jeff quadratic root 2

Percentage Accurate: 72.5% → 89.8%

Time: 17.8s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ \mathbf{if}\;b \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-281}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-1, \frac{b}{c}, \frac{a}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{+91}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot \left(-2\right)}{b + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{a \cdot -2}{b}}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0))))))
   (if (<= b -1e+154)
     (if (>= b 0.0) (/ c (- b)) (/ b (- a)))
     (if (<= b 2.8e-281)
       (if (>= b 0.0)
         (/ 1.0 (fma -1.0 (/ b c) (/ a b)))
         (/ (- t_0 b) (* a 2.0)))
       (if (<= b 9e+91)
         (if (>= b 0.0) (/ (* c (- 2.0)) (+ b t_0)) (/ (* b -2.0) (* a 2.0)))
         (if (>= b 0.0)
           (/ (* c 2.0) (* 2.0 (fma a (/ c b) (- b))))
           (/ (* c (/ (* a -2.0) b)) (* a 2.0))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -1e+154) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c / -b;
		} else {
			tmp_2 = b / -a;
		}
		tmp_1 = tmp_2;
	} else if (b <= 2.8e-281) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = 1.0 / fma(-1.0, (b / c), (a / b));
		} else {
			tmp_3 = (t_0 - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b <= 9e+91) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c * -2.0) / (b + t_0);
		} else {
			tmp_4 = (b * -2.0) / (a * 2.0);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c * 2.0) / (2.0 * fma(a, (c / b), -b));
	} else {
		tmp_1 = (c * ((a * -2.0) / b)) / (a * 2.0);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	tmp_1 = 0.0
	if (b <= -1e+154)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c / Float64(-b));
		else
			tmp_2 = Float64(b / Float64(-a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 2.8e-281)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(1.0 / fma(-1.0, Float64(b / c), Float64(a / b)));
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(a * 2.0));
		end
		tmp_1 = tmp_3;
	elseif (b <= 9e+91)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(c * Float64(-2.0)) / Float64(b + t_0));
		else
			tmp_4 = Float64(Float64(b * -2.0) / Float64(a * 2.0));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c * 2.0) / Float64(2.0 * fma(a, Float64(c / b), Float64(-b))));
	else
		tmp_1 = Float64(Float64(c * Float64(Float64(a * -2.0) / b)) / Float64(a * 2.0));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1e+154], If[GreaterEqual[b, 0.0], N[(c / (-b)), $MachinePrecision], N[(b / (-a)), $MachinePrecision]], If[LessEqual[b, 2.8e-281], If[GreaterEqual[b, 0.0], N[(1.0 / N[(-1.0 * N[(b / c), $MachinePrecision] + N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 9e+91], If[GreaterEqual[b, 0.0], N[(N[(c * (-2.0)), $MachinePrecision] / N[(b + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(b * -2.0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[(2.0 * N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(a * -2.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.00000000000000004e154

