jeff quadratic root 2

Percentage Accurate: 71.6% → 90.3%

Time: 16.1s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 71.6% 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: 90.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot b - c \cdot \left(4 \cdot a\right)\\ t_1 := \frac{2 \cdot c}{\left(-b\right) - \sqrt{t_0}}\\ \mathbf{if}\;b \leq -2.8 \cdot 10^{+157}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, a \cdot \frac{c}{b}, b \cdot -2\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{+96}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{{t_0}^{0.5} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c \cdot 0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (* b b) (* c (* 4.0 a))))
        (t_1 (/ (* 2.0 c) (- (- b) (sqrt t_0)))))
   (if (<= b -2.8e+157)
     (if (>= b 0.0) t_1 (/ (fma 2.0 (* a (/ c b)) (* b -2.0)) (* 2.0 a)))
     (if (<= b 5.5e+96)
       (if (>= b 0.0) t_1 (/ (- (pow t_0 0.5) b) (* 2.0 a)))
       (if (>= b 0.0) (* -2.0 (/ (* c 0.5) b)) (* (/ -0.5 a) (+ b b)))))))

C

double code(double a, double b, double c) {
	double t_0 = (b * b) - (c * (4.0 * a));
	double t_1 = (2.0 * c) / (-b - sqrt(t_0));
	double tmp_1;
	if (b <= -2.8e+157) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = fma(2.0, (a * (c / b)), (b * -2.0)) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 5.5e+96) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = (pow(t_0, 0.5) - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -2.0 * ((c * 0.5) / b);
	} else {
		tmp_1 = (-0.5 / a) * (b + b);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))
	t_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - sqrt(t_0)))
	tmp_1 = 0.0
	if (b <= -2.8e+157)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(fma(2.0, Float64(a * Float64(c / b)), Float64(b * -2.0)) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 5.5e+96)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = Float64(Float64((t_0 ^ 0.5) - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(-2.0 * Float64(Float64(c * 0.5) / b));
	else
		tmp_1 = Float64(Float64(-0.5 / a) * Float64(b + b));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.8e+157], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(2.0 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(b * -2.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 5.5e+96], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(N[Power[t$95$0, 0.5], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(N[(c * 0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + b), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.8000000000000003e157

   1. Initial program 37.9%

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

      1. pow1/2 37.9%

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

      2. *-commutative 37.9%

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

      3. *-commutative 37.9%

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

   3. Applied egg-rr 37.9%

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

   4. Taylor expanded in b around -inf 88.7%

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

      1. fma-def 88.7%

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

      2. associate-*l/ 96.9%

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

      3. *-commutative 96.9%

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

      4. *-commutative 96.9%

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

   6. Simplified 96.9%

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

   if -2.8000000000000003e157 < b < 5.5000000000000002e96

   1. Initial program 85.4%

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

      1. pow1/2 85.4%

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

      2. *-commutative 85.4%

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

      3. *-commutative 85.4%

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

   3. Applied egg-rr 85.4%

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

   if 5.5000000000000002e96 < b

   1. Initial program 56.5%

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

   3. Simplified 56.5%

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

   4. Taylor expanded in b around inf 88.8%

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

      1. associate-/l* 95.4%

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

   6. Simplified 95.4%

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

   7. Taylor expanded in b around -inf 95.4%

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

   8. Taylor expanded in c around 0 95.5%

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

      1. *-commutative 95.5%

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

      2. associate-*l/ 95.5%

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

   10. Simplified 95.5%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+157}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, a \cdot \frac{c}{b}, b \cdot -2\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{+96}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(b \cdot b - c \cdot \left(4 \cdot a\right)\right)}^{0.5} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c \cdot 0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + b\right)\\ \end{array} \]

