jeff quadratic root 2

Percentage Accurate: 72.6% → 90.7%

Time: 22.4s

Alternatives: 11

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

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

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 (fma -2.0 (* c (/ a b)) b))))
       (* (fma 2.0 (/ b a) (* -2.0 (/ c b))) -0.5))
     (if (<= b 1.4e+115)
       (if (>= b 0.0) (/ 2.0 (/ (- (- b) t_0) c)) (/ (- t_0 b) (* a 2.0)))
       (if (>= b 0.0)
         (/ (* c 2.0) (fma -2.0 b (* 2.0 (/ a (/ b c)))))
         (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)))))))

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 + fma(-2.0, (c * (a / b)), b)));
		} else {
			tmp_2 = fma(2.0, (b / a), (-2.0 * (c / b))) * -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) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c * 2.0) / fma(-2.0, b, (2.0 * (a / (b / c))));
	} else {
		tmp_1 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	}
	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 + fma(-2.0, Float64(c * Float64(a / b)), b))));
		else
			tmp_2 = Float64(fma(2.0, Float64(b / a), Float64(-2.0 * Float64(c / b))) * -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(a * 2.0));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c * 2.0) / fma(-2.0, b, Float64(2.0 * Float64(a / Float64(b / c)))));
	else
		tmp_1 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	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 + N[(-2.0 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(b / a), $MachinePrecision] + N[(-2.0 * N[(c / b), $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[(a * 2.0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[(-2.0 * b + N[(2.0 * N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $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%

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

      1. +-commutative 47.0%

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

      2. fma-def 47.0%

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

      3. associate-/l* 47.0%

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

      4. associate-/r/ 47.0%

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

   6. Simplified 47.0%

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

   7. Taylor expanded in b around -inf 94.7%

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

      1. +-commutative 94.7%

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

      2. fma-def 94.7%

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

   9. Simplified 94.7%

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

   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%

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

      1. associate-*r* 92.9%

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

      2. metadata-eval 92.9%

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

      3. fma-def 92.9%

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

      4. associate-/l* 96.5%

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

   6. Simplified 96.5%

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

4. Final simplification 91.4%

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

Alternative 2: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ t_1 := b + \mathsf{fma}\left(-2, c \cdot \frac{a}{b}, b\right)\\ \mathbf{if}\;b \leq -7.4 \cdot 10^{+106}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{b}{a}, -2 \cdot \frac{c}{b}\right) \cdot -0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{+115}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{t_1}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b - \sqrt{\left(c \cdot a\right) \cdot -4}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0)))))
        (t_1 (+ b (fma -2.0 (* c (/ a b)) b))))
   (if (<= b -7.4e+106)
     (if (>= b 0.0)
       (* c (/ -2.0 t_1))
       (* (fma 2.0 (/ b a) (* -2.0 (/ c b))) -0.5))
     (if (<= b 1.2e+115)
       (if (>= b 0.0) (/ (* c 2.0) (- (- b) t_0)) (/ (- t_0 b) (* a 2.0)))
       (if (>= b 0.0)
         (/ (* c -2.0) t_1)
         (* -0.5 (/ (- b (sqrt (* (* c a) -4.0))) a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double t_1 = b + fma(-2.0, (c * (a / b)), b);
	double tmp_1;
	if (b <= -7.4e+106) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / t_1);
		} else {
			tmp_2 = fma(2.0, (b / a), (-2.0 * (c / b))) * -0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.2e+115) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c * 2.0) / (-b - t_0);
		} else {
			tmp_3 = (t_0 - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c * -2.0) / t_1;
	} else {
		tmp_1 = -0.5 * ((b - sqrt(((c * a) * -4.0))) / a);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	t_1 = Float64(b + fma(-2.0, Float64(c * Float64(a / b)), b))
	tmp_1 = 0.0
	if (b <= -7.4e+106)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * Float64(-2.0 / t_1));
		else
			tmp_2 = Float64(fma(2.0, Float64(b / a), Float64(-2.0 * Float64(c / b))) * -0.5);
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.2e+115)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(c * 2.0) / Float64(Float64(-b) - t_0));
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(a * 2.0));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c * -2.0) / t_1);
	else
		tmp_1 = Float64(-0.5 * Float64(Float64(b - sqrt(Float64(Float64(c * a) * -4.0))) / a));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(b + N[(-2.0 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -7.4e+106], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(b / a), $MachinePrecision] + N[(-2.0 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], If[LessEqual[b, 1.2e+115], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c * -2.0), $MachinePrecision] / t$95$1), $MachinePrecision], 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 3 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%