   1. Initial program 49.0%

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

   3. Simplified 49.1%

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

   5. Taylor expanded in b around inf 49.1%

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

   6. Taylor expanded in b around -inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around 0 100.0%

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

       1. associate-*r/ 100.0%

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

       2. neg-mul-1 100.0%

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

       3. associate-*r/ 100.0%

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

       4. neg-mul-1 100.0%

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

   11. Simplified 100.0%

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

   if -1.00000000000000004e154 < b < 2.80000000000000005e-281

   1. Initial program 90.4%

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

      1. add-sqr-sqrt 90.2%

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

      2. pow2 90.2%

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

      3. *-commutative 90.2%

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

   4. Applied egg-rr 90.2%

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

      1. clear-num 90.2%

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

      2. inv-pow 90.2%

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

      3. pow2 90.2%

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

      4. *-commutative 90.2%

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

      5. *-commutative 90.2%

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

      6. unpow2 90.2%

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

      7. add-sqr-sqrt 90.4%

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

   6. Applied egg-rr 90.4%

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

      1. unpow-1 90.4%

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

      2. associate-*r* 90.4%

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

      3. *-commutative 90.4%

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

      4. *-commutative 90.4%

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

      5. cancel-sign-sub-inv 90.4%

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

      6. metadata-eval 90.4%

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

      7. +-commutative 90.4%

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

      8. fma-define 90.4%

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

      9. *-commutative 90.4%

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

   8. Simplified 90.4%

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

   9. Taylor expanded in a around 0 90.4%

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

       1. fma-define 90.4%

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

   11. Simplified 90.4%

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

   if 2.80000000000000005e-281 < b < 9e91

   1. Initial program 87.1%

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

   3. Taylor expanded in b around -inf 87.1%

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

      1. *-commutative 87.1%

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

   5. Simplified 87.1%

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

   if 9e91 < b

   1. Initial program 59.5%

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

   3. Taylor expanded in a around 0 92.4%

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

      1. distribute-lft-out-- 92.4%

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

      2. associate-/l* 98.5%

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

      3. fma-neg 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in b around inf 98.5%

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

      1. associate-*r/ 98.5%

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

      2. associate-*r* 98.5%

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

      3. *-rgt-identity 98.5%

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

      4. times-frac 98.5%

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

      5. /-rgt-identity 98.5%

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

   8. Simplified 98.5%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-281}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-1, \frac{b}{c}, \frac{a}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{+91}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot \left(-2\right)}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{a \cdot -2}{b}}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \mathbf{if}\;b \leq -2 \cdot 10^{+153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-281}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-1, \frac{b}{c}, \frac{a}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-67}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -1.0 (/ b a) (/ c b))))
   (if (<= b -2e+153)
     (if (>= b 0.0) (/ c (- b)) (/ b (- a)))
     (if (<= b 2.8e-281)
       (if (>= b 0.0)
         (/ 1.0 (fma -1.0 (/ b c) (/ a b)))
         (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)))
       (if (<= b 2.6e-67)
         (if (>= b 0.0) (/ (* c 2.0) (- (- b) (sqrt (* c (* a -4.0))))) t_0)
         (if (>= b 0.0) (/ (* c 2.0) (* 2.0 (- (* a (/ c b)) b))) t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-1.0, (b / a), (c / b));
	double tmp_1;
	if (b <= -2e+153) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c / -b;
		} else {
			tmp_2 = b / -a;
		}
		tmp_1 = tmp_2;
	} else if (b <= 2.8e-281) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = 1.0 / fma(-1.0, (b / c), (a / b));
		} else {
			tmp_3 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b <= 2.6e-67) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c * 2.0) / (-b - sqrt((c * (a * -4.0))));
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c * 2.0) / (2.0 * ((a * (c / b)) - b));
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = fma(-1.0, Float64(b / a), Float64(c / b))
	tmp_1 = 0.0
	if (b <= -2e+153)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c / Float64(-b));
		else
			tmp_2 = Float64(b / Float64(-a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 2.8e-281)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(1.0 / fma(-1.0, Float64(b / c), Float64(a / b)));
		else
			tmp_3 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
		end
		tmp_1 = tmp_3;
	elseif (b <= 2.6e-67)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(c * 2.0) / Float64(Float64(-b) - sqrt(Float64(c * Float64(a * -4.0)))));
		else
			tmp_4 = t_0;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c * 2.0) / Float64(2.0 * Float64(Float64(a * Float64(c / b)) - b)));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-1.0 * N[(b / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2e+153], If[GreaterEqual[b, 0.0], N[(c / (-b)), $MachinePrecision], N[(b / (-a)), $MachinePrecision]], If[LessEqual[b, 2.8e-281], If[GreaterEqual[b, 0.0], N[(1.0 / N[(-1.0 * N[(b / c), $MachinePrecision] + N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 2.6e-67], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[(2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2e153