Alternative 2: 90.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\\ \mathbf{if}\;b \leq -2.8 \cdot 10^{+157}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \left(b + -2 \cdot \frac{c}{\frac{b}{a}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+97}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c \cdot 0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* 4.0 a))))))
   (if (<= b -2.8e+157)
     (if (>= b 0.0)
       (* -2.0 (/ c (+ b (+ b (* -2.0 (/ c (/ b a)))))))
       (- (/ c b) (/ b a)))
     (if (<= b 4.4e+97)
       (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_0)) (/ (- t_0 b) (* 2.0 a)))
       (if (>= b 0.0) (* -2.0 (/ (* c 0.5) b)) (* (/ -0.5 a) (+ b b)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (4.0 * a))));
	double tmp_1;
	if (b <= -2.8e+157) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -2.0 * (c / (b + (b + (-2.0 * (c / (b / a))))));
		} else {
			tmp_2 = (c / b) - (b / a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 4.4e+97) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_3 = (t_0 - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -2.0 * ((c * 0.5) / b);
	} else {
		tmp_1 = (-0.5 / a) * (b + b);
	}
	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
    real(8) :: tmp_3
    t_0 = sqrt(((b * b) - (c * (4.0d0 * a))))
    if (b <= (-2.8d+157)) then
        if (b >= 0.0d0) then
            tmp_2 = (-2.0d0) * (c / (b + (b + ((-2.0d0) * (c / (b / a))))))
        else
            tmp_2 = (c / b) - (b / a)
        end if
        tmp_1 = tmp_2
    else if (b <= 4.4d+97) then
        if (b >= 0.0d0) then
            tmp_3 = (2.0d0 * c) / (-b - t_0)
        else
            tmp_3 = (t_0 - b) / (2.0d0 * a)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = (-2.0d0) * ((c * 0.5d0) / b)
    else
        tmp_1 = ((-0.5d0) / a) * (b + b)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (c * (4.0 * a))));
	double tmp_1;
	if (b <= -2.8e+157) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -2.0 * (c / (b + (b + (-2.0 * (c / (b / a))))));
		} else {
			tmp_2 = (c / b) - (b / a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 4.4e+97) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_3 = (t_0 - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -2.0 * ((c * 0.5) / b);
	} else {
		tmp_1 = (-0.5 / a) * (b + b);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (c * (4.0 * a))))
	tmp_1 = 0
	if b <= -2.8e+157:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -2.0 * (c / (b + (b + (-2.0 * (c / (b / a))))))
		else:
			tmp_2 = (c / b) - (b / a)
		tmp_1 = tmp_2
	elif b <= 4.4e+97:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (2.0 * c) / (-b - t_0)
		else:
			tmp_3 = (t_0 - b) / (2.0 * a)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = -2.0 * ((c * 0.5) / b)
	else:
		tmp_1 = (-0.5 / a) * (b + b)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a))))
	tmp_1 = 0.0
	if (b <= -2.8e+157)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-2.0 * Float64(c / Float64(b + Float64(b + Float64(-2.0 * Float64(c / Float64(b / a)))))));
		else
			tmp_2 = Float64(Float64(c / b) - Float64(b / a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 4.4e+97)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0));
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(-2.0 * Float64(Float64(c * 0.5) / b));
	else
		tmp_1 = Float64(Float64(-0.5 / a) * Float64(b + b));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) - (c * (4.0 * a))));
	tmp_2 = 0.0;
	if (b <= -2.8e+157)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -2.0 * (c / (b + (b + (-2.0 * (c / (b / a))))));
		else
			tmp_3 = (c / b) - (b / a);
		end
		tmp_2 = tmp_3;
	elseif (b <= 4.4e+97)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (2.0 * c) / (-b - t_0);
		else
			tmp_4 = (t_0 - b) / (2.0 * a);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = -2.0 * ((c * 0.5) / b);
	else
		tmp_2 = (-0.5 / a) * (b + b);
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2.8e+157], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(c / N[(b + N[(b + N[(-2.0 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 4.4e+97], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(N[(c * 0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.8000000000000003e157

   1. Initial program 37.9%

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

   3. Simplified 37.9%

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

   4. Taylor expanded in b around inf 37.9%

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

      1. associate-/l* 37.9%

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

   6. Simplified 37.9%

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

   7. Taylor expanded in b around -inf 96.9%

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

      1. mul-1-neg 96.9%

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

      2. unsub-neg 96.9%

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

   9. Simplified 96.9%

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

   if -2.8000000000000003e157 < b < 4.4000000000000002e97

   1. Initial program 85.4%

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

   if 4.4000000000000002e97 < b

   1. Initial program 56.5%

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

   3. Simplified 56.5%

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

   4. Taylor expanded in b around inf 88.8%

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

      1. associate-/l* 95.4%

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

   6. Simplified 95.4%

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

   7. Taylor expanded in b around -inf 95.4%

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

   8. Taylor expanded in c around 0 95.5%

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

      1. *-commutative 95.5%

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

      2. associate-*l/ 95.5%

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

   10. Simplified 95.5%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+157}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \left(b + -2 \cdot \frac{c}{\frac{b}{a}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+97}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c \cdot 0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + b\right)\\ \end{array} \]

