jeff quadratic root 2

Percentage Accurate: 71.9% → 89.3%

Time: 22.0s

Alternatives: 5

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: 71.9% 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: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ \mathbf{if}\;b \leq -1.6 \cdot 10^{+22}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;b \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.28 \cdot 10^{+65}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0))))))
   (if (<= b -1.6e+22)
     (if (>= b 0.0) (* b (/ 1.0 a)) (- (/ c b) (/ b a)))
     (if (<= b 1.28e+65)
       (if (>= b 0.0) (/ (* c 2.0) (- (- b) t_0)) (/ (- t_0 b) (* a 2.0)))
       (if (>= b 0.0) (/ (- c) b) (/ (- b) a))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -1.6e+22) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b * (1.0 / a);
		} else {
			tmp_2 = (c / b) - (b / a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.28e+65) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c * 2.0) / (-b - t_0);
		} else {
			tmp_3 = (t_0 - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -c / b;
	} else {
		tmp_1 = -b / a;
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = sqrt(((b * b) - (c * (a * 4.0d0))))
    if (b <= (-1.6d+22)) then
        if (b >= 0.0d0) then
            tmp_2 = b * (1.0d0 / a)
        else
            tmp_2 = (c / b) - (b / a)
        end if
        tmp_1 = tmp_2
    else if (b <= 1.28d+65) then
        if (b >= 0.0d0) then
            tmp_3 = (c * 2.0d0) / (-b - t_0)
        else
            tmp_3 = (t_0 - b) / (a * 2.0d0)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = -c / b
    else
        tmp_1 = -b / a
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -1.6e+22) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b * (1.0 / a);
		} else {
			tmp_2 = (c / b) - (b / a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.28e+65) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c * 2.0) / (-b - t_0);
		} else {
			tmp_3 = (t_0 - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -c / b;
	} else {
		tmp_1 = -b / a;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (c * (a * 4.0))))
	tmp_1 = 0
	if b <= -1.6e+22:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = b * (1.0 / a)
		else:
			tmp_2 = (c / b) - (b / a)
		tmp_1 = tmp_2
	elif b <= 1.28e+65:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (c * 2.0) / (-b - t_0)
		else:
			tmp_3 = (t_0 - b) / (a * 2.0)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = -c / b
	else:
		tmp_1 = -b / a
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	tmp_1 = 0.0
	if (b <= -1.6e+22)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(b * Float64(1.0 / a));
		else
			tmp_2 = Float64(Float64(c / b) - Float64(b / a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.28e+65)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(c * 2.0) / Float64(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) / b);
	else
		tmp_1 = Float64(Float64(-b) / a);
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	tmp_2 = 0.0;
	if (b <= -1.6e+22)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = b * (1.0 / a);
		else
			tmp_3 = (c / b) - (b / a);
		end
		tmp_2 = tmp_3;
	elseif (b <= 1.28e+65)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (c * 2.0) / (-b - t_0);
		else
			tmp_4 = (t_0 - b) / (a * 2.0);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = -c / b;
	else
		tmp_2 = -b / a;
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.6e+22], If[GreaterEqual[b, 0.0], N[(b * N[(1.0 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.28e+65], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[((-c) / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.6e22

