jeff quadratic root 1

Percentage Accurate: 72.1% → 90.9%

Time: 17.1s

Alternatives: 4

Speedup: 0.9×

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: 72.1% 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: 90.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b \leq -7.2 \cdot 10^{+155} \lor \neg \left(b \leq 2 \cdot 10^{+132}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{t\_0 - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* 4.0 (* a c))))))
   (if (or (<= b -7.2e+155) (not (<= b 2e+132)))
     (if (>= b 0.0) (/ b (- a)) (/ (- c) b))
     (if (>= b 0.0) (/ (- (- b) t_0) (* a 2.0)) (* 2.0 (/ c (- t_0 b)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (4.0 * (a * c))));
	double tmp_1;
	if ((b <= -7.2e+155) || !(b <= 2e+132)) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b / -a;
		} else {
			tmp_2 = -c / b;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (-b - t_0) / (a * 2.0);
	} else {
		tmp_1 = 2.0 * (c / (t_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) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = sqrt(((b * b) - (4.0d0 * (a * c))))
    if ((b <= (-7.2d+155)) .or. (.not. (b <= 2d+132))) then
        if (b >= 0.0d0) then
            tmp_2 = b / -a
        else
            tmp_2 = -c / b
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (-b - t_0) / (a * 2.0d0)
    else
        tmp_1 = 2.0d0 * (c / (t_0 - b))
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (4.0 * (a * c))));
	double tmp_1;
	if ((b <= -7.2e+155) || !(b <= 2e+132)) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b / -a;
		} else {
			tmp_2 = -c / b;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (-b - t_0) / (a * 2.0);
	} else {
		tmp_1 = 2.0 * (c / (t_0 - b));
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (4.0 * (a * c))))
	tmp_1 = 0
	if (b <= -7.2e+155) or not (b <= 2e+132):
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = b / -a
		else:
			tmp_2 = -c / b
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = (-b - t_0) / (a * 2.0)
	else:
		tmp_1 = 2.0 * (c / (t_0 - b))
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))
	tmp_1 = 0.0
	if ((b <= -7.2e+155) || !(b <= 2e+132))
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(b / Float64(-a));
		else
			tmp_2 = Float64(Float64(-c) / b);
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(-b) - t_0) / Float64(a * 2.0));
	else
		tmp_1 = Float64(2.0 * Float64(c / Float64(t_0 - b)));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = sqrt(((b * b) - (4.0 * (a * c))));
	tmp_2 = 0.0;
	if ((b <= -7.2e+155) || ~((b <= 2e+132)))
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = b / -a;
		else
			tmp_3 = -c / b;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = (-b - t_0) / (a * 2.0);
	else
		tmp_2 = 2.0 * (c / (t_0 - b));
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[b, -7.2e+155], N[Not[LessEqual[b, 2e+132]], $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(b / (-a)), $MachinePrecision], N[((-c) / b), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(c / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < -7.20000000000000015e155 or 1.99999999999999998e132 < b