Alternative 3: 90.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\\ t_1 := \frac{2 \cdot c}{\left(-b\right) - t_0}\\ \mathbf{if}\;b \leq -2.8 \cdot 10^{+157}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, a \cdot \frac{c}{b}, b \cdot -2\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{+94}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c \cdot 0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* 4.0 a)))))
        (t_1 (/ (* 2.0 c) (- (- b) t_0))))
   (if (<= b -2.8e+157)
     (if (>= b 0.0) t_1 (/ (fma 2.0 (* a (/ c b)) (* b -2.0)) (* 2.0 a)))
     (if (<= b 1e+94)
       (if (>= b 0.0) t_1 (/ (- t_0 b) (* 2.0 a)))
       (if (>= b 0.0) (* -2.0 (/ (* c 0.5) b)) (* (/ -0.5 a) (+ b b)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (4.0 * a))));
	double t_1 = (2.0 * c) / (-b - t_0);
	double tmp_1;
	if (b <= -2.8e+157) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = fma(2.0, (a * (c / b)), (b * -2.0)) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1e+94) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = (t_0 - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -2.0 * ((c * 0.5) / b);
	} else {
		tmp_1 = (-0.5 / a) * (b + b);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a))))
	t_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0))
	tmp_1 = 0.0
	if (b <= -2.8e+157)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(fma(2.0, Float64(a * Float64(c / b)), Float64(b * -2.0)) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1e+94)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(-2.0 * Float64(Float64(c * 0.5) / b));
	else
		tmp_1 = Float64(Float64(-0.5 / a) * Float64(b + b));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.8e+157], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(2.0 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(b * -2.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1e+94], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(t$95$0 - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(N[(c * 0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + b), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.8000000000000003e157

   1. Initial program 37.9%

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

      1. pow1/2 37.9%

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

      2. *-commutative 37.9%

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

      3. *-commutative 37.9%

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

   3. Applied egg-rr 37.9%

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

   4. Taylor expanded in b around -inf 88.7%

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

      1. fma-def 88.7%

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

      2. associate-*l/ 96.9%

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

      3. *-commutative 96.9%

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

      4. *-commutative 96.9%

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

   6. Simplified 96.9%

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

   if -2.8000000000000003e157 < b < 1e94

   1. Initial program 85.4%

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

   if 1e94 < b

   1. Initial program 56.5%

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

   3. Simplified 56.5%

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

   4. Taylor expanded in b around inf 88.8%

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

      1. associate-/l* 95.4%

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

   6. Simplified 95.4%

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

   7. Taylor expanded in b around -inf 95.4%

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

   8. Taylor expanded in c around 0 95.5%

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

      1. *-commutative 95.5%

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

      2. associate-*l/ 95.5%

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

   10. Simplified 95.5%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+157}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, a \cdot \frac{c}{b}, b \cdot -2\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{+94}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c \cdot 0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + b\right)\\ \end{array} \]