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

      1. +-commutative 47.0%

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

      2. fma-def 47.0%

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

      3. associate-/l* 47.0%

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

      4. associate-/r/ 47.0%

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

   6. Simplified 47.0%

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

   7. Taylor expanded in b around -inf 94.7%

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

      1. +-commutative 94.7%

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

      2. fma-def 94.7%

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

   9. Simplified 94.7%

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

   if -7.3999999999999999e106 < b < 1.2e115

   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 1.2e115 < b

   1. Initial program 50.0%

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

   3. Simplified 50.0%

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

   4. Taylor expanded in b around inf 92.6%

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

      1. +-commutative 92.6%

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

      2. fma-def 92.6%

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

      3. associate-/l* 96.2%

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

      4. associate-/r/ 96.2%

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

   6. Simplified 96.2%

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

   7. Taylor expanded in b around 0 96.2%

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

      1. expm1-log1p-u 88.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot \frac{-2}{b + \mathsf{fma}\left(-2, \frac{a}{b} \cdot c, b\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \cdot -0.5\\ \end{array} \]

      2. expm1-udef 46.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{e^{\mathsf{log1p}\left(c \cdot \frac{-2}{b + \mathsf{fma}\left(-2, \frac{a}{b} \cdot c, b\right)}\right)} - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \cdot -0.5\\ \end{array} \]

      3. *-commutative 46.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;e^{\mathsf{log1p}\left(c \cdot \frac{-2}{b + \mathsf{fma}\left(-2, \color{blue}{c \cdot \frac{a}{b}}, b\right)}\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \cdot -0.5\\ \end{array} \]

   9. Applied egg-rr 46.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{e^{\mathsf{log1p}\left(c \cdot \frac{-2}{b + \mathsf{fma}\left(-2, c \cdot \frac{a}{b}, b\right)}\right)} - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \cdot -0.5\\ \end{array} \]
   10. Step-by-step derivation

       1. expm1-def 88.6%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot \frac{-2}{b + \mathsf{fma}\left(-2, c \cdot \frac{a}{b}, b\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \cdot -0.5\\ \end{array} \]

       2. expm1-log1p 96.2%

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

       3. associate-*r/ 96.5%

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

   11. Simplified 96.5%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.4 \cdot 10^{+106}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \mathsf{fma}\left(-2, c \cdot \frac{a}{b}, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{b}{a}, -2 \cdot \frac{c}{b}\right) \cdot -0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{+115}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + \mathsf{fma}\left(-2, c \cdot \frac{a}{b}, b\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b - \sqrt{\left(c \cdot a\right) \cdot -4}}{a}\\ \end{array} \]

Alternative 3: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ t_1 := \frac{t_0 - b}{a \cdot 2}\\ \mathbf{if}\;b \leq -8.2 \cdot 10^{+107}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \mathsf{fma}\left(-2, c \cdot \frac{a}{b}, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{b}{a}, -2 \cdot \frac{c}{b}\right) \cdot -0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+114}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(-2, b, 2 \cdot \frac{a}{\frac{b}{c}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0))))) (t_1 (/ (- t_0 b) (* a 2.0))))
   (if (<= b -8.2e+107)
     (if (>= b 0.0)
       (* c (/ -2.0 (+ b (fma -2.0 (* c (/ a b)) b))))
       (* (fma 2.0 (/ b a) (* -2.0 (/ c b))) -0.5))
     (if (<= b 4e+114)
       (if (>= b 0.0) (/ (* c 2.0) (- (- b) t_0)) t_1)
       (if (>= b 0.0) (/ (* c 2.0) (fma -2.0 b (* 2.0 (/ a (/ b c))))) t_1)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double t_1 = (t_0 - b) / (a * 2.0);
	double tmp_1;
	if (b <= -8.2e+107) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / (b + fma(-2.0, (c * (a / b)), b)));
		} else {
			tmp_2 = fma(2.0, (b / a), (-2.0 * (c / b))) * -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_1;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c * 2.0) / fma(-2.0, b, (2.0 * (a / (b / c))));
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Julia

