jeff quadratic root 2

Percentage Accurate: 72.7% → 90.9%

Time: 27.3s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ \mathbf{if}\;b \leq -1.35 \cdot 10^{+159}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{-2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 3.3 \cdot 10^{+128}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot \left(-2\right)}{b + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - b}{-2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0))))))
   (if (<= b -1.35e+159)
     (if (>= b 0.0) (* c (/ -2.0 (+ b b))) (/ (+ b b) (* -2.0 a)))
     (if (<= b 3.3e+128)
       (if (>= b 0.0) (/ (* c (- 2.0)) (+ b t_0)) (/ (- t_0 b) (* a 2.0)))
       (if (>= b 0.0) (/ c (- b)) (/ (- b b) (* -2.0 a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -1.35e+159) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / (b + b));
		} else {
			tmp_2 = (b + b) / (-2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 3.3e+128) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c * -2.0) / (b + t_0);
		} else {
			tmp_3 = (t_0 - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = c / -b;
	} else {
		tmp_1 = (b - b) / (-2.0 * a);
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = sqrt(((b * b) - (c * (a * 4.0d0))))
    if (b <= (-1.35d+159)) then
        if (b >= 0.0d0) then
            tmp_2 = c * ((-2.0d0) / (b + b))
        else
            tmp_2 = (b + b) / ((-2.0d0) * a)
        end if
        tmp_1 = tmp_2
    else if (b <= 3.3d+128) then
        if (b >= 0.0d0) then
            tmp_3 = (c * -2.0d0) / (b + t_0)
        else
            tmp_3 = (t_0 - b) / (a * 2.0d0)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = c / -b
    else
        tmp_1 = (b - b) / ((-2.0d0) * a)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -1.35e+159) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / (b + b));
		} else {
			tmp_2 = (b + b) / (-2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 3.3e+128) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c * -2.0) / (b + t_0);
		} else {
			tmp_3 = (t_0 - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = c / -b;
	} else {
		tmp_1 = (b - b) / (-2.0 * a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (c * (a * 4.0))))
	tmp_1 = 0
	if b <= -1.35e+159:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = c * (-2.0 / (b + b))
		else:
			tmp_2 = (b + b) / (-2.0 * a)
		tmp_1 = tmp_2
	elif b <= 3.3e+128:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (c * -2.0) / (b + t_0)
		else:
			tmp_3 = (t_0 - b) / (a * 2.0)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = c / -b
	else:
		tmp_1 = (b - b) / (-2.0 * a)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	tmp_1 = 0.0
	if (b <= -1.35e+159)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * Float64(-2.0 / Float64(b + b)));
		else
			tmp_2 = Float64(Float64(b + b) / Float64(-2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 3.3e+128)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(c * Float64(-2.0)) / Float64(b + t_0));
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(a * 2.0));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(c / Float64(-b));
	else
		tmp_1 = Float64(Float64(b - b) / Float64(-2.0 * a));
	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 <= -1.35e+159)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = c * (-2.0 / (b + b));
		else
			tmp_3 = (b + b) / (-2.0 * a);
		end
		tmp_2 = tmp_3;
	elseif (b <= 3.3e+128)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (c * -2.0) / (b + t_0);
		else
			tmp_4 = (t_0 - b) / (a * 2.0);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = c / -b;
	else
		tmp_2 = (b - b) / (-2.0 * a);
	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, -1.35e+159], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + b), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 3.3e+128], If[GreaterEqual[b, 0.0], N[(N[(c * (-2.0)), $MachinePrecision] / N[(b + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(c / (-b)), $MachinePrecision], N[(N[(b - b), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.35000000000000004e159

