jeff quadratic root 1

Percentage Accurate: 72.4% → 90.5%

Time: 33.0s

Alternatives: 6

Speedup: 1.0×

Specification

Math

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

FPCore

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

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 = (-b - t_0) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + t_0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - ((4.0d0 * a) * c)))
    if (b >= 0.0d0) then
        tmp = (-b - t_0) / (2.0d0 * a)
    else
        tmp = (2.0d0 * c) / (-b + t_0)
    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 = (-b - t_0) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + t_0);
	}
	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 = (-b - t_0) / (2.0 * a)
	else:
		tmp = (2.0 * c) / (-b + t_0)
	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(Float64(-b) - t_0) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_0));
	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 = (-b - t_0) / (2.0 * a);
	else
		tmp = (2.0 * c) / (-b + t_0);
	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[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Initial Program: 72.4% accurate, 1.0× speedup

Math

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

FPCore

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

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 = (-b - t_0) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + t_0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - ((4.0d0 * a) * c)))
    if (b >= 0.0d0) then
        tmp = (-b - t_0) / (2.0d0 * a)
    else
        tmp = (2.0d0 * c) / (-b + t_0)
    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 = (-b - t_0) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + t_0);
	}
	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 = (-b - t_0) / (2.0 * a)
	else:
		tmp = (2.0 * c) / (-b + t_0)
	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(Float64(-b) - t_0) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_0));
	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 = (-b - t_0) / (2.0 * a);
	else
		tmp = (2.0 * c) / (-b + t_0);
	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[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 90.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ \mathbf{if}\;b \leq -4.5 \cdot 10^{+149}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{b \cdot \left(-1 - -2 \cdot \left(a \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot {b}^{-2}\right)\right)\right)\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.42 \cdot 10^{+38}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0))))))
   (if (<= b -4.5e+149)
     (if (>= b 0.0)
       (* (/ -0.5 a) (+ b (sqrt (fma c (* a -4.0) (* b b)))))
       (*
        c
        (/
         2.0
         (-
          (* b (- -1.0 (* -2.0 (* a (expm1 (log1p (* c (pow b -2.0))))))))
          b))))
     (if (<= b 1.42e+38)
       (if (>= b 0.0) (/ (- (- b) t_0) (* a 2.0)) (/ (* c 2.0) (- t_0 b)))
       (if (>= b 0.0) (- (/ c b) (/ b a)) (/ (* c 2.0) (* b -2.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -4.5e+149) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-0.5 / a) * (b + sqrt(fma(c, (a * -4.0), (b * b))));
		} else {
			tmp_2 = c * (2.0 / ((b * (-1.0 - (-2.0 * (a * expm1(log1p((c * pow(b, -2.0)))))))) - b));
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.42e+38) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_0) / (a * 2.0);
		} else {
			tmp_3 = (c * 2.0) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = (c * 2.0) / (b * -2.0);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	tmp_1 = 0.0
	if (b <= -4.5e+149)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-0.5 / a) * Float64(b + sqrt(fma(c, Float64(a * -4.0), Float64(b * b)))));
		else
			tmp_2 = Float64(c * Float64(2.0 / Float64(Float64(b * Float64(-1.0 - Float64(-2.0 * Float64(a * expm1(log1p(Float64(c * (b ^ -2.0)))))))) - b)));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.42e+38)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(-b) - t_0) / Float64(a * 2.0));
		else
			tmp_3 = Float64(Float64(c * 2.0) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp_1 = Float64(Float64(c * 2.0) / Float64(b * -2.0));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -4.5e+149], If[GreaterEqual[b, 0.0], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(2.0 / N[(N[(b * N[(-1.0 - N[(-2.0 * N[(a * N[(Exp[N[Log[1 + N[(c * N[Power[b, -2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.42e+38], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(b * -2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.49999999999999982e149

   1. Initial program 24.9%

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

   3. Simplified 25.1%

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

   5. Taylor expanded in b around -inf 89.2%

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

      1. mul-1-neg 89.2%

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

      2. *-commutative 89.2%

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

      3. distribute-rgt-neg-in 89.2%

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

      4. associate-/l* 94.3%

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

   7. Simplified 94.3%

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

      1. expm1-log1p-u 94.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\left(1 + -2 \cdot \left(a \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c}{{b}^{2}}\right)\right)\right)\right) \cdot \left(-b\right) - b}\\ \end{array} \]