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

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -8.2e+107], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[(b + N[(-2.0 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(b / a), $MachinePrecision] + N[(-2.0 * N[(c / b), $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], t$95$1], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[(-2.0 * b + N[(2.0 * N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

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%

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

      1. +-commutative 47.0%

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

      2. fma-def 47.0%

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

      3. associate-/l* 47.0%

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

      4. associate-/r/ 47.0%

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

   6. Simplified 47.0%

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

   7. Taylor expanded in b around -inf 94.7%

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

      1. +-commutative 94.7%

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

      2. fma-def 94.7%

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

   9. Simplified 94.7%

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

   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%

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

      1. associate-*r* 92.9%

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

      2. metadata-eval 92.9%

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

      3. fma-def 92.9%

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

      4. associate-/l* 96.5%

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

   6. Simplified 96.5%

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

4. Final simplification 91.1%

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

Alternative 4: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left(c \cdot a\right) \cdot -4}\\ t_1 := -0.5 \cdot \frac{b - t_0}{a}\\ t_2 := b + \mathsf{fma}\left(-2, c \cdot \frac{a}{b}, b\right)\\ \mathbf{if}\;b \leq -3.4 \cdot 10^{-99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{b}{a}, -2 \cdot \frac{c}{b}\right) \cdot -0.5\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{\frac{b}{a}}{c}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{-106}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + t_0}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* (* c a) -4.0)))
        (t_1 (* -0.5 (/ (- b t_0) a)))
        (t_2 (+ b (fma -2.0 (* c (/ a b)) b))))
   (if (<= b -3.4e-99)
     (if (>= b 0.0)
       (* c (/ -2.0 t_2))
       (* (fma 2.0 (/ b a) (* -2.0 (/ c b))) -0.5))
     (if (<= b -2e-310)
       (if (>= b 0.0) (* c (/ (/ b a) c)) t_1)
       (if (<= b 1e-106)
         (if (>= b 0.0) (* c (/ -2.0 (+ b t_0))) (* -0.5 (/ (+ b b) a)))
         (if (>= b 0.0) (/ (* c -2.0) t_2) t_1))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((c * a) * -4.0));
	double t_1 = -0.5 * ((b - t_0) / a);
	double t_2 = b + fma(-2.0, (c * (a / b)), b);
	double tmp_1;
	if (b <= -3.4e-99) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / t_2);
		} else {
			tmp_2 = fma(2.0, (b / a), (-2.0 * (c / b))) * -0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = c * ((b / a) / c);
		} else {
			tmp_3 = t_1;
		}
		tmp_1 = tmp_3;
	} else if (b <= 1e-106) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = c * (-2.0 / (b + t_0));
		} else {
			tmp_4 = -0.5 * ((b + b) / a);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c * -2.0) / t_2;
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(c * a) * -4.0))
	t_1 = Float64(-0.5 * Float64(Float64(b - t_0) / a))
	t_2 = Float64(b + fma(-2.0, Float64(c * Float64(a / b)), b))
	tmp_1 = 0.0
	if (b <= -3.4e-99)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * Float64(-2.0 / t_2));
		else
			tmp_2 = Float64(fma(2.0, Float64(b / a), Float64(-2.0 * Float64(c / b))) * -0.5);
		end
		tmp_1 = tmp_2;
	elseif (b <= -2e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(c * Float64(Float64(b / a) / c));
		else
			tmp_3 = t_1;
		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 + t_0)));
		else
			tmp_4 = Float64(-0.5 * Float64(Float64(b + b) / a));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c * -2.0) / t_2);
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

Wolfram

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

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%

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

      1. +-commutative 67.2%

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

      2. fma-def 67.2%

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

      3. associate-/l* 67.2%

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

      4. associate-/r/ 67.2%

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

   6. Simplified 67.2%

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

   7. Taylor expanded in b around -inf 83.0%

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

      1. +-commutative 83.0%

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

      2. fma-def 83.0%

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

   9. Simplified 83.0%

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

   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%

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

      1. +-commutative 80.2%

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

      2. fma-def 80.2%

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

      3. associate-/l* 80.2%

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

      4. associate-/r/ 80.2%

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

   6. Simplified 80.2%

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

   7. Taylor expanded in b around 0 83.6%

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

   8. Taylor expanded in b around 0 83.6%

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

      1. associate-/r* 83.6%

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

   10. Simplified 83.6%

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

   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%

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

   5. Taylor expanded in b around 0 70.7%

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

   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 85.1%

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

      1. +-commutative 85.1%

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

      2. fma-def 85.1%

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

      3. associate-/l* 87.0%

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

      4. associate-/r/ 87.0%

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

   6. Simplified 87.0%

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

   7. Taylor expanded in b around 0 87.0%

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

      1. expm1-log1p-u 79.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot \frac{-2}{b + \mathsf{fma}\left(-2, \frac{a}{b} \cdot c, b\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \cdot -0.5\\ \end{array} \]

