jeff quadratic root 1

Percentage Accurate: 71.6% → 90.2%

Time: 28.3s

Alternatives: 4

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 71.6% accurate, 1.0× speedup

Math

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

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: 90.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\\ \mathbf{if}\;b \leq -5 \cdot 10^{+90} \lor \neg \left(b \leq 5 \cdot 10^{+133}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{t\_0 - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* 4.0 (* c a))))))
   (if (or (<= b -5e+90) (not (<= b 5e+133)))
     (if (>= b 0.0) (- (/ c b) (/ b a)) (/ (* c 2.0) (* b -2.0)))
     (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 * (c * a))));
	double tmp_1;
	if ((b <= -5e+90) || !(b <= 5e+133)) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c / b) - (b / a);
		} else {
			tmp_2 = (c * 2.0) / (b * -2.0);
		}
		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 * (c * a))))
    if ((b <= (-5d+90)) .or. (.not. (b <= 5d+133))) then
        if (b >= 0.0d0) then
            tmp_2 = (c / b) - (b / a)
        else
            tmp_2 = (c * 2.0d0) / (b * (-2.0d0))
        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 * (c * a))));
	double tmp_1;
	if ((b <= -5e+90) || !(b <= 5e+133)) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c / b) - (b / a);
		} else {
			tmp_2 = (c * 2.0) / (b * -2.0);
		}
		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 * (c * a))))
	tmp_1 = 0
	if (b <= -5e+90) or not (b <= 5e+133):
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (c / b) - (b / a)
		else:
			tmp_2 = (c * 2.0) / (b * -2.0)
		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(c * a))))
	tmp_1 = 0.0
	if ((b <= -5e+90) || !(b <= 5e+133))
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(c / b) - Float64(b / a));
		else
			tmp_2 = Float64(Float64(c * 2.0) / Float64(b * -2.0));
		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 * (c * a))));
	tmp_2 = 0.0;
	if ((b <= -5e+90) || ~((b <= 5e+133)))
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (c / b) - (b / a);
		else
			tmp_3 = (c * 2.0) / (b * -2.0);
		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[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[b, -5e+90], N[Not[LessEqual[b, 5e+133]], $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(b * -2.0), $MachinePrecision]), $MachinePrecision]], 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 < -5.0000000000000004e90 or 4.99999999999999961e133 < b

   1. Initial program 45.4%

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

      1. sqr-neg 45.4%

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

      2. sqr-neg 45.4%

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

      3. associate-*l* 45.4%

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

      4. *-commutative 45.4%

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

      5. associate-/l* 45.4%

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

      6. sqr-neg 45.4%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in a around 0 65.0%

