jeff quadratic root 1

Percentage Accurate: 72.4% → 89.9%

Time: 31.8s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ t_1 := \frac{-c}{b}\\ \mathbf{if}\;b \leq -1.5 \cdot 10^{+139}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{-1}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\\ \mathbf{elif}\;b \leq 2.55 \cdot 10^{+25}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + t\_0}{a \cdot \left(-2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \left(-2 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0))))) (t_1 (/ (- c) b)))
   (if (<= b -1.5e+139)
     (if (>= b 0.0) (* -0.5 (/ -1.0 a)) t_1)
     (if (<= b 2.55e+25)
       (if (>= b 0.0) (/ (+ b t_0) (* a (- 2.0))) (/ (* c 2.0) (- t_0 b)))
       (if (>= b 0.0) (* -0.5 (+ (* -2.0 (/ c b)) (* 2.0 (/ b a)))) t_1)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double t_1 = -c / b;
	double tmp_1;
	if (b <= -1.5e+139) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -0.5 * (-1.0 / a);
		} else {
			tmp_2 = t_1;
		}
		tmp_1 = tmp_2;
	} else if (b <= 2.55e+25) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (b + t_0) / (a * -2.0);
		} else {
			tmp_3 = (c * 2.0) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -0.5 * ((-2.0 * (c / b)) + (2.0 * (b / a)));
	} else {
		tmp_1 = t_1;
	}
	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
    t_0 = sqrt(((b * b) - (c * (a * 4.0d0))))
    t_1 = -c / b
    if (b <= (-1.5d+139)) then
        if (b >= 0.0d0) then
            tmp_2 = (-0.5d0) * ((-1.0d0) / a)
        else
            tmp_2 = t_1
        end if
        tmp_1 = tmp_2
    else if (b <= 2.55d+25) then
        if (b >= 0.0d0) then
            tmp_3 = (b + t_0) / (a * -2.0d0)
        else
            tmp_3 = (c * 2.0d0) / (t_0 - b)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = (-0.5d0) * (((-2.0d0) * (c / b)) + (2.0d0 * (b / a)))
    else
        tmp_1 = t_1
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (c * (a * 4.0))));
	double t_1 = -c / b;
	double tmp_1;
	if (b <= -1.5e+139) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -0.5 * (-1.0 / a);
		} else {
			tmp_2 = t_1;
		}
		tmp_1 = tmp_2;
	} else if (b <= 2.55e+25) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (b + t_0) / (a * -2.0);
		} else {
			tmp_3 = (c * 2.0) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -0.5 * ((-2.0 * (c / b)) + (2.0 * (b / a)));
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (c * (a * 4.0))))
	t_1 = -c / b
	tmp_1 = 0
	if b <= -1.5e+139:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -0.5 * (-1.0 / a)
		else:
			tmp_2 = t_1
		tmp_1 = tmp_2
	elif b <= 2.55e+25:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (b + t_0) / (a * -2.0)
		else:
			tmp_3 = (c * 2.0) / (t_0 - b)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = -0.5 * ((-2.0 * (c / b)) + (2.0 * (b / a)))
	else:
		tmp_1 = t_1
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	t_1 = Float64(Float64(-c) / b)
	tmp_1 = 0.0
	if (b <= -1.5e+139)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-0.5 * Float64(-1.0 / a));
		else
			tmp_2 = t_1;
		end
		tmp_1 = tmp_2;
	elseif (b <= 2.55e+25)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(b + t_0) / Float64(a * Float64(-2.0)));
		else
			tmp_3 = Float64(Float64(c * 2.0) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(-0.5 * Float64(Float64(-2.0 * Float64(c / b)) + Float64(2.0 * Float64(b / a))));
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	t_1 = -c / b;
	tmp_2 = 0.0;
	if (b <= -1.5e+139)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -0.5 * (-1.0 / a);
		else
			tmp_3 = t_1;
		end
		tmp_2 = tmp_3;
	elseif (b <= 2.55e+25)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (b + t_0) / (a * -2.0);
		else
			tmp_4 = (c * 2.0) / (t_0 - b);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = -0.5 * ((-2.0 * (c / b)) + (2.0 * (b / a)));
	else
		tmp_2 = t_1;
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-c) / b), $MachinePrecision]}, If[LessEqual[b, -1.5e+139], If[GreaterEqual[b, 0.0], N[(-0.5 * N[(-1.0 / a), $MachinePrecision]), $MachinePrecision], t$95$1], If[LessEqual[b, 2.55e+25], If[GreaterEqual[b, 0.0], N[(N[(b + t$95$0), $MachinePrecision] / N[(a * (-2.0)), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(-0.5 * N[(N[(-2.0 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.5e139