      2. expm1-udef 35.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{e^{\mathsf{log1p}\left(c \cdot \frac{-2}{b + \mathsf{fma}\left(-2, \frac{a}{b} \cdot c, b\right)}\right)} - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \cdot -0.5\\ \end{array} \]

      3. *-commutative 35.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;e^{\mathsf{log1p}\left(c \cdot \frac{-2}{b + \mathsf{fma}\left(-2, \color{blue}{c \cdot \frac{a}{b}}, b\right)}\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \cdot -0.5\\ \end{array} \]

   9. Applied egg-rr 35.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{e^{\mathsf{log1p}\left(c \cdot \frac{-2}{b + \mathsf{fma}\left(-2, c \cdot \frac{a}{b}, b\right)}\right)} - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \cdot -0.5\\ \end{array} \]
   10. Step-by-step derivation

       1. expm1-def 79.6%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot \frac{-2}{b + \mathsf{fma}\left(-2, c \cdot \frac{a}{b}, b\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \cdot -0.5\\ \end{array} \]

       2. expm1-log1p 87.0%

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

       3. associate-*r/ 87.3%

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

   11. Simplified 87.3%

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

4. Final simplification 83.5%

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

Alternative 5: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b + \mathsf{fma}\left(-2, c \cdot \frac{a}{b}, b\right)\\ t_1 := c \cdot \frac{-2}{t_0}\\ t_2 := \sqrt{\left(c \cdot a\right) \cdot -4}\\ \mathbf{if}\;b \leq -5 \cdot 10^{+106}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{b}{a}, -2 \cdot \frac{c}{b}\right) \cdot -0.5\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 8.4 \cdot 10^{-107}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + t_2}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b - t_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ b (fma -2.0 (* c (/ a b)) b)))
        (t_1 (* c (/ -2.0 t_0)))
        (t_2 (sqrt (* (* c a) -4.0))))
   (if (<= b -5e+106)
     (if (>= b 0.0) t_1 (* (fma 2.0 (/ b a) (* -2.0 (/ c b))) -0.5))
     (if (<= b -2e-310)
       (if (>= b 0.0)
         t_1
         (* -0.5 (/ (- b (sqrt (+ (* b b) (* c (* a -4.0))))) a)))
       (if (<= b 8.4e-107)
         (if (>= b 0.0) (* c (/ -2.0 (+ b t_2))) (* -0.5 (/ (+ b b) a)))
         (if (>= b 0.0) (/ (* c -2.0) t_0) (* -0.5 (/ (- b t_2) a))))))))

C

double code(double a, double b, double c) {
	double t_0 = b + fma(-2.0, (c * (a / b)), b);
	double t_1 = c * (-2.0 / t_0);
	double t_2 = sqrt(((c * a) * -4.0));
	double tmp_1;
	if (b <= -5e+106) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = fma(2.0, (b / a), (-2.0 * (c / b))) * -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(((b * b) + (c * (a * -4.0))))) / a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 8.4e-107) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = c * (-2.0 / (b + t_2));
		} else {
			tmp_4 = -0.5 * ((b + b) / a);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c * -2.0) / t_0;
	} else {
		tmp_1 = -0.5 * ((b - t_2) / a);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(b + fma(-2.0, Float64(c * Float64(a / b)), b))
	t_1 = Float64(c * Float64(-2.0 / t_0))
	t_2 = sqrt(Float64(Float64(c * a) * -4.0))
	tmp_1 = 0.0
	if (b <= -5e+106)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(fma(2.0, Float64(b / a), Float64(-2.0 * Float64(c / b))) * -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(b * b) + Float64(c * Float64(a * -4.0))))) / a));
		end
		tmp_1 = tmp_3;
	elseif (b <= 8.4e-107)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(c * Float64(-2.0 / Float64(b + t_2)));
		else
			tmp_4 = Float64(-0.5 * Float64(Float64(b + b) / a));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c * -2.0) / t_0);
	else
		tmp_1 = Float64(-0.5 * Float64(Float64(b - t_2) / a));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b + N[(-2.0 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(-2.0 / t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -5e+106], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(2.0 * N[(b / a), $MachinePrecision] + N[(-2.0 * N[(c / b), $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[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 8.4e-107], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[(b + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c * -2.0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(-0.5 * N[(N[(b - t$95$2), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 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%