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

      1. distribute-lft-out-- 65.0%

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

      2. associate-/l* 68.2%

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

      3. fma-neg 68.2%

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

   7. Simplified 68.2%

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

   8. Taylor expanded in b around -inf 96.9%

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

   9. Taylor expanded in a around inf 96.9%

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

       1. +-commutative 96.9%

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

       2. mul-1-neg 96.9%

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

       3. unsub-neg 96.9%

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

   11. Simplified 96.9%

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

   12. Taylor expanded in c around 0 96.9%

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

       1. associate-*r/ 97.0%

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

       2. sub-neg 97.0%

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

       3. neg-mul-1 97.0%

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

       4. distribute-rgt-out 97.0%

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

       5. metadata-eval 97.0%

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

   14. Simplified 97.0%

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

   if -5.0000000000000004e90 < b < 4.99999999999999961e133

   1. Initial program 84.4%

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

      1. sqr-neg 84.4%

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

      2. sqr-neg 84.4%

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

      3. associate-*l* 85.0%

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

      4. *-commutative 85.0%

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

      5. associate-/l* 85.0%

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

      6. sqr-neg 85.0%

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

   3. Simplified 85.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+90} \lor \neg \left(b \leq 5 \cdot 10^{+133}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2}\\ \end{array}\\ t_1 := \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\\ \mathbf{if}\;b \leq -5.8 \cdot 10^{+91}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{t\_1 - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+134}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{c}{b} \cdot -0.5\right)\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (if (>= b 0.0) (- (/ c b) (/ b a)) (/ (* c 2.0) (* b -2.0))))
        (t_1 (sqrt (- (* b b) (* 4.0 (* c a))))))
   (if (<= b -5.8e+91)
     t_0
     (if (<= b -4e-310)
       (if (>= b 0.0) (/ (- b) a) (* 2.0 (/ c (- t_1 b))))
       (if (<= b 3.1e+134)
         (if (>= b 0.0) (/ (- (- b) t_1) (* a 2.0)) (* 2.0 (* (/ c b) -0.5)))
         t_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);
	}
	double t_0 = tmp;
	double t_1 = sqrt(((b * b) - (4.0 * (c * a))));
	double tmp_1;
	if (b <= -5.8e+91) {
		tmp_1 = t_0;
	} else if (b <= -4e-310) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -b / a;
		} else {
			tmp_2 = 2.0 * (c / (t_1 - b));
		}
		tmp_1 = tmp_2;
	} else if (b <= 3.1e+134) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_1) / (a * 2.0);
		} else {
			tmp_3 = 2.0 * ((c / b) * -0.5);
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    if (b >= 0.0d0) then
        tmp = (c / b) - (b / a)
    else
        tmp = (c * 2.0d0) / (b * (-2.0d0))
    end if
    t_0 = tmp
    t_1 = sqrt(((b * b) - (4.0d0 * (c * a))))
    if (b <= (-5.8d+91)) then
        tmp_1 = t_0
    else if (b <= (-4d-310)) then
        if (b >= 0.0d0) then
            tmp_2 = -b / a
        else
            tmp_2 = 2.0d0 * (c / (t_1 - b))
        end if
        tmp_1 = tmp_2
    else if (b <= 3.1d+134) then
        if (b >= 0.0d0) then
            tmp_3 = (-b - t_1) / (a * 2.0d0)
        else
            tmp_3 = 2.0d0 * ((c / b) * (-0.5d0))
        end if
        tmp_1 = tmp_3
    else
        tmp_1 = t_0
    end if
    code = tmp_1
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);
	}
	double t_0 = tmp;
	double t_1 = Math.sqrt(((b * b) - (4.0 * (c * a))));
	double tmp_1;
	if (b <= -5.8e+91) {
		tmp_1 = t_0;
	} else if (b <= -4e-310) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -b / a;
		} else {
			tmp_2 = 2.0 * (c / (t_1 - b));
		}
		tmp_1 = tmp_2;
	} else if (b <= 3.1e+134) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_1) / (a * 2.0);
		} else {
			tmp_3 = 2.0 * ((c / b) * -0.5);
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

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)
	t_0 = tmp
	t_1 = math.sqrt(((b * b) - (4.0 * (c * a))))
	tmp_1 = 0
	if b <= -5.8e+91:
		tmp_1 = t_0
	elif b <= -4e-310:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -b / a
		else:
			tmp_2 = 2.0 * (c / (t_1 - b))
		tmp_1 = tmp_2
	elif b <= 3.1e+134:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (-b - t_1) / (a * 2.0)
		else:
			tmp_3 = 2.0 * ((c / b) * -0.5)
		tmp_1 = tmp_3
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(Float64(c * 2.0) / Float64(b * -2.0));
	end
	t_0 = tmp
	t_1 = sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a))))
	tmp_1 = 0.0
	if (b <= -5.8e+91)
		tmp_1 = t_0;
	elseif (b <= -4e-310)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-b) / a);
		else
			tmp_2 = Float64(2.0 * Float64(c / Float64(t_1 - b)));
		end
		tmp_1 = tmp_2;
	elseif (b <= 3.1e+134)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(-b) - t_1) / Float64(a * 2.0));
		else
			tmp_3 = Float64(2.0 * Float64(Float64(c / b) * -0.5));
		end
		tmp_1 = tmp_3;
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_5 = 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
	t_0 = tmp;
	t_1 = sqrt(((b * b) - (4.0 * (c * a))));
	tmp_2 = 0.0;
	if (b <= -5.8e+91)
		tmp_2 = t_0;
	elseif (b <= -4e-310)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -b / a;
		else
			tmp_3 = 2.0 * (c / (t_1 - b));
		end
		tmp_2 = tmp_3;
	elseif (b <= 3.1e+134)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (-b - t_1) / (a * 2.0);
		else
			tmp_4 = 2.0 * ((c / b) * -0.5);
		end
		tmp_2 = tmp_4;
	else
		tmp_2 = t_0;
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = If[GreaterEqual[b, 0.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(b * -2.0), $MachinePrecision]), $MachinePrecision]]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -5.8e+91], t$95$0, If[LessEqual[b, -4e-310], If[GreaterEqual[b, 0.0], N[((-b) / a), $MachinePrecision], N[(2.0 * N[(c / N[(t$95$1 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 3.1e+134], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$1), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.80000000000000028e91 or 3.09999999999999982e134 < b