      2. expm1-undefine 94.3%

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

      3. div-inv 94.3%

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

      4. pow-flip 94.3%

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

      5. metadata-eval 94.3%

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

   9. Applied egg-rr 94.3%

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

       1. expm1-define 95.2%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\left(1 + -2 \cdot \left(a \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot {b}^{-2}\right)\right)\right)\right) \cdot \left(-b\right) - b}\\ \end{array} \]

   11. Simplified 95.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\left(1 + -2 \cdot \left(a \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot {b}^{-2}\right)\right)\right)\right) \cdot \left(-b\right) - b}\\ \end{array} \]

   if -4.49999999999999982e149 < b < 1.4200000000000001e38

   1. Initial program 83.2%

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

   if 1.4200000000000001e38 < b

   1. Initial program 64.0%

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

   3. Taylor expanded in a around 0 90.1%

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

      1. distribute-lft-out-- 90.1%

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

      2. associate-/l* 96.4%

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

      3. fmm-def 96.4%

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

   5. Simplified 96.4%

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

   6. Taylor expanded in b around -inf 96.4%

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

      1. *-commutative 96.4%

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

   8. Simplified 96.4%

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

   9. Taylor expanded in a around inf 96.5%

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

       1. +-commutative 96.5%

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

       2. mul-1-neg 96.5%

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

       3. unsub-neg 96.5%

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

   11. Simplified 96.5%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+149}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{b \cdot \left(-1 - -2 \cdot \left(a \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot {b}^{-2}\right)\right)\right)\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.42 \cdot 10^{+38}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ \mathbf{if}\;b \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + b\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{b \cdot \left(-1 + -2 \cdot \left(a \cdot \left(\frac{c}{b} \cdot \frac{-1}{b}\right)\right)\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.42 \cdot 10^{+38}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0))))))
   (if (<= b -1e+154)
     (if (>= b 0.0)
       (* (/ -0.5 a) (+ b b))
       (*
        c
        (/ 2.0 (- (* b (+ -1.0 (* -2.0 (* a (* (/ c b) (/ -1.0 b)))))) b))))
     (if (<= b 1.42e+38)
       (if (>= b 0.0) (/ (- (- b) t_0) (* a 2.0)) (/ (* c 2.0) (- t_0 b)))
       (if (>= b 0.0) (- (/ c b) (/ b a)) (/ (* c 2.0) (* b -2.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -1e+154) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-0.5 / a) * (b + b);
		} else {
			tmp_2 = c * (2.0 / ((b * (-1.0 + (-2.0 * (a * ((c / b) * (-1.0 / b)))))) - b));