Alternative 4: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+157}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \left(b + -2 \cdot \frac{c}{\frac{b}{a}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \left(b + -2 \cdot \left(c \cdot \frac{a}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.8e+157)
   (if (>= b 0.0)
     (* -2.0 (/ c (+ b (+ b (* -2.0 (/ c (/ b a)))))))
     (- (/ c b) (/ b a)))
   (if (<= b -4e-310)
     (if (>= b 0.0)
       (- (- (/ c b)) (/ (* c c) (/ (pow b 3.0) a)))
       (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* 2.0 a)))
     (if (>= b 0.0)
       (* -2.0 (/ c (+ b (+ b (* -2.0 (* c (/ a b)))))))
       (* (/ -0.5 a) (+ b b))))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -2.8e+157) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -2.0 * (c / (b + (b + (-2.0 * (c / (b / a))))));
		} else {
			tmp_2 = (c / b) - (b / a);
		}
		tmp_1 = tmp_2;
	} else if (b <= -4e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -(c / b) - ((c * c) / (pow(b, 3.0) / a));
		} else {
			tmp_3 = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -2.0 * (c / (b + (b + (-2.0 * (c * (a / b))))));
	} else {
		tmp_1 = (-0.5 / a) * (b + b);
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    if (b <= (-2.8d+157)) then
        if (b >= 0.0d0) then
            tmp_2 = (-2.0d0) * (c / (b + (b + ((-2.0d0) * (c / (b / a))))))
        else
            tmp_2 = (c / b) - (b / a)
        end if
        tmp_1 = tmp_2
    else if (b <= (-4d-310)) then
        if (b >= 0.0d0) then
            tmp_3 = -(c / b) - ((c * c) / ((b ** 3.0d0) / a))
        else
            tmp_3 = (sqrt(((b * b) - (c * (4.0d0 * a)))) - b) / (2.0d0 * a)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = (-2.0d0) * (c / (b + (b + ((-2.0d0) * (c * (a / b))))))
    else
        tmp_1 = ((-0.5d0) / a) * (b + b)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -2.8e+157) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -2.0 * (c / (b + (b + (-2.0 * (c / (b / a))))));
		} else {
			tmp_2 = (c / b) - (b / a);
		}
		tmp_1 = tmp_2;
	} else if (b <= -4e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -(c / b) - ((c * c) / (Math.pow(b, 3.0) / a));
		} else {
			tmp_3 = (Math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -2.0 * (c / (b + (b + (-2.0 * (c * (a / b))))));
	} else {
		tmp_1 = (-0.5 / a) * (b + b);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if b <= -2.8e+157:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -2.0 * (c / (b + (b + (-2.0 * (c / (b / a))))))
		else:
			tmp_2 = (c / b) - (b / a)
		tmp_1 = tmp_2
	elif b <= -4e-310:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = -(c / b) - ((c * c) / (math.pow(b, 3.0) / a))
		else:
			tmp_3 = (math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (2.0 * a)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = -2.0 * (c / (b + (b + (-2.0 * (c * (a / b))))))
	else:
		tmp_1 = (-0.5 / a) * (b + b)
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -2.8e+157)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-2.0 * Float64(c / Float64(b + Float64(b + Float64(-2.0 * Float64(c / Float64(b / a)))))));
		else
			tmp_2 = Float64(Float64(c / b) - Float64(b / a));
		end
		tmp_1 = tmp_2;
	elseif (b <= -4e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-Float64(c / b)) - Float64(Float64(c * c) / Float64((b ^ 3.0) / a)));
		else
			tmp_3 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(-2.0 * Float64(c / Float64(b + Float64(b + Float64(-2.0 * Float64(c * Float64(a / b)))))));
	else
		tmp_1 = Float64(Float64(-0.5 / a) * Float64(b + b));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	tmp_2 = 0.0;
	if (b <= -2.8e+157)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -2.0 * (c / (b + (b + (-2.0 * (c / (b / a))))));
		else
			tmp_3 = (c / b) - (b / a);
		end
		tmp_2 = tmp_3;
	elseif (b <= -4e-310)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = -(c / b) - ((c * c) / ((b ^ 3.0) / a));
		else
			tmp_4 = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (2.0 * a);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = -2.0 * (c / (b + (b + (-2.0 * (c * (a / b))))));
	else
		tmp_2 = (-0.5 / a) * (b + b);
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.8e+157], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(c / N[(b + N[(b + N[(-2.0 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -4e-310], If[GreaterEqual[b, 0.0], N[((-N[(c / b), $MachinePrecision]) - N[(N[(c * c), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(c / N[(b + N[(b + N[(-2.0 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.8000000000000003e157

   1. Initial program 37.9%

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

   3. Simplified 37.9%

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

   4. Taylor expanded in b around inf 37.9%

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

      1. associate-/l* 37.9%

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

   6. Simplified 37.9%

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

   7. Taylor expanded in b around -inf 96.9%

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

      1. mul-1-neg 96.9%

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

      2. unsub-neg 96.9%

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

   9. Simplified 96.9%

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

   if -2.8000000000000003e157 < b < -3.999999999999988e-310

   1. Initial program 89.0%

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

   2. Taylor expanded in c around 0 89.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. distribute-lft-out 89.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \left(\frac{{c}^{2} \cdot a}{{b}^{3}} + \frac{c}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. associate-/l* 89.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \left(\color{blue}{\frac{{c}^{2}}{\frac{{b}^{3}}{a}}} + \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      3. unpow2 89.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \left(\frac{\color{blue}{c \cdot c}}{\frac{{b}^{3}}{a}} + \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Simplified 89.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \left(\frac{c \cdot c}{\frac{{b}^{3}}{a}} + \frac{c}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   if -3.999999999999988e-310 < b

   1. Initial program 68.2%

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

   3. Simplified 68.2%

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

   4. Taylor expanded in b around inf 65.5%

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

      1. associate-/l* 68.8%

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

   6. Simplified 68.8%

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

   7. Taylor expanded in b around -inf 68.8%

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

      1. div-inv 68.8%

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

   9. Applied egg-rr 68.8%

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

   10. Taylor expanded in b around 0 68.8%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+157}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \left(b + -2 \cdot \frac{c}{\frac{b}{a}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-\frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \left(b + -2 \cdot \left(c \cdot \frac{a}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + b\right)\\ \end{array} \]