   1. Initial program 49.0%

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

   3. Simplified 49.1%

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

   5. Taylor expanded in b around inf 49.1%

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

   6. Taylor expanded in b around -inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around 0 100.0%

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

       1. associate-*r/ 100.0%

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

       2. neg-mul-1 100.0%

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

       3. associate-*r/ 100.0%

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

       4. neg-mul-1 100.0%

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

   11. Simplified 100.0%

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

   if -2e153 < b < 2.80000000000000005e-281

   1. Initial program 90.4%

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

      1. add-sqr-sqrt 90.2%

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

      2. pow2 90.2%

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

      3. *-commutative 90.2%

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

   4. Applied egg-rr 90.2%

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

      1. clear-num 90.2%

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

      2. inv-pow 90.2%

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

      3. pow2 90.2%

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

      4. *-commutative 90.2%

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

      5. *-commutative 90.2%

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

      6. unpow2 90.2%

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

      7. add-sqr-sqrt 90.4%

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

   6. Applied egg-rr 90.4%

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

      1. unpow-1 90.4%

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

      2. associate-*r* 90.4%

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

      3. *-commutative 90.4%

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

      4. *-commutative 90.4%

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

      5. cancel-sign-sub-inv 90.4%

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

      6. metadata-eval 90.4%

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

      7. +-commutative 90.4%

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

      8. fma-define 90.4%

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

      9. *-commutative 90.4%

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

   8. Simplified 90.4%

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

   9. Taylor expanded in a around 0 90.4%

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

       1. fma-define 90.4%

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

   11. Simplified 90.4%

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

   if 2.80000000000000005e-281 < b < 2.5999999999999999e-67

   1. Initial program 80.3%

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

   3. Taylor expanded in b around -inf 80.3%

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

      1. mul-1-neg 80.3%

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

      2. *-commutative 80.3%

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

      3. distribute-rgt-neg-in 80.3%

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

      4. associate-/l* 80.3%

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

   5. Simplified 80.3%

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

   6. Taylor expanded in a around inf 80.3%

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

      1. fma-define 80.3%

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

   8. Simplified 80.3%

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

   9. Taylor expanded in b around 0 67.8%

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

       1. associate-*r* 67.8%

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

       2. *-commutative 67.8%

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

   11. Simplified 67.8%

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

   if 2.5999999999999999e-67 < b

   1. Initial program 71.8%

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

   3. Taylor expanded in b around -inf 71.8%

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

      1. mul-1-neg 71.8%

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

      2. *-commutative 71.8%

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

      3. distribute-rgt-neg-in 71.8%

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

      4. associate-/l* 71.8%

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

   5. Simplified 71.8%

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

   6. Taylor expanded in a around inf 71.8%

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

      1. fma-define 71.8%

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

   8. Simplified 71.8%

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

   9. Taylor expanded in a around 0 80.7%

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

       1. distribute-lft-out-- 80.7%

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

       2. associate-/l* 84.7%

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

   11. Simplified 84.7%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-281}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-1, \frac{b}{c}, \frac{a}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-67}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ \mathbf{if}\;b \leq -4 \cdot 10^{+153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+92}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot \left(-2\right)}{b + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{a \cdot -2}{b}}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0))))))
   (if (<= b -4e+153)
     (if (>= b 0.0) (/ c (- b)) (/ b (- a)))
     (if (<= b 1.6e+92)
       (if (>= b 0.0) (/ (* c (- 2.0)) (+ b t_0)) (/ (- t_0 b) (* a 2.0)))
       (if (>= b 0.0)
         (/ (* c 2.0) (* 2.0 (fma a (/ c b) (- b))))
         (/ (* c (/ (* a -2.0) b)) (* a 2.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -4e+153) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c / -b;
		} else {
			tmp_2 = b / -a;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.6e+92) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c * -2.0) / (b + t_0);
		} else {
			tmp_3 = (t_0 - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c * 2.0) / (2.0 * fma(a, (c / b), -b));
	} else {
		tmp_1 = (c * ((a * -2.0) / b)) / (a * 2.0);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	tmp_1 = 0.0
	if (b <= -4e+153)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c / Float64(-b));
		else
			tmp_2 = Float64(b / Float64(-a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.6e+92)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(c * Float64(-2.0)) / Float64(b + t_0));
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(a * 2.0));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c * 2.0) / Float64(2.0 * fma(a, Float64(c / b), Float64(-b))));
	else
		tmp_1 = Float64(Float64(c * Float64(Float64(a * -2.0) / b)) / Float64(a * 2.0));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -4e+153], If[GreaterEqual[b, 0.0], N[(c / (-b)), $MachinePrecision], N[(b / (-a)), $MachinePrecision]], If[LessEqual[b, 1.6e+92], If[GreaterEqual[b, 0.0], N[(N[(c * (-2.0)), $MachinePrecision] / N[(b + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[(2.0 * N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(a * -2.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -4e153