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.42e+38) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_0) / (a * 2.0);
		} else {
			tmp_3 = (c * 2.0) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = (c * 2.0) / (b * -2.0);
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = sqrt(((b * b) - (c * (a * 4.0d0))))
    if (b <= (-1d+154)) then
        if (b >= 0.0d0) then
            tmp_2 = ((-0.5d0) / a) * (b + b)
        else
            tmp_2 = c * (2.0d0 / ((b * ((-1.0d0) + ((-2.0d0) * (a * ((c / b) * ((-1.0d0) / b)))))) - b))
        end if
        tmp_1 = tmp_2
    else if (b <= 1.42d+38) then
        if (b >= 0.0d0) then
            tmp_3 = (-b - t_0) / (a * 2.0d0)
        else
            tmp_3 = (c * 2.0d0) / (t_0 - b)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = (c / b) - (b / a)
    else
        tmp_1 = (c * 2.0d0) / (b * (-2.0d0))
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -1e+154) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-0.5 / a) * (b + b);
		} else {
			tmp_2 = c * (2.0 / ((b * (-1.0 + (-2.0 * (a * ((c / b) * (-1.0 / b)))))) - b));
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.42e+38) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_0) / (a * 2.0);
		} else {
			tmp_3 = (c * 2.0) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c / b) - (b / a);
	} else {
		tmp_1 = (c * 2.0) / (b * -2.0);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (c * (a * 4.0))))
	tmp_1 = 0
	if b <= -1e+154:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (-0.5 / a) * (b + b)
		else:
			tmp_2 = c * (2.0 / ((b * (-1.0 + (-2.0 * (a * ((c / b) * (-1.0 / b)))))) - b))
		tmp_1 = tmp_2
	elif b <= 1.42e+38:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (-b - t_0) / (a * 2.0)
		else:
			tmp_3 = (c * 2.0) / (t_0 - b)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = (c / b) - (b / a)
	else:
		tmp_1 = (c * 2.0) / (b * -2.0)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	tmp_1 = 0.0
	if (b <= -1e+154)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-0.5 / a) * Float64(b + b));
		else
			tmp_2 = Float64(c * Float64(2.0 / Float64(Float64(b * Float64(-1.0 + Float64(-2.0 * Float64(a * Float64(Float64(c / b) * Float64(-1.0 / b)))))) - b)));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.42e+38)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(-b) - t_0) / Float64(a * 2.0));
		else
			tmp_3 = Float64(Float64(c * 2.0) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp_1 = Float64(Float64(c * 2.0) / Float64(b * -2.0));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	tmp_2 = 0.0;
	if (b <= -1e+154)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (-0.5 / a) * (b + b);
		else
			tmp_3 = c * (2.0 / ((b * (-1.0 + (-2.0 * (a * ((c / b) * (-1.0 / b)))))) - b));
		end
		tmp_2 = tmp_3;
	elseif (b <= 1.42e+38)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (-b - t_0) / (a * 2.0);
		else
			tmp_4 = (c * 2.0) / (t_0 - b);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = (c / b) - (b / a);
	else
		tmp_2 = (c * 2.0) / (b * -2.0);
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1e+154], If[GreaterEqual[b, 0.0], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + b), $MachinePrecision]), $MachinePrecision], N[(c * N[(2.0 / N[(N[(b * N[(-1.0 + N[(-2.0 * N[(a * N[(N[(c / b), $MachinePrecision] * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.42e+38], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(b * -2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.00000000000000004e154