Alternative 5: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\\ \mathbf{if}\;b \leq -2.8 \cdot 10^{+157}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \left(b + -2 \cdot \frac{c}{\frac{b}{a}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-103}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \left(b + -2 \cdot \left(c \cdot \frac{a}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* 4.0 a))))))
   (if (<= b -2.8e+157)
     (if (>= b 0.0)
       (* -2.0 (/ c (+ b (+ b (* -2.0 (/ c (/ b a)))))))
       (- (/ c b) (/ b a)))
     (if (<= b 9.2e-103)
       (if (>= b 0.0) (* c (/ 2.0 (- b t_0))) (/ (- t_0 b) (* 2.0 a)))
       (if (>= b 0.0)
         (* -2.0 (/ c (+ b (+ b (* -2.0 (* c (/ a b)))))))
         (* (/ -0.5 a) (+ b b)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (4.0 * a))));
	double tmp_1;
	if (b <= -2.8e+157) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -2.0 * (c / (b + (b + (-2.0 * (c / (b / a))))));
		} else {
			tmp_2 = (c / b) - (b / a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 9.2e-103) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = c * (2.0 / (b - t_0));
		} else {
			tmp_3 = (t_0 - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -2.0 * (c / (b + (b + (-2.0 * (c * (a / b))))));
	} else {
		tmp_1 = (-0.5 / a) * (b + b);
	}
	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
    real(8) :: tmp_3
    t_0 = sqrt(((b * b) - (c * (4.0d0 * a))))
    if (b <= (-2.8d+157)) then
        if (b >= 0.0d0) then
            tmp_2 = (-2.0d0) * (c / (b + (b + ((-2.0d0) * (c / (b / a))))))
        else
            tmp_2 = (c / b) - (b / a)
        end if
        tmp_1 = tmp_2
    else if (b <= 9.2d-103) then
        if (b >= 0.0d0) then
            tmp_3 = c * (2.0d0 / (b - t_0))
        else
            tmp_3 = (t_0 - b) / (2.0d0 * a)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = (-2.0d0) * (c / (b + (b + ((-2.0d0) * (c * (a / b))))))
    else
        tmp_1 = ((-0.5d0) / a) * (b + b)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (c * (4.0 * a))));
	double tmp_1;
	if (b <= -2.8e+157) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -2.0 * (c / (b + (b + (-2.0 * (c / (b / a))))));
		} else {
			tmp_2 = (c / b) - (b / a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 9.2e-103) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = c * (2.0 / (b - t_0));
		} else {
			tmp_3 = (t_0 - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -2.0 * (c / (b + (b + (-2.0 * (c * (a / b))))));
	} else {
		tmp_1 = (-0.5 / a) * (b + b);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (c * (4.0 * a))))
	tmp_1 = 0
	if b <= -2.8e+157:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -2.0 * (c / (b + (b + (-2.0 * (c / (b / a))))))
		else:
			tmp_2 = (c / b) - (b / a)
		tmp_1 = tmp_2
	elif b <= 9.2e-103:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = c * (2.0 / (b - t_0))
		else:
			tmp_3 = (t_0 - b) / (2.0 * a)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = -2.0 * (c / (b + (b + (-2.0 * (c * (a / b))))))
	else:
		tmp_1 = (-0.5 / a) * (b + b)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a))))
	tmp_1 = 0.0
	if (b <= -2.8e+157)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-2.0 * Float64(c / Float64(b + Float64(b + Float64(-2.0 * Float64(c / Float64(b / a)))))));
		else
			tmp_2 = Float64(Float64(c / b) - Float64(b / a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 9.2e-103)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(c * Float64(2.0 / Float64(b - t_0)));
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(-2.0 * Float64(c / Float64(b + Float64(b + Float64(-2.0 * Float64(c * Float64(a / b)))))));
	else
		tmp_1 = Float64(Float64(-0.5 / a) * Float64(b + b));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) - (c * (4.0 * a))));
	tmp_2 = 0.0;
	if (b <= -2.8e+157)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -2.0 * (c / (b + (b + (-2.0 * (c / (b / a))))));
		else
			tmp_3 = (c / b) - (b / a);
		end
		tmp_2 = tmp_3;
	elseif (b <= 9.2e-103)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = c * (2.0 / (b - t_0));
		else
			tmp_4 = (t_0 - b) / (2.0 * a);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = -2.0 * (c / (b + (b + (-2.0 * (c * (a / b))))));
	else
		tmp_2 = (-0.5 / a) * (b + b);
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2.8e+157], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(c / N[(b + N[(b + N[(-2.0 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 9.2e-103], If[GreaterEqual[b, 0.0], N[(c * N[(2.0 / N[(b - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(c / N[(b + N[(b + N[(-2.0 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.8000000000000003e157