   1. Initial program 45.4%

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

      1. sqr-neg 45.4%

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

      2. sqr-neg 45.4%

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

      3. associate-*l* 45.4%

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

      4. *-commutative 45.4%

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

      5. associate-/l* 45.4%

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

      6. sqr-neg 45.4%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in a around 0 65.0%

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

      1. distribute-lft-out-- 65.0%

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

      2. associate-/l* 68.2%

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

      3. fma-neg 68.2%

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

   7. Simplified 68.2%

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

   8. Taylor expanded in b around -inf 96.9%

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

   9. Taylor expanded in a around inf 96.9%

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

       1. +-commutative 96.9%

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

       2. mul-1-neg 96.9%

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

       3. unsub-neg 96.9%

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

   11. Simplified 96.9%

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

   12. Taylor expanded in c around 0 96.9%

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

       1. associate-*r/ 97.0%

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

       2. sub-neg 97.0%

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

       3. neg-mul-1 97.0%

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

       4. distribute-rgt-out 97.0%

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

       5. metadata-eval 97.0%

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

   14. Simplified 97.0%

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

   if -5.80000000000000028e91 < b < -3.999999999999988e-310

   1. Initial program 81.7%

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

      1. sqr-neg 81.7%

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

      2. sqr-neg 81.7%

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

      3. associate-*l* 81.7%

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

      4. *-commutative 81.7%

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

      5. associate-/l* 81.7%

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

      6. sqr-neg 81.7%

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

   3. Simplified 81.7%

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

   5. Taylor expanded in a around 0 81.7%

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

      1. distribute-lft-out-- 81.7%

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

      2. associate-/l* 81.7%

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

      3. fma-neg 81.7%

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

   7. Simplified 81.7%

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

   8. Taylor expanded in a around 0 81.7%

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

      1. associate-*r/ 81.7%

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

      2. mul-1-neg 81.7%

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

   10. Simplified 81.7%

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

   if -3.999999999999988e-310 < b < 3.09999999999999982e134

   1. Initial program 86.7%

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

      1. sqr-neg 86.7%

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

      2. sqr-neg 86.7%

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

      3. associate-*l* 87.9%

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

      4. *-commutative 87.9%

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

      5. associate-/l* 87.9%

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

      6. sqr-neg 87.9%

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

   3. Simplified 87.9%

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

   5. Taylor expanded in b around -inf 87.9%

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

      1. *-commutative 87.9%

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

   7. Simplified 87.9%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{+91}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2}\\ \end{array}\\ \mathbf{elif}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+134}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{c}{b} \cdot -0.5\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{+91}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.5e+91)
   (if (>= b 0.0) (- (/ c b) (/ b a)) (/ (* c 2.0) (* b -2.0)))
   (if (>= b 0.0)
     (/ (- b) a)
     (* 2.0 (/ c (- (sqrt (- (* b b) (* 4.0 (* c a)))) b))))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -6.5e+91) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c / b) - (b / a);
		} else {
			tmp_2 = (c * 2.0) / (b * -2.0);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = -b / a;
	} else {
		tmp_1 = 2.0 * (c / (sqrt(((b * b) - (4.0 * (c * a)))) - b));
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    if (b <= (-6.5d+91)) then
        if (b >= 0.0d0) then
            tmp_2 = (c / b) - (b / a)
        else
            tmp_2 = (c * 2.0d0) / (b * (-2.0d0))
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = -b / a
    else
        tmp_1 = 2.0d0 * (c / (sqrt(((b * b) - (4.0d0 * (c * a)))) - b))
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -6.5e+91) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c / b) - (b / a);
		} else {
			tmp_2 = (c * 2.0) / (b * -2.0);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = -b / a;
	} else {
		tmp_1 = 2.0 * (c / (Math.sqrt(((b * b) - (4.0 * (c * a)))) - b));
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if b <= -6.5e+91:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (c / b) - (b / a)
		else:
			tmp_2 = (c * 2.0) / (b * -2.0)
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = -b / a
	else:
		tmp_1 = 2.0 * (c / (math.sqrt(((b * b) - (4.0 * (c * a)))) - b))
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -6.5e+91)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(c / b) - Float64(b / a));
		else
			tmp_2 = Float64(Float64(c * 2.0) / Float64(b * -2.0));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-b) / a);
	else
		tmp_1 = Float64(2.0 * Float64(c / Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a)))) - b)));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp_2 = 0.0;
	if (b <= -6.5e+91)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (c / b) - (b / a);
		else
			tmp_3 = (c * 2.0) / (b * -2.0);
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = -b / a;
	else
		tmp_2 = 2.0 * (c / (sqrt(((b * b) - (4.0 * (c * a)))) - b));
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.5e+91], If[GreaterEqual[b, 0.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(b * -2.0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[((-b) / a), $MachinePrecision], N[(2.0 * N[(c / N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -6.4999999999999997e91