   1. Initial program 24.9%

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

   3. Simplified 25.1%

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

   5. Taylor expanded in b around -inf 89.2%

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

      1. mul-1-neg 89.2%

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

      2. *-commutative 89.2%

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

      3. distribute-rgt-neg-in 89.2%

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

      4. associate-/l* 94.3%

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

   7. Simplified 94.3%

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

      1. *-un-lft-identity 94.3%

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

      2. pow2 94.3%

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

      3. times-frac 95.1%

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

   9. Applied egg-rr 95.1%

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

       1. *-commutative 95.1%

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

   11. Simplified 95.1%

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

   12. Taylor expanded in c around 0 95.1%

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

   if -1.00000000000000004e154 < b < 1.4200000000000001e38

   1. Initial program 83.2%

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

   if 1.4200000000000001e38 < b

   1. Initial program 64.0%

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

   3. Taylor expanded in a around 0 90.1%

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

      1. distribute-lft-out-- 90.1%

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

      2. associate-/l* 96.4%

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

      3. fmm-def 96.4%

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

   5. Simplified 96.4%

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

   6. Taylor expanded in b around -inf 96.4%

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

      1. *-commutative 96.4%

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

   8. Simplified 96.4%

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

   9. Taylor expanded in a around inf 96.5%

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

       1. +-commutative 96.5%

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

       2. mul-1-neg 96.5%

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

       3. unsub-neg 96.5%

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

   11. Simplified 96.5%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + b\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{b \cdot \left(-1 + -2 \cdot \left(a \cdot \left(\frac{c}{b} \cdot \frac{-1}{b}\right)\right)\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.42 \cdot 10^{+38}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{+147}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + b\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{b \cdot \left(-1 + -2 \cdot \left(a \cdot \left(\frac{c}{b} \cdot \frac{-1}{b}\right)\right)\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.2e+147)
   (if (>= b 0.0)
     (* (/ -0.5 a) (+ b b))
     (* c (/ 2.0 (- (* b (+ -1.0 (* -2.0 (* a (* (/ c b) (/ -1.0 b)))))) b))))
   (if (>= b 0.0)
     (* b (+ (/ c (pow b 2.0)) (/ -1.0 a)))
     (/ (* c 2.0) (- (sqrt (- (* b b) (* c (* a 4.0)))) b)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -7.2e+147) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-0.5 / a) * (b + b);
		} else {
			tmp_2 = c * (2.0 / ((b * (-1.0 + (-2.0 * (a * ((c / b) * (-1.0 / b)))))) - b));
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = b * ((c / pow(b, 2.0)) + (-1.0 / a));
	} else {
		tmp_1 = (c * 2.0) / (sqrt(((b * b) - (c * (a * 4.0)))) - 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
    if (b <= (-7.2d+147)) then
        if (b >= 0.0d0) then
            tmp_2 = ((-0.5d0) / a) * (b + b)
        else
            tmp_2 = c * (2.0d0 / ((b * ((-1.0d0) + ((-2.0d0) * (a * ((c / b) * ((-1.0d0) / b)))))) - b))
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = b * ((c / (b ** 2.0d0)) + ((-1.0d0) / a))
    else
        tmp_1 = (c * 2.0d0) / (sqrt(((b * b) - (c * (a * 4.0d0)))) - b)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -7.2e+147) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-0.5 / a) * (b + b);
		} else {
			tmp_2 = c * (2.0 / ((b * (-1.0 + (-2.0 * (a * ((c / b) * (-1.0 / b)))))) - b));
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = b * ((c / Math.pow(b, 2.0)) + (-1.0 / a));
	} else {
		tmp_1 = (c * 2.0) / (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if b <= -7.2e+147:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (-0.5 / a) * (b + b)
		else:
			tmp_2 = c * (2.0 / ((b * (-1.0 + (-2.0 * (a * ((c / b) * (-1.0 / b)))))) - b))
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = b * ((c / math.pow(b, 2.0)) + (-1.0 / a))
	else:
		tmp_1 = (c * 2.0) / (math.sqrt(((b * b) - (c * (a * 4.0)))) - b)
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -7.2e+147)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-0.5 / a) * Float64(b + b));
		else
			tmp_2 = Float64(c * Float64(2.0 / Float64(Float64(b * Float64(-1.0 + Float64(-2.0 * Float64(a * Float64(Float64(c / b) * Float64(-1.0 / b)))))) - b)));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(b * Float64(Float64(c / (b ^ 2.0)) + Float64(-1.0 / a)));
	else
		tmp_1 = Float64(Float64(c * 2.0) / Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp_2 = 0.0;
	if (b <= -7.2e+147)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (-0.5 / a) * (b + b);
		else
			tmp_3 = c * (2.0 / ((b * (-1.0 + (-2.0 * (a * ((c / b) * (-1.0 / b)))))) - b));
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = b * ((c / (b ^ 2.0)) + (-1.0 / a));
	else
		tmp_2 = (c * 2.0) / (sqrt(((b * b) - (c * (a * 4.0)))) - b);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.2e+147], If[GreaterEqual[b, 0.0], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + b), $MachinePrecision]), $MachinePrecision], N[(c * N[(2.0 / N[(N[(b * N[(-1.0 + N[(-2.0 * N[(a * N[(N[(c / b), $MachinePrecision] * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(b * N[(N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -7.20000000000000041e147