   1. Initial program 63.8%

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

   3. Simplified 63.8%

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

   4. Taylor expanded in b around inf 63.8%

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

      1. +-commutative 63.8%

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

      2. *-commutative 63.8%

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

      3. fma-def 63.8%

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

      4. associate-/l* 63.8%

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

      5. associate-*r/ 63.8%

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

      6. *-commutative 63.8%

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

   6. Simplified 63.8%

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

   7. Taylor expanded in b around 0 63.8%

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

   8. Taylor expanded in b around -inf 96.4%

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

      1. mul-1-neg 96.4%

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

      2. unsub-neg 96.4%

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

   10. Simplified 96.4%

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

       1. associate-/r* 96.4%

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

       2. metadata-eval 96.4%

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

       3. clear-num 96.4%

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

       4. div-inv 96.4%

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

   12. Applied egg-rr 96.4%

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

   if -1.6e22 < b < 1.28000000000000007e65

   1. Initial program 90.4%

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

   if 1.28000000000000007e65 < b

   1. Initial program 66.1%

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

      1. div-inv 66.0%

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

      2. *-commutative 66.0%

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

      3. *-commutative 66.0%

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

      4. *-commutative 66.0%

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

   3. Applied egg-rr 66.0%

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

   4. Taylor expanded in b around inf 84.9%

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

      1. +-commutative 84.9%

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

      2. *-commutative 84.9%

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

      3. fma-def 84.9%

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

      4. associate-/l* 95.3%

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

      5. associate-*r/ 95.3%

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

      6. *-commutative 95.3%

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

      7. *-rgt-identity 95.3%

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

      8. associate-*r/ 95.3%

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

      9. associate-*l* 95.3%

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

      10. associate-*r/ 95.3%

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

      11. metadata-eval 95.3%

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

      12. associate-/l* 95.3%

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

      13. associate-*r/ 95.3%

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

   6. Simplified 95.3%

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

   7. Taylor expanded in c around 0 95.6%

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

      1. associate-*r/ 95.6%

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

      2. neg-mul-1 95.6%

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

   9. Simplified 95.6%

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

   10. Taylor expanded in b around -inf 95.6%

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

       1. neg-mul-1 95.6%

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

       2. distribute-neg-frac 95.6%

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

   12. Simplified 95.6%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+22}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;b \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.28 \cdot 10^{+65}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 2: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ \mathbf{if}\;b \leq -1.6 \cdot 10^{+22}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;b \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.28 \cdot 10^{+65}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - t_0}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0))))))
   (if (<= b -1.6e+22)
     (if (>= b 0.0) (* b (/ 1.0 a)) (- (/ c b) (/ b a)))
     (if (<= b 1.28e+65)
       (if (>= b 0.0) (/ 2.0 (/ (- (- b) t_0) c)) (/ (- t_0 b) (* a 2.0)))
       (if (>= b 0.0) (/ (- c) b) (/ (- b) a))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -1.6e+22) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b * (1.0 / a);
		} else {
			tmp_2 = (c / b) - (b / a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.28e+65) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = 2.0 / ((-b - t_0) / c);
		} else {
			tmp_3 = (t_0 - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -c / b;
	} else {
		tmp_1 = -b / a;
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = sqrt(((b * b) - (c * (a * 4.0d0))))
    if (b <= (-1.6d+22)) then
        if (b >= 0.0d0) then
            tmp_2 = b * (1.0d0 / a)
        else
            tmp_2 = (c / b) - (b / a)
        end if
        tmp_1 = tmp_2
    else if (b <= 1.28d+65) then
        if (b >= 0.0d0) then
            tmp_3 = 2.0d0 / ((-b - t_0) / c)
        else
            tmp_3 = (t_0 - b) / (a * 2.0d0)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = -c / b
    else
        tmp_1 = -b / a
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -1.6e+22) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b * (1.0 / a);
		} else {
			tmp_2 = (c / b) - (b / a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.28e+65) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = 2.0 / ((-b - t_0) / c);
		} else {
			tmp_3 = (t_0 - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -c / b;
	} else {
		tmp_1 = -b / a;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (c * (a * 4.0))))
	tmp_1 = 0
	if b <= -1.6e+22:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = b * (1.0 / a)
		else:
			tmp_2 = (c / b) - (b / a)
		tmp_1 = tmp_2
	elif b <= 1.28e+65:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = 2.0 / ((-b - t_0) / c)
		else:
			tmp_3 = (t_0 - b) / (a * 2.0)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = -c / b
	else:
		tmp_1 = -b / a
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	tmp_1 = 0.0
	if (b <= -1.6e+22)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(b * Float64(1.0 / a));
		else
			tmp_2 = Float64(Float64(c / b) - Float64(b / a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.28e+65)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(2.0 / Float64(Float64(Float64(-b) - t_0) / c));
		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) / b);
	else
		tmp_1 = Float64(Float64(-b) / a);
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	tmp_2 = 0.0;
	if (b <= -1.6e+22)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = b * (1.0 / a);
		else
			tmp_3 = (c / b) - (b / a);
		end
		tmp_2 = tmp_3;
	elseif (b <= 1.28e+65)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = 2.0 / ((-b - t_0) / c);
		else
			tmp_4 = (t_0 - b) / (a * 2.0);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = -c / b;
	else
		tmp_2 = -b / a;
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.6e+22], If[GreaterEqual[b, 0.0], N[(b * N[(1.0 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.28e+65], If[GreaterEqual[b, 0.0], N[(2.0 / N[(N[((-b) - t$95$0), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[((-c) / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.6e22