   1. Initial program 46.8%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in b around -inf 98.3%

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

      1. mul-1-neg 98.3%

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

      2. distribute-neg-frac2 98.3%

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

   7. Simplified 98.3%

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

      1. clear-num 98.3%

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

      2. inv-pow 98.3%

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

      3. fma-undefine 98.3%

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

      4. pow2 98.3%

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

      5. fma-undefine 98.3%

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

   9. Applied egg-rr 98.3%

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

       1. unpow-1 98.3%

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

   11. Simplified 98.3%

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

   12. Taylor expanded in c around 0 98.3%

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

       1. clear-num 98.3%

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

       2. *-un-lft-identity 98.3%

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

       3. flip-+ 98.3%

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

       4. +-inverses 98.3%

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

       5. +-inverses 98.3%

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

   14. Applied egg-rr 98.3%

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

   16. Simplified 98.3%

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

   if -1.5e139 < b < 2.5500000000000002e25

   1. Initial program 88.7%

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

   if 2.5500000000000002e25 < b

   1. Initial program 65.7%

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

   3. Simplified 65.8%

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

   5. Taylor expanded in b around -inf 65.8%

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

      1. mul-1-neg 65.8%

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

      2. distribute-neg-frac2 65.8%

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

   7. Simplified 65.8%

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

   8. Taylor expanded in c around 0 95.9%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+139}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{-1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.55 \cdot 10^{+25}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot \left(-2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \left(-2 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{+139}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{-1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.02e+139)
   (if (>= b 0.0) (* -0.5 (/ -1.0 a)) (/ (- c) b))
   (if (>= b 0.0)
     (* b (+ (/ c (pow b 2.0)) (/ -1.0 a)))
     (/ (* c 2.0) (- (sqrt (- (* b b) (* c (* a 4.0)))) b)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1.02e+139) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -0.5 * (-1.0 / a);
		} else {
			tmp_2 = -c / b;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = b * ((c / pow(b, 2.0)) + (-1.0 / a));
	} else {
		tmp_1 = (c * 2.0) / (sqrt(((b * b) - (c * (a * 4.0)))) - b);
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    if (b <= (-1.02d+139)) then
        if (b >= 0.0d0) then
            tmp_2 = (-0.5d0) * ((-1.0d0) / a)
        else
            tmp_2 = -c / b
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = b * ((c / (b ** 2.0d0)) + ((-1.0d0) / a))
    else
        tmp_1 = (c * 2.0d0) / (sqrt(((b * b) - (c * (a * 4.0d0)))) - b)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1.02e+139) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -0.5 * (-1.0 / a);
		} else {
			tmp_2 = -c / b;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = b * ((c / Math.pow(b, 2.0)) + (-1.0 / a));
	} else {
		tmp_1 = (c * 2.0) / (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if b <= -1.02e+139:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -0.5 * (-1.0 / a)
		else:
			tmp_2 = -c / b
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = b * ((c / math.pow(b, 2.0)) + (-1.0 / a))
	else:
		tmp_1 = (c * 2.0) / (math.sqrt(((b * b) - (c * (a * 4.0)))) - b)
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -1.02e+139)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-0.5 * Float64(-1.0 / a));
		else
			tmp_2 = Float64(Float64(-c) / b);
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(b * Float64(Float64(c / (b ^ 2.0)) + Float64(-1.0 / a)));
	else
		tmp_1 = Float64(Float64(c * 2.0) / Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp_2 = 0.0;
	if (b <= -1.02e+139)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -0.5 * (-1.0 / a);
		else
			tmp_3 = -c / b;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = b * ((c / (b ^ 2.0)) + (-1.0 / a));
	else
		tmp_2 = (c * 2.0) / (sqrt(((b * b) - (c * (a * 4.0)))) - b);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.02e+139], If[GreaterEqual[b, 0.0], N[(-0.5 * N[(-1.0 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(b * N[(N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.02e139