   1. Initial program 44.2%

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

   3. Simplified 44.3%

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

   5. Taylor expanded in c around 0 44.3%

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

   6. Taylor expanded in b around -inf 94.3%

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

      1. mul-1-neg 94.3%

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

   8. Simplified 94.3%

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

   if -1.35000000000000004e159 < b < 3.3000000000000001e128

   1. Initial program 85.9%

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

   if 3.3000000000000001e128 < b

   1. Initial program 38.7%

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

   3. Simplified 38.9%

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

   5. Taylor expanded in c around 0 95.5%

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

   6. Taylor expanded in c around 0 95.5%

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

   7. Taylor expanded in c around 0 95.8%

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

      1. associate-*r/ 95.8%

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

      2. neg-mul-1 95.8%

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

   9. Simplified 95.8%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+159}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{-2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 3.3 \cdot 10^{+128}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot \left(-2\right)}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - b}{-2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+159}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{-2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq -6.2 \cdot 10^{-250}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(a, \frac{c}{b}, b\right)}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{b + \sqrt{a \cdot \left(c \cdot -4\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.35e+159)
   (if (>= b 0.0) (* c (/ -2.0 (+ b b))) (/ (+ b b) (* -2.0 a)))
   (if (<= b -6.2e-250)
     (if (>= b 0.0)
       (/ 1.0 (/ (fma a (/ c b) b) c))
       (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)))
     (if (>= b 0.0)
       (/ (* c 2.0) (* 2.0 (fma a (/ c b) (- b))))
       (/ 1.0 (/ (* a 2.0) (+ b (sqrt (* a (* c -4.0))))))))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1.35e+159) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / (b + b));
		} else {
			tmp_2 = (b + b) / (-2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= -6.2e-250) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = 1.0 / (fma(a, (c / b), b) / c);
		} else {
			tmp_3 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c * 2.0) / (2.0 * fma(a, (c / b), -b));
	} else {
		tmp_1 = 1.0 / ((a * 2.0) / (b + sqrt((a * (c * -4.0)))));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -1.35e+159)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * Float64(-2.0 / Float64(b + b)));
		else
			tmp_2 = Float64(Float64(b + b) / Float64(-2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= -6.2e-250)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(1.0 / Float64(fma(a, Float64(c / b), b) / c));
		else
			tmp_3 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c * 2.0) / Float64(2.0 * fma(a, Float64(c / b), Float64(-b))));
	else
		tmp_1 = Float64(1.0 / Float64(Float64(a * 2.0) / Float64(b + sqrt(Float64(a * Float64(c * -4.0))))));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.35e+159], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + b), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -6.2e-250], If[GreaterEqual[b, 0.0], N[(1.0 / N[(N[(a * N[(c / b), $MachinePrecision] + b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[(2.0 * N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a * 2.0), $MachinePrecision] / N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.35000000000000004e159

   1. Initial program 44.2%

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

   3. Simplified 44.3%

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

   5. Taylor expanded in c around 0 44.3%

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

   6. Taylor expanded in b around -inf 94.3%

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

      1. mul-1-neg 94.3%

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

   8. Simplified 94.3%

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

   if -1.35000000000000004e159 < b < -6.2000000000000002e-250

   1. Initial program 87.4%

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

   3. Taylor expanded in a around 0 87.4%

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

      1. distribute-lft-out-- 87.4%

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

      2. associate-/l* 87.4%

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

      3. fma-neg 87.4%

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

   5. Simplified 87.4%

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

      1. clear-num 87.4%

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

      2. inv-pow 87.4%

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

      3. times-frac 87.4%

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

      4. metadata-eval 87.4%

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

      5. add-sqr-sqrt 87.4%

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

      6. sqrt-unprod 87.4%

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

      7. sqr-neg 87.4%

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

      8. sqrt-prod 87.4%

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

      9. add-sqr-sqrt 87.4%

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

   7. Applied egg-rr 87.4%

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

      1. unpow-1 87.4%

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

      2. *-lft-identity 87.4%

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

   9. Simplified 87.4%

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

   if -6.2000000000000002e-250 < b

   1. Initial program 68.9%

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

   3. Taylor expanded in a around 0 62.1%

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

      1. distribute-lft-out-- 62.1%

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

      2. associate-/l* 65.1%

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

      3. fma-neg 65.1%

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

   5. Simplified 65.1%

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

      1. clear-num 65.2%

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

      2. inv-pow 65.2%

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

      3. *-commutative 65.2%

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

      4. add-sqr-sqrt 65.2%

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

      5. sqrt-unprod 65.2%

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

      6. sqr-neg 65.2%

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

      7. sqrt-prod 56.0%

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

      8. add-sqr-sqrt 65.2%

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

      9. pow2 65.2%

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

      10. associate-*l* 65.2%

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

   7. Applied egg-rr 65.2%

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

      1. unpow-1 65.2%

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

      2. unpow2 65.2%

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

      3. fma-neg 65.2%

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

      4. distribute-lft-neg-in 65.2%

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

      5. metadata-eval 65.2%

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

      6. *-commutative 65.2%

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

   9. Simplified 65.2%

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

   10. Taylor expanded in b around 0 65.2%

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

       1. *-commutative 65.2%

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

       2. associate-*r* 65.2%

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

   12. Simplified 65.2%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+159}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{-2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq -6.2 \cdot 10^{-250}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(a, \frac{c}{b}, b\right)}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{b + \sqrt{a \cdot \left(c \cdot -4\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+159}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{-2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.35e+159)
   (if (>= b 0.0) (* c (/ -2.0 (+ b b))) (/ (+ b b) (* -2.0 a)))
   (if (>= b 0.0)
     (/ (* c 2.0) (* 2.0 (fma a (/ c b) (- b))))
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1.35e+159) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / (b + b));
		} else {
			tmp_2 = (b + b) / (-2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (c * 2.0) / (2.0 * fma(a, (c / b), -b));
	} else {
		tmp_1 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -1.35e+159)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * Float64(-2.0 / Float64(b + b)));
		else
			tmp_2 = Float64(Float64(b + b) / Float64(-2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c * 2.0) / Float64(2.0 * fma(a, Float64(c / b), Float64(-b))));
	else
		tmp_1 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.35e+159], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + b), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[(2.0 * N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.35000000000000004e159