   1. Initial program 47.2%

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

   3. Simplified 47.3%

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

   5. Taylor expanded in b around -inf 73.1%

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

      1. *-commutative 73.1%

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

   7. Simplified 73.1%

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

   8. Taylor expanded in c around 0 96.5%

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

   9. Taylor expanded in b around 0 96.8%

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

   11. Simplified 96.8%

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

   if -7.20000000000000015e155 < b < 1.99999999999999998e132

   1. Initial program 85.2%

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

   3. Simplified 85.7%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{+155} \lor \neg \left(b \leq 2 \cdot 10^{+132}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{-63} \lor \neg \left(b \leq 2 \cdot 10^{+132}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{b}{4} + \sqrt{a \cdot \frac{c}{-4}}}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (or (<= b -1.3e-63) (not (<= b 2e+132)))
   (if (>= b 0.0) (/ b (- a)) (/ (- c) b))
   (if (>= b 0.0)
     (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* a 2.0))
     (* 2.0 (/ (+ (/ b 4.0) (sqrt (* a (/ c -4.0)))) (- a))))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if ((b <= -1.3e-63) || !(b <= 2e+132)) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b / -a;
		} else {
			tmp_2 = -c / b;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (a * 2.0);
	} else {
		tmp_1 = 2.0 * (((b / 4.0) + sqrt((a * (c / -4.0)))) / -a);
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    if ((b <= (-1.3d-63)) .or. (.not. (b <= 2d+132))) then
        if (b >= 0.0d0) then
            tmp_2 = b / -a
        else
            tmp_2 = -c / b
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (a * 2.0d0)
    else
        tmp_1 = 2.0d0 * (((b / 4.0d0) + sqrt((a * (c / (-4.0d0))))) / -a)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if ((b <= -1.3e-63) || !(b <= 2e+132)) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b / -a;
		} else {
			tmp_2 = -c / b;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (-b - Math.sqrt(((b * b) - (4.0 * (a * c))))) / (a * 2.0);
	} else {
		tmp_1 = 2.0 * (((b / 4.0) + Math.sqrt((a * (c / -4.0)))) / -a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if (b <= -1.3e-63) or not (b <= 2e+132):
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = b / -a
		else:
			tmp_2 = -c / b
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = (-b - math.sqrt(((b * b) - (4.0 * (a * c))))) / (a * 2.0)
	else:
		tmp_1 = 2.0 * (((b / 4.0) + math.sqrt((a * (c / -4.0)))) / -a)
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if ((b <= -1.3e-63) || !(b <= 2e+132))
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(b / Float64(-a));
		else
			tmp_2 = Float64(Float64(-c) / b);
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(a * 2.0));
	else
		tmp_1 = Float64(2.0 * Float64(Float64(Float64(b / 4.0) + sqrt(Float64(a * Float64(c / -4.0)))) / Float64(-a)));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp_2 = 0.0;
	if ((b <= -1.3e-63) || ~((b <= 2e+132)))
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = b / -a;
		else
			tmp_3 = -c / b;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (a * 2.0);
	else
		tmp_2 = 2.0 * (((b / 4.0) + sqrt((a * (c / -4.0)))) / -a);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[Or[LessEqual[b, -1.3e-63], N[Not[LessEqual[b, 2e+132]], $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(b / (-a)), $MachinePrecision], N[((-c) / b), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(b / 4.0), $MachinePrecision] + N[Sqrt[N[(a * N[(c / -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-a)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.3000000000000001e-63 or 1.99999999999999998e132 < b

   1. Initial program 66.3%

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

   3. Simplified 66.2%

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

   5. Taylor expanded in b around -inf 76.5%

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

      1. *-commutative 76.5%

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

   7. Simplified 76.5%

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

   8. Taylor expanded in c around 0 90.6%

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

   9. Taylor expanded in b around 0 90.8%

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

   11. Simplified 90.8%

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

   if -1.3000000000000001e-63 < b < 1.99999999999999998e132

   1. Initial program 80.5%

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

   3. Simplified 81.4%

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

      1. add-cube-cbrt 81.2%

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

      2. pow3 81.3%

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

      3. *-commutative 81.3%

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

      4. associate-*r* 81.3%

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

   6. Applied egg-rr 81.3%

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

   7. Taylor expanded in a around -inf 62.0%

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

      1. mul-1-neg 62.0%

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

      2. +-commutative 62.0%

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

      3. rem-cube-cbrt 62.0%

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

      4. unpow2 62.0%

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

      5. rem-square-sqrt 79.3%

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

      6. metadata-eval 79.3%

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

      7. rem-cube-cbrt 79.4%

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

      8. associate-/l* 79.4%

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

   9. Simplified 79.4%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{-63} \lor \neg \left(b \leq 2 \cdot 10^{+132}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{b}{4} + \sqrt{a \cdot \frac{c}{-4}}}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.6% accurate, 10.1× speedup

Math

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

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(c * Float64(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[(c * N[(2.0 / N[(b * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 72.7%

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

3. Simplified 72.6%

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

5. Taylor expanded in b around -inf 71.2%

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

   1. *-commutative 71.2%

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

7. Simplified 71.2%

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

8. Taylor expanded in c around 0 67.9%

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

   1. +-commutative 67.9%

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

   2. mul-1-neg 67.9%

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

   3. unsub-neg 67.9%

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

10. Simplified 67.9%

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

Alternative 4: 67.5% accurate, 13.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

3. Simplified 72.6%

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

5. Taylor expanded in b around -inf 71.2%

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

   1. *-commutative 71.2%

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

7. Simplified 71.2%

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

8. Taylor expanded in c around 0 67.7%

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

9. Taylor expanded in b around 0 67.9%

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

11. Simplified 67.9%

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

Reproduce

herbie shell --seed 2024172 
(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)))))))