   1. Initial program 24.9%

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

   3. Simplified 25.1%

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

   5. Taylor expanded in b around -inf 89.2%

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

      1. mul-1-neg 89.2%

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

      2. *-commutative 89.2%

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

      3. distribute-rgt-neg-in 89.2%

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

      4. associate-/l* 94.3%

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

   7. Simplified 94.3%

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

      1. *-un-lft-identity 94.3%

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

      2. pow2 94.3%

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

      3. times-frac 95.1%

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

   9. Applied egg-rr 95.1%

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

       1. *-commutative 95.1%

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

   11. Simplified 95.1%

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

   12. Taylor expanded in c around 0 95.1%

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

   if -7.20000000000000041e147 < b

   1. Initial program 77.0%

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

   3. Taylor expanded in a around 0 71.7%

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

      1. distribute-lft-out-- 71.7%

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

      2. associate-/l* 73.8%

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

      3. fmm-def 73.8%

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

   5. Simplified 73.8%

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

   6. Taylor expanded in b around -inf 73.5%

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

      1. mul-1-neg 73.5%

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

      2. *-commutative 73.5%

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

      3. distribute-rgt-neg-in 73.5%

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

      4. +-commutative 73.5%

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

      5. mul-1-neg 73.5%

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

      6. unsub-neg 73.5%

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

   8. Simplified 73.5%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{+147}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + b\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{b \cdot \left(-1 + -2 \cdot \left(a \cdot \left(\frac{c}{b} \cdot \frac{-1}{b}\right)\right)\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + b\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{b \cdot \left(-1 + -2 \cdot \left(a \cdot \left(\frac{c}{b} \cdot \frac{-1}{b}\right)\right)\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+154)
   (if (>= b 0.0)
     (* (/ -0.5 a) (+ b b))
     (* c (/ 2.0 (- (* b (+ -1.0 (* -2.0 (* a (* (/ c b) (/ -1.0 b)))))) b))))
   (if (>= b 0.0)
     (/ b (- a))
     (/ (* c 2.0) (- (sqrt (- (* b b) (* c (* a 4.0)))) b)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1e+154) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-0.5 / a) * (b + b);
		} else {
			tmp_2 = c * (2.0 / ((b * (-1.0 + (-2.0 * (a * ((c / b) * (-1.0 / b)))))) - b));
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = b / -a;
	} else {
		tmp_1 = (c * 2.0) / (sqrt(((b * b) - (c * (a * 4.0)))) - 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
    if (b <= (-1d+154)) then
        if (b >= 0.0d0) then
            tmp_2 = ((-0.5d0) / a) * (b + b)
        else
            tmp_2 = c * (2.0d0 / ((b * ((-1.0d0) + ((-2.0d0) * (a * ((c / b) * ((-1.0d0) / b)))))) - b))
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = b / -a
    else
        tmp_1 = (c * 2.0d0) / (sqrt(((b * b) - (c * (a * 4.0d0)))) - b)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1e+154) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-0.5 / a) * (b + b);
		} else {
			tmp_2 = c * (2.0 / ((b * (-1.0 + (-2.0 * (a * ((c / b) * (-1.0 / b)))))) - b));
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = b / -a;
	} else {
		tmp_1 = (c * 2.0) / (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if b <= -1e+154:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (-0.5 / a) * (b + b)
		else:
			tmp_2 = c * (2.0 / ((b * (-1.0 + (-2.0 * (a * ((c / b) * (-1.0 / b)))))) - b))
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = b / -a
	else:
		tmp_1 = (c * 2.0) / (math.sqrt(((b * b) - (c * (a * 4.0)))) - b)
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -1e+154)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-0.5 / a) * Float64(b + b));
		else
			tmp_2 = Float64(c * Float64(2.0 / Float64(Float64(b * Float64(-1.0 + Float64(-2.0 * Float64(a * Float64(Float64(c / b) * Float64(-1.0 / b)))))) - b)));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(b / Float64(-a));
	else
		tmp_1 = Float64(Float64(c * 2.0) / Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp_2 = 0.0;
	if (b <= -1e+154)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (-0.5 / a) * (b + b);
		else
			tmp_3 = c * (2.0 / ((b * (-1.0 + (-2.0 * (a * ((c / b) * (-1.0 / b)))))) - b));
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = b / -a;
	else
		tmp_2 = (c * 2.0) / (sqrt(((b * b) - (c * (a * 4.0)))) - b);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+154], If[GreaterEqual[b, 0.0], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + b), $MachinePrecision]), $MachinePrecision], N[(c * N[(2.0 / N[(N[(b * N[(-1.0 + N[(-2.0 * N[(a * N[(N[(c / b), $MachinePrecision] * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(b / (-a)), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.00000000000000004e154