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

      1. +-commutative 47.0%

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

      2. fma-def 47.0%

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

      3. associate-/l* 47.0%

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

      4. associate-/r/ 47.0%

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

   6. Simplified 47.0%

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

   7. Taylor expanded in b around -inf 94.7%

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

      1. +-commutative 94.7%

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

      2. fma-def 94.7%

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

   9. Simplified 94.7%

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

   if -4.9999999999999998e106 < 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%

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

      1. +-commutative 86.1%

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

      2. fma-def 86.1%

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

      3. associate-/l* 86.1%

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

      4. associate-/r/ 86.1%

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

   6. Simplified 86.1%

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

      1. fma-udef 86.1%

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

   8. Applied egg-rr 86.1%

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

   if -1.999999999999994e-310 < b < 8.3999999999999997e-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%

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

   5. Taylor expanded in b around 0 70.7%

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

   if 8.3999999999999997e-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 85.1%

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

      1. +-commutative 85.1%

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

      2. fma-def 85.1%

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

      3. associate-/l* 87.0%

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

      4. associate-/r/ 87.0%

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

   6. Simplified 87.0%

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

   7. Taylor expanded in b around 0 87.0%

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

      1. expm1-log1p-u 79.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot \frac{-2}{b + \mathsf{fma}\left(-2, \frac{a}{b} \cdot c, b\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \cdot -0.5\\ \end{array} \]

      2. expm1-udef 35.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{e^{\mathsf{log1p}\left(c \cdot \frac{-2}{b + \mathsf{fma}\left(-2, \frac{a}{b} \cdot c, b\right)}\right)} - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \cdot -0.5\\ \end{array} \]

      3. *-commutative 35.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;e^{\mathsf{log1p}\left(c \cdot \frac{-2}{b + \mathsf{fma}\left(-2, \color{blue}{c \cdot \frac{a}{b}}, b\right)}\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \cdot -0.5\\ \end{array} \]

   9. Applied egg-rr 35.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{e^{\mathsf{log1p}\left(c \cdot \frac{-2}{b + \mathsf{fma}\left(-2, c \cdot \frac{a}{b}, b\right)}\right)} - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \cdot -0.5\\ \end{array} \]
   10. Step-by-step derivation

       1. expm1-def 79.6%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot \frac{-2}{b + \mathsf{fma}\left(-2, c \cdot \frac{a}{b}, b\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \sqrt{-4 \cdot \left(a \cdot c\right)}}{a} \cdot -0.5\\ \end{array} \]

       2. expm1-log1p 87.0%

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

       3. associate-*r/ 87.3%

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

   11. Simplified 87.3%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+106}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \mathsf{fma}\left(-2, c \cdot \frac{a}{b}, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{b}{a}, -2 \cdot \frac{c}{b}\right) \cdot -0.5\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \mathsf{fma}\left(-2, c \cdot \frac{a}{b}, b\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 8.4 \cdot 10^{-107}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\left(c \cdot a\right) \cdot -4}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + \mathsf{fma}\left(-2, c \cdot \frac{a}{b}, b\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b - \sqrt{\left(c \cdot a\right) \cdot -4}}{a}\\ \end{array} \]

Alternative 6: 80.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = (-0.5d0) * ((b + b) / a)
    t_1 = sqrt(((c * a) * (-4.0d0)))
    if (b <= (-3.4d-99)) then
        tmp = -b / a
    else if (b <= (-2d-310)) then
        if (b >= 0.0d0) then
            tmp_1 = c * ((b / a) / c)
        else
            tmp_1 = (-0.5d0) * ((b - t_1) / a)
        end if
        tmp = tmp_1
    else if (b <= 1d-106) then
        if (b >= 0.0d0) then
            tmp_2 = c * ((-2.0d0) / (b + t_1))
        else
            tmp_2 = t_0
        end if
        tmp = tmp_2
    else if (b >= 0.0d0) then
        tmp = -c / b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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