   1. Initial program 46.8%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in b around -inf 98.3%

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

      1. mul-1-neg 98.3%

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

      2. distribute-neg-frac2 98.3%

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

   7. Simplified 98.3%

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

      1. clear-num 98.3%

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

      2. inv-pow 98.3%

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

      3. fma-undefine 98.3%

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

      4. pow2 98.3%

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

      5. fma-undefine 98.3%

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

   9. Applied egg-rr 98.3%

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

       1. unpow-1 98.3%

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

   11. Simplified 98.3%

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

   12. Taylor expanded in c around 0 98.3%

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

       1. clear-num 98.3%

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

       2. *-un-lft-identity 98.3%

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

       3. flip-+ 98.3%

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

       4. +-inverses 98.3%

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

       5. +-inverses 98.3%

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

   14. Applied egg-rr 98.3%

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

   16. Simplified 98.3%

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

   if -1.02e139 < b

   1. Initial program 81.1%

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

   3. Taylor expanded in a around 0 75.0%

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

      1. distribute-lft-out-- 75.0%

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

      2. associate-/l* 77.5%

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

   5. Simplified 77.5%

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

   6. Taylor expanded in b around -inf 76.3%

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

      1. mul-1-neg 76.3%

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

      2. *-commutative 76.3%

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

      3. distribute-rgt-neg-in 76.3%

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

      4. +-commutative 76.3%

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

      5. mul-1-neg 76.3%

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

      6. unsub-neg 76.3%

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

   8. Simplified 76.3%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{+139}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{-1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+140}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{-1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+140)
   (if (>= b 0.0) (* -0.5 (/ -1.0 a)) (/ (- c) b))
   (if (>= b 0.0)
     (/ (* 2.0 (- (* a (/ c b)) b)) (* a 2.0))
     (/ (* c 2.0) (- (sqrt (- (* b b) (* c (* a 4.0)))) b)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1e+140) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -0.5 * (-1.0 / a);
		} else {
			tmp_2 = -c / b;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * ((a * (c / b)) - b)) / (a * 2.0);
	} else {
		tmp_1 = (c * 2.0) / (sqrt(((b * b) - (c * (a * 4.0)))) - b);
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    if (b <= (-1d+140)) then
        if (b >= 0.0d0) then
            tmp_2 = (-0.5d0) * ((-1.0d0) / a)
        else
            tmp_2 = -c / b
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (2.0d0 * ((a * (c / b)) - b)) / (a * 2.0d0)
    else
        tmp_1 = (c * 2.0d0) / (sqrt(((b * b) - (c * (a * 4.0d0)))) - b)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1e+140) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -0.5 * (-1.0 / a);
		} else {
			tmp_2 = -c / b;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * ((a * (c / b)) - b)) / (a * 2.0);
	} else {
		tmp_1 = (c * 2.0) / (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if b <= -1e+140:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -0.5 * (-1.0 / a)
		else:
			tmp_2 = -c / b
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = (2.0 * ((a * (c / b)) - b)) / (a * 2.0)
	else:
		tmp_1 = (c * 2.0) / (math.sqrt(((b * b) - (c * (a * 4.0)))) - b)
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -1e+140)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-0.5 * Float64(-1.0 / a));
		else
			tmp_2 = Float64(Float64(-c) / b);
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * Float64(Float64(a * Float64(c / b)) - b)) / Float64(a * 2.0));
	else
		tmp_1 = Float64(Float64(c * 2.0) / Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp_2 = 0.0;
	if (b <= -1e+140)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -0.5 * (-1.0 / a);
		else
			tmp_3 = -c / b;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = (2.0 * ((a * (c / b)) - b)) / (a * 2.0);
	else
		tmp_2 = (c * 2.0) / (sqrt(((b * b) - (c * (a * 4.0)))) - b);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+140], If[GreaterEqual[b, 0.0], N[(-0.5 * N[(-1.0 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.00000000000000006e140