   1. Initial program 48.7%

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

      1. sqr-neg 48.7%

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

      2. sqr-neg 48.7%

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

      3. associate-*l* 48.7%

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

      4. *-commutative 48.7%

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

      5. associate-/l* 48.7%

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

      6. sqr-neg 48.7%

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

   3. Simplified 48.7%

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

   5. Taylor expanded in a around 0 48.7%

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

      1. distribute-lft-out-- 48.7%

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

      2. associate-/l* 48.7%

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

      3. fma-neg 48.7%

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

   7. Simplified 48.7%

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

   8. Taylor expanded in b around -inf 95.1%

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

   9. Taylor expanded in a around inf 95.1%

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

       1. +-commutative 95.1%

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

       2. mul-1-neg 95.1%

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

       3. unsub-neg 95.1%

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

   11. Simplified 95.1%

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

   12. Taylor expanded in c around 0 95.1%

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

       1. associate-*r/ 95.1%

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

       2. sub-neg 95.1%

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

       3. neg-mul-1 95.1%

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

       4. distribute-rgt-out 95.1%

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

       5. metadata-eval 95.1%

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

   14. Simplified 95.1%

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

   if -6.4999999999999997e91 < b

   1. Initial program 76.6%

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

      1. sqr-neg 76.6%

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

      2. sqr-neg 76.6%

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

      3. associate-*l* 77.1%

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

      4. *-commutative 77.1%

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

      5. associate-/l* 77.1%

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

      6. sqr-neg 77.1%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in a around 0 67.9%

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

      1. distribute-lft-out-- 67.9%

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

      2. associate-/l* 69.3%

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

      3. fma-neg 69.3%

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

   7. Simplified 69.3%

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

   8. Taylor expanded in a around 0 69.3%

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

      1. associate-*r/ 69.3%

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

      2. mul-1-neg 69.3%

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

   10. Simplified 69.3%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{+91}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.3% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.4%

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

   1. sqr-neg 70.4%

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

   2. sqr-neg 70.4%

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

   3. associate-*l* 70.8%

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

   4. *-commutative 70.8%

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

   5. associate-/l* 70.8%

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

   6. sqr-neg 70.8%

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

3. Simplified 70.8%

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

5. Taylor expanded in a around 0 63.6%

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

   1. distribute-lft-out-- 63.6%

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

   2. associate-/l* 64.7%

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

   3. fma-neg 64.7%

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

7. Simplified 64.7%

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

8. Taylor expanded in b around -inf 62.9%

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

9. Taylor expanded in a around inf 63.0%

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

    1. +-commutative 63.0%

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

    2. mul-1-neg 63.0%

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

    3. unsub-neg 63.0%

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

11. Simplified 63.0%

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

12. Taylor expanded in c around 0 63.0%

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

    1. associate-*r/ 63.0%

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

    2. sub-neg 63.0%

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

    3. neg-mul-1 63.0%

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

    4. distribute-rgt-out 63.0%

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

    5. metadata-eval 63.0%

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

14. Simplified 63.0%

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

15. Final simplification 63.0%

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

Reproduce

herbie shell --seed 2024101 
(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)))))))