   1. Initial program 63.8%

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

   3. Simplified 63.8%

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

   4. Taylor expanded in b around inf 63.8%

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

      1. +-commutative 63.8%

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

      2. *-commutative 63.8%

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

      3. fma-def 63.8%

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

      4. associate-/l* 63.8%

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

      5. associate-*r/ 63.8%

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

      6. *-commutative 63.8%

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

   6. Simplified 63.8%

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

   7. Taylor expanded in b around 0 63.8%

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

   8. Taylor expanded in b around -inf 96.4%

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

      1. mul-1-neg 96.4%

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

      2. unsub-neg 96.4%

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

   10. Simplified 96.4%

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

       1. associate-/r* 96.4%

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

       2. metadata-eval 96.4%

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

       3. clear-num 96.4%

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

       4. div-inv 96.4%

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

   12. Applied egg-rr 96.4%

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

   if -1.6e22 < b < 1.28000000000000007e65

   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. Step-by-step derivation

      1. expm1-log1p-u 77.9%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. expm1-udef 56.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{e^{\mathsf{log1p}\left(\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)} - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      3. associate-/l* 56.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;e^{\mathsf{log1p}\left(\color{blue}{\frac{2}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{c}}}\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      4. *-commutative 56.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{2}{\frac{\left(-b\right) - \sqrt{b \cdot b - \color{blue}{c \cdot \left(4 \cdot a\right)}}}{c}}\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      5. *-commutative 56.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{2}{\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \color{blue}{\left(a \cdot 4\right)}}}{c}}\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   3. Applied egg-rr 56.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{c}}\right)} - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   4. Step-by-step derivation

      1. expm1-def 77.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{c}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. expm1-log1p 90.3%

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

   5. Simplified 90.3%

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

   if 1.28000000000000007e65 < b

   1. Initial program 66.1%

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

      1. div-inv 66.0%

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

      2. *-commutative 66.0%

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

      3. *-commutative 66.0%

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

      4. *-commutative 66.0%

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

   3. Applied egg-rr 66.0%

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

   4. Taylor expanded in b around inf 84.9%

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

      1. +-commutative 84.9%

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

      2. *-commutative 84.9%

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

      3. fma-def 84.9%

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

      4. associate-/l* 95.3%

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

      5. associate-*r/ 95.3%

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

      6. *-commutative 95.3%

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

      7. *-rgt-identity 95.3%

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

      8. associate-*r/ 95.3%

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

      9. associate-*l* 95.3%

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

      10. associate-*r/ 95.3%

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

      11. metadata-eval 95.3%

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

      12. associate-/l* 95.3%

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

      13. associate-*r/ 95.3%

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

   6. Simplified 95.3%

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

   7. Taylor expanded in c around 0 95.6%

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

      1. associate-*r/ 95.6%

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

      2. neg-mul-1 95.6%

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

   9. Simplified 95.6%

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

   10. Taylor expanded in b around -inf 95.6%

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

       1. neg-mul-1 95.6%

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

       2. distribute-neg-frac 95.6%

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

   12. Simplified 95.6%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+22}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;b \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.28 \cdot 10^{+65}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 3: 78.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+22}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;b \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\left(c \cdot 2\right) \cdot \frac{1}{\mathsf{fma}\left(b, -2, c \cdot \left(2 \cdot \frac{a}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.6e+22)
   (if (>= b 0.0) (* b (/ 1.0 a)) (- (/ c b) (/ b a)))
   (if (>= b 0.0)
     (* (* c 2.0) (/ 1.0 (fma b -2.0 (* c (* 2.0 (/ a b))))))
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1.6e+22) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b * (1.0 / a);
		} else {
			tmp_2 = (c / b) - (b / a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (c * 2.0) * (1.0 / fma(b, -2.0, (c * (2.0 * (a / b)))));
	} else {
		tmp_1 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -1.6e+22)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(b * Float64(1.0 / a));
		else
			tmp_2 = Float64(Float64(c / b) - Float64(b / a));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c * 2.0) * Float64(1.0 / fma(b, -2.0, Float64(c * Float64(2.0 * Float64(a / b))))));
	else
		tmp_1 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.6e+22], If[GreaterEqual[b, 0.0], N[(b * N[(1.0 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] * N[(1.0 / N[(b * -2.0 + N[(c * N[(2.0 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.6e22