   1. Initial program 24.9%

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

   3. Simplified 25.1%

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

   5. Taylor expanded in b around -inf 89.2%

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

      1. mul-1-neg 89.2%

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

      2. *-commutative 89.2%

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

      3. distribute-rgt-neg-in 89.2%

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

      4. associate-/l* 94.3%

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

   7. Simplified 94.3%

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

      1. *-un-lft-identity 94.3%

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

      2. pow2 94.3%

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

      3. times-frac 95.1%

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

   9. Applied egg-rr 95.1%

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

       1. *-commutative 95.1%

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

   11. Simplified 95.1%

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

   12. Taylor expanded in c around 0 95.1%

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

   if -1.00000000000000004e154 < b

   1. Initial program 77.0%

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

   3. Taylor expanded in a around 0 71.7%

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

      1. distribute-lft-out-- 71.7%

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

      2. associate-/l* 73.8%

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

      3. fmm-def 73.8%

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

   5. Simplified 73.8%

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

   6. Taylor expanded in a around 0 73.5%

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

      1. associate-*r/ 73.5%

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

      2. mul-1-neg 73.5%

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

   8. Simplified 73.5%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + b\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{b \cdot \left(-1 + -2 \cdot \left(a \cdot \left(\frac{c}{b} \cdot \frac{-1}{b}\right)\right)\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+147}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + b\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{b \cdot \left(-1 + -2 \cdot \left(a \cdot \left(\frac{c}{b} \cdot \frac{-1}{b}\right)\right)\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e+147)
   (if (>= b 0.0)
     (* (/ -0.5 a) (+ b b))
     (* c (/ 2.0 (- (* b (+ -1.0 (* -2.0 (* a (* (/ c b) (/ -1.0 b)))))) b))))
   (if (>= b 0.0)
     (/ (* 2.0 (fma a (/ c b) (- b))) (* a 2.0))
     (/ (* c 2.0) (- (sqrt (- (* b b) (* c (* a 4.0)))) b)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -4.5e+147) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-0.5 / a) * (b + b);
		} else {
			tmp_2 = c * (2.0 / ((b * (-1.0 + (-2.0 * (a * ((c / b) * (-1.0 / b)))))) - b));
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * fma(a, (c / b), -b)) / (a * 2.0);
	} else {
		tmp_1 = (c * 2.0) / (sqrt(((b * b) - (c * (a * 4.0)))) - b);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -4.5e+147)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-0.5 / a) * Float64(b + b));
		else
			tmp_2 = Float64(c * Float64(2.0 / Float64(Float64(b * Float64(-1.0 + Float64(-2.0 * Float64(a * Float64(Float64(c / b) * Float64(-1.0 / b)))))) - b)));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * fma(a, Float64(c / b), Float64(-b))) / Float64(a * 2.0));
	else
		tmp_1 = Float64(Float64(c * 2.0) / Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e+147], If[GreaterEqual[b, 0.0], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + b), $MachinePrecision]), $MachinePrecision], N[(c * N[(2.0 / N[(N[(b * N[(-1.0 + N[(-2.0 * N[(a * N[(N[(c / b), $MachinePrecision] * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.50000000000000008e147

   1. Initial program 24.9%

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

   3. Simplified 25.1%

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

   5. Taylor expanded in b around -inf 89.2%

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

      1. mul-1-neg 89.2%

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

      2. *-commutative 89.2%

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

      3. distribute-rgt-neg-in 89.2%

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

      4. associate-/l* 94.3%

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

   7. Simplified 94.3%

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

      1. *-un-lft-identity 94.3%

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

      2. pow2 94.3%

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

      3. times-frac 95.1%

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

   9. Applied egg-rr 95.1%

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

       1. *-commutative 95.1%

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

   11. Simplified 95.1%

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

   12. Taylor expanded in c around 0 95.1%

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

   if -4.50000000000000008e147 < b

   1. Initial program 77.0%

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

   3. Taylor expanded in a around 0 71.7%

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

      1. distribute-lft-out-- 71.7%

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

      2. associate-/l* 73.8%

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

      3. fmm-def 73.8%

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

   5. Simplified 73.8%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+147}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + b\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{b \cdot \left(-1 + -2 \cdot \left(a \cdot \left(\frac{c}{b} \cdot \frac{-1}{b}\right)\right)\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.6% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.5%

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

3. Taylor expanded in a around 0 64.1%

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

   1. distribute-lft-out-- 64.1%

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

   2. associate-/l* 65.8%

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

   3. fmm-def 65.8%

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

5. Simplified 65.8%

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

6. Taylor expanded in b around -inf 64.6%

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

   1. *-commutative 64.6%

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

8. Simplified 64.6%

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

9. Taylor expanded in a around inf 64.6%

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

    1. +-commutative 64.6%

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

    2. mul-1-neg 64.6%

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

    3. unsub-neg 64.6%

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

11. Simplified 64.6%

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

12. Final simplification 64.6%

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

Reproduce

herbie shell --seed 2024112 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  :precision binary64
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))