Python

def code(a, b, c):
	t_0 = -0.5 * ((b + b) / a)
	t_1 = math.sqrt(((c * a) * -4.0))
	tmp = 0
	if b <= -3.4e-99:
		tmp = -b / a
	elif b <= -2e-310:
		tmp_1 = 0
		if b >= 0.0:
			tmp_1 = c * ((b / a) / c)
		else:
			tmp_1 = -0.5 * ((b - t_1) / a)
		tmp = tmp_1
	elif b <= 1e-106:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = c * (-2.0 / (b + t_1))
		else:
			tmp_2 = t_0
		tmp = tmp_2
	elif b >= 0.0:
		tmp = -c / b
	else:
		tmp = t_0
	return tmp

Julia

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

MATLAB

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

Wolfram

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

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 82.4%

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

   5. Taylor expanded in b around -inf 82.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a} \cdot -0.5\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-*r/ 82.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-1 \cdot b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a} \cdot -0.5\\ \end{array} \]

      2. neg-mul-1 82.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{-b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a} \cdot -0.5\\ \end{array} \]

   7. Simplified 82.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a} \cdot -0.5\\ \end{array} \]

   8. Taylor expanded in b around 0 82.4%

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

      1. associate-*r/ 82.4%

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

      2. mul-1-neg 82.4%

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

   10. Simplified 82.4%

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

   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%

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

      1. +-commutative 80.2%

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

      2. fma-def 80.2%

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

      3. associate-/l* 80.2%

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

      4. associate-/r/ 80.2%

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

   6. Simplified 80.2%

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

   7. Taylor expanded in b around 0 83.6%

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

   8. Taylor expanded in b around 0 83.6%

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

      1. associate-/r* 83.6%

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

   10. Simplified 83.6%

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

   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%

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

   5. Taylor expanded in b around 0 70.7%

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

   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 71.7%

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

   5. Taylor expanded in c around 0 87.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a} \cdot -0.5\\ \end{array} \]
   6. Step-by-step derivation

      1. mul-1-neg 87.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-\frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a} \cdot -0.5\\ \end{array} \]

      2. distribute-neg-frac 87.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a} \cdot -0.5\\ \end{array} \]

   7. Simplified 87.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a} \cdot -0.5\\ \end{array} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.2%

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

Alternative 7: 80.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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%

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

      1. +-commutative 67.2%

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

      2. fma-def 67.2%

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

      3. associate-/l* 67.2%

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

      4. associate-/r/ 67.2%

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

   6. Simplified 67.2%

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

   7. Taylor expanded in b around -inf 81.9%

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

      1. +-commutative 81.9%

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

      2. *-commutative 81.9%

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

      3. fma-def 81.9%

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

      4. *-commutative 81.9%

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

      5. associate-/l* 82.9%

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

      6. associate-*l/ 82.9%

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

   9. Simplified 82.9%

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

   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%

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

      1. +-commutative 80.2%

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

      2. fma-def 80.2%

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

      3. associate-/l* 80.2%

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

      4. associate-/r/ 80.2%

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

   6. Simplified 80.2%

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

   7. Taylor expanded in b around 0 83.6%

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

   8. Taylor expanded in b around 0 83.6%

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

      1. associate-/r* 83.6%

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

   10. Simplified 83.6%

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

   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%

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

   5. Taylor expanded in b around 0 70.7%

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

   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 71.7%

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

   5. Taylor expanded in c around 0 87.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a} \cdot -0.5\\ \end{array} \]
   6. Step-by-step derivation

      1. mul-1-neg 87.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-\frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a} \cdot -0.5\\ \end{array} \]

      2. distribute-neg-frac 87.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a} \cdot -0.5\\ \end{array} \]

   7. Simplified 87.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a} \cdot -0.5\\ \end{array} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.4%

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

Alternative 8: 80.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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%