   1. Initial program 63.8%

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

   3. Simplified 63.8%

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

   4. Taylor expanded in b around inf 63.8%

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

      1. +-commutative 63.8%

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

      2. *-commutative 63.8%

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

      3. fma-def 63.8%

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

      4. associate-/l* 63.8%

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

      5. associate-*r/ 63.8%

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

      6. *-commutative 63.8%

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

   6. Simplified 63.8%

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

   7. Taylor expanded in b around 0 63.8%

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

   8. Taylor expanded in b around -inf 96.4%

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

      1. mul-1-neg 96.4%

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

      2. unsub-neg 96.4%

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

   10. Simplified 96.4%

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

       1. associate-/r* 96.4%

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

       2. metadata-eval 96.4%

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

       3. clear-num 96.4%

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

       4. div-inv 96.4%

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

   12. Applied egg-rr 96.4%

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

   if -1.6e22 < b

   1. Initial program 81.5%

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

      1. div-inv 81.4%

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

      2. *-commutative 81.4%

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

      3. *-commutative 81.4%

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

      4. *-commutative 81.4%

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

   3. Applied egg-rr 81.4%

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

   4. Taylor expanded in b around inf 69.7%

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

      1. +-commutative 69.7%

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

      2. *-commutative 69.7%

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

      3. fma-def 69.7%

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

      4. associate-/l* 73.4%

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

      5. associate-*r/ 73.4%

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

      6. *-commutative 73.4%

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

      7. *-rgt-identity 73.4%

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

      8. associate-*r/ 73.4%

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

      9. associate-*l* 73.4%

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

      10. associate-*r/ 73.4%

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

      11. metadata-eval 73.4%

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

      12. associate-/l* 73.4%

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

      13. associate-*r/ 73.4%

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

   6. Simplified 73.4%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+22}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;b \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\left(c \cdot 2\right) \cdot \frac{1}{\mathsf{fma}\left(b, -2, c \cdot \left(2 \cdot \frac{a}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array} \]

Alternative 4: 35.3% accurate, 19.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.2%

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

   1. div-inv 76.1%

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

   2. *-commutative 76.1%

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

   3. *-commutative 76.1%

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

   4. *-commutative 76.1%

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

3. Applied egg-rr 76.1%

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

4. Taylor expanded in b around inf 67.9%

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

   1. +-commutative 67.9%

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

   2. *-commutative 67.9%

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

   3. fma-def 67.9%

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

   4. associate-/l* 70.5%

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

   5. associate-*r/ 70.5%

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

   6. *-commutative 70.5%

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

   7. *-rgt-identity 70.5%

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

   8. associate-*r/ 70.5%

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

   9. associate-*l* 70.5%

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

   10. associate-*r/ 70.5%

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

   11. metadata-eval 70.5%

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

   12. associate-/l* 70.5%

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

   13. associate-*r/ 70.5%

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

6. Simplified 70.5%

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

7. Taylor expanded in c around inf 39.5%

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

8. Taylor expanded in b around -inf 38.8%

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

   1. neg-mul-1 69.9%

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

   2. distribute-neg-frac 69.9%

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

10. Simplified 38.8%

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

11. Final simplification 38.8%

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

Alternative 5: 67.2% accurate, 19.5× 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(Float64(-c) / b);
	else
		tmp = Float64(Float64(-b) / 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 76.2%

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

   1. div-inv 76.1%

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

   2. *-commutative 76.1%

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

   3. *-commutative 76.1%

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

   4. *-commutative 76.1%

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

3. Applied egg-rr 76.1%

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

4. Taylor expanded in b around inf 67.9%

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

   1. +-commutative 67.9%

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

   2. *-commutative 67.9%

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

   3. fma-def 67.9%

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

   4. associate-/l* 70.5%

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

   5. associate-*r/ 70.5%

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

   6. *-commutative 70.5%

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

   7. *-rgt-identity 70.5%

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

   8. associate-*r/ 70.5%

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

   9. associate-*l* 70.5%

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

   10. associate-*r/ 70.5%

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

   11. metadata-eval 70.5%

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

   12. associate-/l* 70.5%

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

   13. associate-*r/ 70.5%

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

6. Simplified 70.5%

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

7. Taylor expanded in c around 0 70.7%

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

   1. associate-*r/ 70.7%

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

   2. neg-mul-1 70.7%

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

9. Simplified 70.7%

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

10. Taylor expanded in b around -inf 69.9%

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

    1. neg-mul-1 69.9%

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

    2. distribute-neg-frac 69.9%

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

12. Simplified 69.9%

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

13. Final simplification 69.9%

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

Reproduce

herbie shell --seed 2023274 
(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))))