   1. Initial program 49.0%

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

   3. Simplified 49.1%

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

   5. Taylor expanded in b around inf 49.1%

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

   6. Taylor expanded in b around -inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around 0 100.0%

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

       1. associate-*r/ 100.0%

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

       2. neg-mul-1 100.0%

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

       3. associate-*r/ 100.0%

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

       4. neg-mul-1 100.0%

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

   11. Simplified 100.0%

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

   if -4e153 < b < 1.60000000000000013e92

   1. Initial program 88.8%

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

   if 1.60000000000000013e92 < b

   1. Initial program 59.5%

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

   3. Taylor expanded in a around 0 92.4%

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

      1. distribute-lft-out-- 92.4%

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

      2. associate-/l* 98.5%

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

      3. fma-neg 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in b around inf 98.5%

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

      1. associate-*r/ 98.5%

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

      2. associate-*r* 98.5%

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

      3. *-rgt-identity 98.5%

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

      4. times-frac 98.5%

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

      5. /-rgt-identity 98.5%

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

   8. Simplified 98.5%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+92}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot \left(-2\right)}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{a \cdot -2}{b}}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \mathbf{if}\;b \leq 1.85 \cdot 10^{-67}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -1.0 (/ b a) (/ c b))))
   (if (<= b 1.85e-67)
     (if (>= b 0.0) (/ (* c 2.0) (- (- b) (sqrt (* c (* a -4.0))))) t_0)
     (if (>= b 0.0) (/ (* c 2.0) (* 2.0 (- (* a (/ c b)) b))) t_0))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-1.0, (b / a), (c / b));
	double tmp_1;
	if (b <= 1.85e-67) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c * 2.0) / (-b - sqrt((c * (a * -4.0))));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (c * 2.0) / (2.0 * ((a * (c / b)) - b));
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = fma(-1.0, Float64(b / a), Float64(c / b))
	tmp_1 = 0.0
	if (b <= 1.85e-67)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(c * 2.0) / Float64(Float64(-b) - sqrt(Float64(c * Float64(a * -4.0)))));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c * 2.0) / Float64(2.0 * Float64(Float64(a * Float64(c / b)) - b)));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-1.0 * N[(b / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.85e-67], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[(2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.85e-67

   1. Initial program 77.7%

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

   3. Taylor expanded in b around -inf 69.7%

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

      1. mul-1-neg 69.7%

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

      2. *-commutative 69.7%

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

      3. distribute-rgt-neg-in 69.7%

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

      4. associate-/l* 71.1%

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

   5. Simplified 71.1%

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

   6. Taylor expanded in a around inf 71.8%

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

      1. fma-define 71.8%

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

   8. Simplified 71.8%

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

   9. Taylor expanded in b around 0 68.8%

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

       1. associate-*r* 68.8%

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

       2. *-commutative 68.8%

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

   11. Simplified 68.8%

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

   if 1.85e-67 < b

   1. Initial program 71.8%

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

   3. Taylor expanded in b around -inf 71.8%

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

      1. mul-1-neg 71.8%

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

      2. *-commutative 71.8%

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

      3. distribute-rgt-neg-in 71.8%

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

      4. associate-/l* 71.8%

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

   5. Simplified 71.8%

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

   6. Taylor expanded in a around inf 71.8%

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

      1. fma-define 71.8%

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

   8. Simplified 71.8%

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

   9. Taylor expanded in a around 0 80.7%

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

       1. distribute-lft-out-- 80.7%

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

       2. associate-/l* 84.7%

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

   11. Simplified 84.7%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.85 \cdot 10^{-67}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \mathbf{if}\;b \leq 2.65 \cdot 10^{-67}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{\left(-b\right) - \sqrt{a \cdot \left(c \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -1.0 (/ b a) (/ c b))))
   (if (<= b 2.65e-67)
     (if (>= b 0.0) (* c (/ 2.0 (- (- b) (sqrt (* a (* c -4.0)))))) t_0)
     (if (>= b 0.0) (/ (* c 2.0) (* 2.0 (- (* a (/ c b)) b))) t_0))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-1.0, (b / a), (c / b));
	double tmp_1;
	if (b <= 2.65e-67) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (2.0 / (-b - sqrt((a * (c * -4.0)))));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (c * 2.0) / (2.0 * ((a * (c / b)) - b));
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = fma(-1.0, Float64(b / a), Float64(c / b))
	tmp_1 = 0.0
	if (b <= 2.65e-67)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * Float64(2.0 / Float64(Float64(-b) - sqrt(Float64(a * Float64(c * -4.0))))));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c * 2.0) / Float64(2.0 * Float64(Float64(a * Float64(c / b)) - b)));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-1.0 * N[(b / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 2.65e-67], If[GreaterEqual[b, 0.0], N[(c * N[(2.0 / N[((-b) - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[(2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.64999999999999986e-67