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

      1. +-commutative 67.2%

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

      2. fma-def 67.2%

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

      3. associate-/l* 67.2%

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

      4. associate-/r/ 67.2%

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

   6. Simplified 67.2%

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

   7. Taylor expanded in b around -inf 83.0%

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

      1. +-commutative 83.0%

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

      2. fma-def 83.0%

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

   9. Simplified 83.0%

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

   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%

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

      1. +-commutative 80.2%

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

      2. fma-def 80.2%

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

      3. associate-/l* 80.2%

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

      4. associate-/r/ 80.2%

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

   6. Simplified 80.2%

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

   7. Taylor expanded in b around 0 83.6%

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

   8. Taylor expanded in b around 0 83.6%

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

      1. associate-/r* 83.6%

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

   10. Simplified 83.6%

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

   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%

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

   5. Taylor expanded in b around 0 70.7%

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

   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 71.7%

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

   5. Taylor expanded in c around 0 87.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a} \cdot -0.5\\ \end{array} \]
   6. Step-by-step derivation

      1. mul-1-neg 87.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-\frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a} \cdot -0.5\\ \end{array} \]

      2. distribute-neg-frac 87.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a} \cdot -0.5\\ \end{array} \]

   7. Simplified 87.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a} \cdot -0.5\\ \end{array} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.4%

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

Alternative 9: 74.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = (-0.5d0) * ((b + b) / a)
    if (b <= 1d-106) then
        if (b >= 0.0d0) then
            tmp_2 = c * ((-2.0d0) / (b + sqrt(((c * a) * (-4.0d0)))))
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = -c / b
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -0.5 * ((b + b) / a);
	double tmp_1;
	if (b <= 1e-106) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / (b + Math.sqrt(((c * a) * -4.0))));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = -c / b;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = -0.5 * ((b + b) / a)
	tmp_1 = 0
	if b <= 1e-106:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = c * (-2.0 / (b + math.sqrt(((c * a) * -4.0))))
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = -c / b
	else:
		tmp_1 = t_0
	return tmp_1

Julia

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

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = -0.5 * ((b + b) / a);
	tmp_2 = 0.0;
	if (b <= 1e-106)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = c * (-2.0 / (b + sqrt(((c * a) * -4.0))));
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = -c / b;
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < 9.99999999999999941e-107

   1. Initial program 71.4%

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

   3. Simplified 71.4%

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

   4. Taylor expanded in b around -inf 68.1%

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

   5. Taylor expanded in b around 0 67.0%

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

   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 71.7%

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

   5. Taylor expanded in c around 0 87.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a} \cdot -0.5\\ \end{array} \]
   6. Step-by-step derivation

      1. mul-1-neg 87.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-\frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a} \cdot -0.5\\ \end{array} \]

      2. distribute-neg-frac 87.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a} \cdot -0.5\\ \end{array} \]

   7. Simplified 87.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a} \cdot -0.5\\ \end{array} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.0%

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

Alternative 10: 67.6% accurate, 13.0× speedup

Math

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

FPCore

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

C

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

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = -c / 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) / 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 / 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) / b), $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 69.6%

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

5. Taylor expanded in c around 0 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a} \cdot -0.5\\ \end{array} \]
6. Step-by-step derivation

   1. mul-1-neg 69.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-\frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a} \cdot -0.5\\ \end{array} \]

   2. distribute-neg-frac 69.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a} \cdot -0.5\\ \end{array} \]

7. Simplified 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a} \cdot -0.5\\ \end{array} \]

8. Final simplification 69.4%

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

Alternative 11: 35.3% accurate, 29.5× speedup

Math

\[\begin{array}{l} \\ \frac{-b}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ (- b) a))

C

double code(double a, double b, double c) {
	return -b / a;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = -b / a
end function

Java

public static double code(double a, double b, double c) {
	return -b / a;
}

Python

def code(a, b, c):
	return -b / a

Julia

function code(a, b, c)
	return Float64(Float64(-b) / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = -b / a;
end

Wolfram

code[a_, b_, c_] := N[((-b) / a), $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 69.6%

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

5. Taylor expanded in b around -inf 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a} \cdot -0.5\\ \end{array} \]
6. Step-by-step derivation

   1. associate-*r/ 34.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-1 \cdot b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a} \cdot -0.5\\ \end{array} \]

   2. neg-mul-1 34.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{-b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a} \cdot -0.5\\ \end{array} \]

7. Simplified 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a} \cdot -0.5\\ \end{array} \]

8. Taylor expanded in b around 0 34.3%

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

   1. associate-*r/ 34.3%

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

   2. mul-1-neg 34.3%

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

10. Simplified 34.3%

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

11. Final simplification 34.3%

    \[\leadsto \frac{-b}{a} \]

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))))