded in b around 0 70.7%

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

      1. *-commutative 70.7%

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

      2. associate-*r* 70.7%

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

      3. *-commutative 70.7%

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

   9. Simplified 70.7%

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

       1. associate-*r/ 70.9%

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

       2. *-commutative 70.9%

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

   11. Applied egg-rr 70.9%

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

   if 9.99999999999999941e-107 < b

   1. Initial program 71.9%

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

   3. Simplified 71.7%

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

   4. Taylor expanded in b around inf 86.7%

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

   5. Taylor expanded in b around -inf 86.7%

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

      1. count-2 86.7%

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

   7. Simplified 86.7%

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

      1. associate-*r/ 87.0%

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

   9. Applied egg-rr 87.0%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{-99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right) \cdot -0.5\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b - \sqrt{\left(c \cdot a\right) \cdot -4}}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{-106}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \mathsf{fma}\left(2, \frac{b}{a}, -2 \cdot \frac{c}{b}\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + b}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \end{array} \]

Alternative 7: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \frac{-2}{b + b}\\ \mathbf{if}\;b \leq -3.4 \cdot 10^{-99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right) \cdot -0.5\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b - \sqrt{\left(c \cdot a\right) \cdot -4}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (/ -2.0 (+ b b)))))
   (if (<= b -3.4e-99)
     (if (>= b 0.0) t_0 (* (+ (* -2.0 (/ c b)) (* 2.0 (/ b a))) -0.5))
     (if (>= b 0.0) t_0 (* -0.5 (/ (- b (sqrt (* (* c a) -4.0))) a))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (-2.0 / (b + b));
	double tmp_1;
	if (b <= -3.4e-99) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = ((-2.0 * (c / b)) + (2.0 * (b / a))) * -0.5;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = -0.5 * ((b - sqrt(((c * a) * -4.0))) / 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 = c * ((-2.0d0) / (b + b))
    if (b <= (-3.4d-99)) then
        if (b >= 0.0d0) then
            tmp_2 = t_0
        else
            tmp_2 = (((-2.0d0) * (c / b)) + (2.0d0 * (b / a))) * (-0.5d0)
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = t_0
    else
        tmp_1 = (-0.5d0) * ((b - sqrt(((c * a) * (-4.0d0)))) / a)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (-2.0 / (b + b));
	double tmp_1;
	if (b <= -3.4e-99) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = ((-2.0 * (c / b)) + (2.0 * (b / a))) * -0.5;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = -0.5 * ((b - Math.sqrt(((c * a) * -4.0))) / a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = c * (-2.0 / (b + b))
	tmp_1 = 0
	if b <= -3.4e-99:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_0
		else:
			tmp_2 = ((-2.0 * (c / b)) + (2.0 * (b / a))) * -0.5
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = t_0
	else:
		tmp_1 = -0.5 * ((b - math.sqrt(((c * a) * -4.0))) / a)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(-2.0 / Float64(b + b)))
	tmp_1 = 0.0
	if (b <= -3.4e-99)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(Float64(Float64(-2.0 * Float64(c / b)) + Float64(2.0 * Float64(b / a))) * -0.5);
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = t_0;
	else
		tmp_1 = Float64(-0.5 * Float64(Float64(b - sqrt(Float64(Float64(c * a) * -4.0))) / a));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = c * (-2.0 / (b + b));
	tmp_2 = 0.0;
	if (b <= -3.4e-99)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_0;
		else
			tmp_3 = ((-2.0 * (c / b)) + (2.0 * (b / a))) * -0.5;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = t_0;
	else
		tmp_2 = -0.5 * ((b - sqrt(((c * a) * -4.0))) / a);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(-2.0 / N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.4e-99], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(N[(-2.0 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$0, N[(-0.5 * N[(N[(b - N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.40000000000000007e-99

   1. Initial program 67.2%

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

   3. Simplified 67.2%

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

   4. Taylor expanded in b around inf 67.2%

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

   5. Taylor expanded in b around -inf 83.0%

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

   if -3.40000000000000007e-99 < b

   1. Initial program 74.3%

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

   3. Simplified 74.2%

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

   4. Taylor expanded in b around inf 73.6%

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

   5. Taylor expanded in b around 0 74.3%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{-99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right) \cdot -0.5\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b - \sqrt{\left(c \cdot a\right) \cdot -4}}{a}\\ \end{array} \]

Alternative 8: 67.7% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

3. Simplified 71.5%

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

4. Taylor expanded in b around inf 71.2%

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

5. Taylor expanded in b around -inf 69.4%

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

6. Final simplification 69.4%

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

Alternative 9: 67.5% accurate, 13.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

3. Simplified 71.5%

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

4. Taylor expanded in b around inf 71.2%

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

5. Taylor expanded in b around -inf 69.2%

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

   1. count-2 69.2%

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

7. Simplified 69.2%

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

8. Final simplification 69.2%

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

Alternative 10: 67.6% accurate, 13.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

3. Simplified 71.5%

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

4. Taylor expanded in b around inf 71.2%

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

5. Taylor expanded in b around -inf 69.2%

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

   1. count-2 69.2%

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

7. Simplified 69.2%

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

   1. associate-*r/ 69.4%

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

9. Applied egg-rr 69.4%

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

10. Final simplification 69.4%

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

Reproduce

herbie shell --seed 2023275 
(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))))