   1. Initial program 44.2%

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

   3. Simplified 44.3%

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

   5. Taylor expanded in c around 0 44.3%

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

   6. Taylor expanded in b around -inf 94.3%

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

      1. mul-1-neg 94.3%

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

   8. Simplified 94.3%

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

   if -1.35000000000000004e159 < b

   1. Initial program 76.2%

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

   3. Taylor expanded in a around 0 72.1%

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

      1. distribute-lft-out-- 72.1%

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

      2. associate-/l* 73.9%

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

      3. fma-neg 73.9%

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

   5. Simplified 73.9%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+159}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{-2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{-99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{-2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{b + \sqrt{a \cdot \left(c \cdot -4\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.2e-99)
   (if (>= b 0.0) (* c (/ -2.0 (+ b b))) (/ (+ b b) (* -2.0 a)))
   (if (>= b 0.0)
     (/ (* c 2.0) (* 2.0 (fma a (/ c b) (- b))))
     (/ 1.0 (/ (* a 2.0) (+ b (sqrt (* a (* c -4.0)))))))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -2.2e-99) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / (b + b));
		} else {
			tmp_2 = (b + b) / (-2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (c * 2.0) / (2.0 * fma(a, (c / b), -b));
	} else {
		tmp_1 = 1.0 / ((a * 2.0) / (b + sqrt((a * (c * -4.0)))));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -2.2e-99)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * Float64(-2.0 / Float64(b + b)));
		else
			tmp_2 = Float64(Float64(b + b) / Float64(-2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c * 2.0) / Float64(2.0 * fma(a, Float64(c / b), Float64(-b))));
	else
		tmp_1 = Float64(1.0 / Float64(Float64(a * 2.0) / Float64(b + sqrt(Float64(a * Float64(c * -4.0))))));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.2e-99], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + b), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[(2.0 * N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a * 2.0), $MachinePrecision] / N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.20000000000000004e-99

   1. Initial program 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in c around 0 70.8%

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

   6. Taylor expanded in b around -inf 86.5%

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

      1. mul-1-neg 86.5%

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

   8. Simplified 86.5%

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

   if -2.20000000000000004e-99 < b

   1. Initial program 71.3%

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

   3. Taylor expanded in a around 0 65.8%

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

      1. distribute-lft-out-- 65.8%

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

      2. associate-/l* 68.2%

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

      3. fma-neg 68.2%

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

   5. Simplified 68.2%

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

      1. clear-num 68.2%

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

      2. inv-pow 68.2%

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

      3. *-commutative 68.2%

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

      4. add-sqr-sqrt 68.2%

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

      5. sqrt-unprod 67.9%

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

      6. sqr-neg 67.9%

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

      7. sqrt-prod 45.2%

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

      8. add-sqr-sqrt 64.3%

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

      9. pow2 64.3%

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

      10. associate-*l* 64.3%

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

   7. Applied egg-rr 64.3%

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

      1. unpow-1 64.3%

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

      2. unpow2 64.3%

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

      3. fma-neg 64.3%

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

      4. distribute-lft-neg-in 64.3%

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

      5. metadata-eval 64.3%

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

      6. *-commutative 64.3%

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

   9. Simplified 64.3%

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

   10. Taylor expanded in b around 0 64.2%

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

       1. *-commutative 64.2%

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

       2. associate-*r* 64.2%

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

   12. Simplified 64.2%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{-99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{-2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{b + \sqrt{a \cdot \left(c \cdot -4\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 11.5% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.1%

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

3. Simplified 71.1%

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

5. Taylor expanded in c around 0 69.1%

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

6. Taylor expanded in c around 0 29.9%

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

8. Applied egg-rr 9.3%

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

9. Final simplification 9.3%

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

Alternative 6: 35.1% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.1%

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

3. Simplified 71.1%

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

5. Taylor expanded in c around 0 69.1%

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

6. Taylor expanded in c around 0 29.9%

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

7. Final simplification 29.9%

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

Alternative 7: 35.2% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.1%

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

3. Simplified 71.1%

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

5. Taylor expanded in c around 0 69.1%

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

6. Taylor expanded in c around 0 29.9%

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

7. Taylor expanded in c around 0 29.9%

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

   1. associate-*r/ 29.9%

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

   2. neg-mul-1 29.9%

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

9. Simplified 29.9%

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

10. Final simplification 29.9%

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

Alternative 8: 67.5% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.1%

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

3. Simplified 71.1%

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

5. Taylor expanded in c around 0 69.1%

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

6. Taylor expanded in b around -inf 64.3%

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

   1. mul-1-neg 64.3%

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

8. Simplified 64.3%

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

9. Final simplification 64.3%

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

Reproduce

herbie shell --seed 2024073 
(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))))