   1. Initial program 37.9%

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

   3. Simplified 37.9%

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

   4. Taylor expanded in b around inf 37.9%

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

      1. associate-/l* 37.9%

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

   6. Simplified 37.9%

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

   7. Taylor expanded in b around -inf 96.9%

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

      1. mul-1-neg 96.9%

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

      2. unsub-neg 96.9%

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

   9. Simplified 96.9%

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

   if -2.8000000000000003e157 < b < 9.2000000000000003e-103

   1. Initial program 86.1%

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

      1. expm1-log1p-u 83.3%

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

      2. expm1-udef 73.7%

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

      3. *-commutative 73.7%

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

      4. add-sqr-sqrt 69.6%

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

      5. sqrt-unprod 73.1%

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

      6. sqr-neg 73.1%

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

      7. sqrt-prod 73.2%

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

      8. add-sqr-sqrt 73.2%

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

      9. *-commutative 73.2%

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

      10. *-commutative 73.2%

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

   3. Applied egg-rr 73.2%

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

      1. expm1-def 81.2%

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

      2. expm1-log1p 84.0%

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

      3. associate-*r/ 84.0%

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

   5. Simplified 84.0%

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

   if 9.2000000000000003e-103 < b

   1. Initial program 66.1%

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

   3. Simplified 66.1%

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

   4. Taylor expanded in b around inf 79.3%

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

      1. associate-/l* 83.6%

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

   6. Simplified 83.6%

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

   7. Taylor expanded in b around -inf 83.6%

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

      1. div-inv 83.6%

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

   9. Applied egg-rr 83.6%

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

   10. Taylor expanded in b around 0 83.6%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+157}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \left(b + -2 \cdot \frac{c}{\frac{b}{a}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-103}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \left(b + -2 \cdot \left(c \cdot \frac{a}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + b\right)\\ \end{array} \]

Alternative 6: 68.9% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.7%

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

3. Simplified 70.6%

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

4. Taylor expanded in b around inf 69.4%

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

   1. associate-/l* 70.9%

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

6. Simplified 70.9%

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

7. Taylor expanded in b around -inf 68.2%

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

   1. div-inv 68.2%

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

9. Applied egg-rr 68.2%

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

10. Taylor expanded in b around 0 68.2%

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

11. Final simplification 68.2%

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

Alternative 7: 69.2% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.7%

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

3. Simplified 70.6%

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

4. Taylor expanded in b around inf 69.4%

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

   1. associate-/l* 70.9%

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

6. Simplified 70.9%

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

7. Taylor expanded in b around -inf 68.5%

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

   1. mul-1-neg 68.5%

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

   2. unsub-neg 68.5%

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

9. Simplified 68.5%

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

10. Final simplification 68.5%

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

Alternative 8: 36.1% accurate, 13.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.7%

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

3. Simplified 70.6%

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

4. Taylor expanded in b around inf 69.4%

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

   1. associate-/l* 70.9%

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

6. Simplified 70.9%

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

7. Taylor expanded in b around -inf 68.2%

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

8. Taylor expanded in c around inf 37.8%

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

   1. associate-*r/ 37.9%

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

   2. cancel-sign-sub-inv 37.9%

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

   3. metadata-eval 37.9%

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

   4. *-un-lft-identity 37.9%

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

10. Applied egg-rr 37.9%

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

11. Final simplification 37.9%

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

Alternative 9: 68.8% accurate, 13.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.7%

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

3. Simplified 70.6%

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

4. Taylor expanded in b around inf 69.4%

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

   1. associate-/l* 70.9%

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

6. Simplified 70.9%

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

7. Taylor expanded in b around -inf 68.2%

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

8. Taylor expanded in c around 0 68.0%

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

   1. *-commutative 68.0%

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

   2. associate-*l/ 68.0%

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

10. Simplified 68.0%

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

11. Final simplification 68.0%

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

Reproduce

herbie shell --seed 2023199 
(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))))