   1. Initial program 77.7%

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

   3. Taylor expanded in b around -inf 69.7%

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

      1. mul-1-neg 69.7%

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

      2. *-commutative 69.7%

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

      3. distribute-rgt-neg-in 69.7%

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

      4. associate-/l* 71.1%

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

   5. Simplified 71.1%

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

   6. Taylor expanded in a around inf 71.8%

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

      1. fma-define 71.8%

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

   8. Simplified 71.8%

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

   9. Taylor expanded in b around 0 68.8%

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

       1. associate-*r* 68.8%

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

       2. *-commutative 68.8%

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

   11. Simplified 68.8%

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

       1. *-un-lft-identity 68.8%

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

       2. *-commutative 68.8%

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

   13. Applied egg-rr 68.8%

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

       1. *-lft-identity 68.8%

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

       2. associate-/l* 68.8%

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

       3. associate-*r* 68.8%

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

   15. Simplified 68.8%

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

   if 2.64999999999999986e-67 < b

   1. Initial program 71.8%

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

   3. Taylor expanded in b around -inf 71.8%

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

      1. mul-1-neg 71.8%

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

      2. *-commutative 71.8%

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

      3. distribute-rgt-neg-in 71.8%

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

      4. associate-/l* 71.8%

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

   5. Simplified 71.8%

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

   6. Taylor expanded in a around inf 71.8%

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

      1. fma-define 71.8%

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

   8. Simplified 71.8%

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

   9. Taylor expanded in a around 0 80.7%

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

       1. distribute-lft-out-- 80.7%

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

       2. associate-/l* 84.7%

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

   11. Simplified 84.7%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.65 \cdot 10^{-67}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{\left(-b\right) - \sqrt{a \cdot \left(c \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.8% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0)
   (/ (* c 2.0) (* 2.0 (- (* a (/ c b)) b)))
   (fma -1.0 (/ b a) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (c * 2.0) / (2.0 * ((a * (c / b)) - b));
	} else {
		tmp = fma(-1.0, (b / a), (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(c * 2.0) / Float64(2.0 * Float64(Float64(a * Float64(c / b)) - b)));
	else
		tmp = fma(-1.0, Float64(b / a), Float64(c / b));
	end
	return tmp
end

Wolfram

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

Derivation

1. Initial program 75.4%

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

3. Taylor expanded in b around -inf 70.5%

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

   1. mul-1-neg 70.5%

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

   2. *-commutative 70.5%

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

   3. distribute-rgt-neg-in 70.5%

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

   4. associate-/l* 71.3%

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

5. Simplified 71.3%

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

6. Taylor expanded in a around inf 71.8%

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

   1. fma-define 71.8%

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

8. Simplified 71.8%

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

9. Taylor expanded in a around 0 67.1%

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

    1. distribute-lft-out-- 67.1%

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

    2. associate-/l* 68.8%

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

11. Simplified 68.8%

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

12. Final simplification 68.8%

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

Alternative 7: 68.6% accurate, 13.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.4%

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

3. Simplified 75.4%

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

5. Taylor expanded in b around inf 71.9%

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

6. Taylor expanded in b around -inf 68.3%

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

   1. *-commutative 68.3%

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

8. Simplified 68.3%

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

9. Taylor expanded in b around 0 68.4%

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

    1. associate-*r/ 68.4%

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

    2. neg-mul-1 68.4%

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

    3. associate-*r/ 68.4%

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

    4. neg-mul-1 68.4%

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

11. Simplified 68.4%

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

12. Final simplification 68.4%

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

Reproduce

herbie shell --seed 2024129 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))