   1. Initial program 46.8%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in b around -inf 98.3%

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

      1. mul-1-neg 98.3%

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

      2. distribute-neg-frac2 98.3%

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

   7. Simplified 98.3%

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

      1. clear-num 98.3%

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

      2. inv-pow 98.3%

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

      3. fma-undefine 98.3%

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

      4. pow2 98.3%

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

      5. fma-undefine 98.3%

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

   9. Applied egg-rr 98.3%

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

       1. unpow-1 98.3%

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

   11. Simplified 98.3%

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

   12. Taylor expanded in c around 0 98.3%

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

       1. clear-num 98.3%

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

       2. *-un-lft-identity 98.3%

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

       3. flip-+ 98.3%

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

       4. +-inverses 98.3%

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

       5. +-inverses 98.3%

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

   14. Applied egg-rr 98.3%

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

   16. Simplified 98.3%

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

   if -1.00000000000000006e140 < b

   1. Initial program 81.1%

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

   3. Taylor expanded in a around 0 75.0%

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

      1. distribute-lft-out-- 75.0%

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

      2. associate-/l* 77.5%

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

   5. Simplified 77.5%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+140}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{-1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.1% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(-0.5 * Float64(Float64(-2.0 * Float64(c / b)) + Float64(2.0 * Float64(b / 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 = -0.5 * ((-2.0 * (c / b)) + (2.0 * (b / a)));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Initial program 73.8%

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

3. Simplified 73.8%

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

5. Taylor expanded in b around -inf 69.2%

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

   1. mul-1-neg 69.2%

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

   2. distribute-neg-frac2 69.2%

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

7. Simplified 69.2%

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

8. Taylor expanded in c around 0 66.4%

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

9. Final simplification 66.4%

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

Alternative 5: 67.8% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(-0.5 * Float64(b * Float64(2.0 / 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 = -0.5 * (b * (2.0 / a));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Initial program 73.8%

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

3. Simplified 73.8%

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

5. Taylor expanded in b around -inf 69.2%

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

   1. mul-1-neg 69.2%

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

   2. distribute-neg-frac2 69.2%

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

7. Simplified 69.2%

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

   1. clear-num 69.1%

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

   2. inv-pow 69.1%

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

   3. fma-undefine 69.1%

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

   4. pow2 69.1%

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

   5. fma-undefine 69.1%

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

9. Applied egg-rr 69.1%

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

    1. unpow-1 69.1%

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

11. Simplified 69.1%

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

12. Taylor expanded in c around 0 66.2%

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

    1. associate-/r/ 66.1%

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

    2. count-2 66.1%

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

    3. associate-*r* 66.1%

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

14. Applied egg-rr 66.1%

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

16. Simplified 66.1%

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

17. Final simplification 66.1%

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

Alternative 6: 67.9% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(-0.5 * Float64(Float64(b + b) / 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 = -0.5 * ((b + b) / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Initial program 73.8%

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

3. Simplified 73.8%

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

5. Taylor expanded in b around -inf 69.2%

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

   1. mul-1-neg 69.2%

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

   2. distribute-neg-frac2 69.2%

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

7. Simplified 69.2%

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

8. Taylor expanded in c around 0 66.2%

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

9. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
10. Add Preprocessing

Alternative 7: 34.4% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{-1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0) (* -0.5 (/ -1.0 a)) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -0.5 * (-1.0 / 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 = (-0.5d0) * ((-1.0d0) / 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 = -0.5 * (-1.0 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = -0.5 * (-1.0 / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(-0.5 * Float64(-1.0 / 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 = -0.5 * (-1.0 / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(-0.5 * N[(-1.0 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Initial program 73.8%

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

3. Simplified 73.8%

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

5. Taylor expanded in b around -inf 69.2%

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

   1. mul-1-neg 69.2%

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

   2. distribute-neg-frac2 69.2%

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

7. Simplified 69.2%

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

   1. clear-num 69.1%

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

   2. inv-pow 69.1%

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

   3. fma-undefine 69.1%

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

   4. pow2 69.1%

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

   5. fma-undefine 69.1%

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

9. Applied egg-rr 69.1%

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

    1. unpow-1 69.1%

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

11. Simplified 69.1%

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

12. Taylor expanded in c around 0 66.2%

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

    1. clear-num 66.2%

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

    2. *-un-lft-identity 66.2%

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

    3. flip-+ 34.3%

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

    4. +-inverses 34.3%

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

    5. +-inverses 34.3%

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

14. Applied egg-rr 34.3%

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

16. Simplified 34.9%

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

17. Final simplification 34.9%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{-1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
18. Add Preprocessing

Reproduce

herbie shell --seed 2024080 
(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)))))))