jeff quadratic root 2

Percentage Accurate: 73.0% → 91.1%

Time: 18.5s

Alternatives: 12

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: 73.0% 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: 91.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\\ \mathbf{if}\;b \leq -2.55 \cdot 10^{+154}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a \cdot 2}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* b b) (* a (* c -4.0))))))
   (if (<= b -2.55e+154)
     (- 0.0 (/ b a))
     (if (<= b 2e+139)
       (if (>= b 0.0) (/ (* c -2.0) (+ b t_0)) (/ (- t_0 b) (* a 2.0)))
       (/ c (- 0.0 b))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) + (a * (c * -4.0))));
	double tmp;
	if (b <= -2.55e+154) {
		tmp = 0.0 - (b / a);
	} else if (b <= 2e+139) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = (c * -2.0) / (b + t_0);
		} else {
			tmp_1 = (t_0 - b) / (a * 2.0);
		}
		tmp = tmp_1;
	} else {
		tmp = c / (0.0 - b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    t_0 = sqrt(((b * b) + (a * (c * (-4.0d0)))))
    if (b <= (-2.55d+154)) then
        tmp = 0.0d0 - (b / a)
    else if (b <= 2d+139) then
        if (b >= 0.0d0) then
            tmp_1 = (c * (-2.0d0)) / (b + t_0)
        else
            tmp_1 = (t_0 - b) / (a * 2.0d0)
        end if
        tmp = tmp_1
    else
        tmp = c / (0.0d0 - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) + (a * (c * -4.0))));
	double tmp;
	if (b <= -2.55e+154) {
		tmp = 0.0 - (b / a);
	} else if (b <= 2e+139) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = (c * -2.0) / (b + t_0);
		} else {
			tmp_1 = (t_0 - b) / (a * 2.0);
		}
		tmp = tmp_1;
	} else {
		tmp = c / (0.0 - b);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) + (a * (c * -4.0))))
	tmp = 0
	if b <= -2.55e+154:
		tmp = 0.0 - (b / a)
	elif b <= 2e+139:
		tmp_1 = 0
		if b >= 0.0:
			tmp_1 = (c * -2.0) / (b + t_0)
		else:
			tmp_1 = (t_0 - b) / (a * 2.0)
		tmp = tmp_1
	else:
		tmp = c / (0.0 - b)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0))))
	tmp = 0.0
	if (b <= -2.55e+154)
		tmp = Float64(0.0 - Float64(b / a));
	elseif (b <= 2e+139)
		tmp_1 = 0.0
		if (b >= 0.0)
			tmp_1 = Float64(Float64(c * -2.0) / Float64(b + t_0));
		else
			tmp_1 = Float64(Float64(t_0 - b) / Float64(a * 2.0));
		end
		tmp = tmp_1;
	else
		tmp = Float64(c / Float64(0.0 - b));
	end
	return tmp
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = sqrt(((b * b) + (a * (c * -4.0))));
	tmp = 0.0;
	if (b <= -2.55e+154)
		tmp = 0.0 - (b / a);
	elseif (b <= 2e+139)
		tmp_2 = 0.0;
		if (b >= 0.0)
			tmp_2 = (c * -2.0) / (b + t_0);
		else
			tmp_2 = (t_0 - b) / (a * 2.0);
		end
		tmp = tmp_2;
	else
		tmp = c / (0.0 - b);
	end
	tmp_3 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2.55e+154], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e+139], 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]], N[(c / N[(0.0 - b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.55e154

   1. Initial program 32.3%

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

   3. Simplified 32.3%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(b\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. neg-sub0 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(0 - b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. --lowering--.f64 96.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   7. Simplified 96.3%

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

   8. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{neg}\left(\frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. neg-sub0 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;0 - \color{blue}{\frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. --lowering--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      4. /-lowering-/.f64 96.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   10. Simplified 96.3%

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

       1. associate-/r* N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(0 - b\right) - b}{2}}{a}\\ \end{array} \]

       2. div-inv N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0 - b\right) - b}{2} \cdot \frac{1}{a}\\ \end{array} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(\left(\frac{\left(0 - b\right) - b}{2}\right), \left(\frac{1}{a}\right)\right)\\ \end{array} \]

   12. Applied egg-rr 95.9%

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

       1. associate-/l* N/A

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

       2. metadata-eval N/A

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

       3. *-commutative N/A

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

       4. neg-mul-1 N/A

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

       5. distribute-lft-neg-in N/A

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

       6. div-inv N/A

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

       7. sub0-neg N/A

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

       8. if-same N/A

          \[\leadsto 0 - \color{blue}{\frac{b}{a}} \]

       9. sub0-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

       10. neg-lowering-neg.f64 N/A

           \[\leadsto \mathsf{neg.f64}\left(\left(\frac{b}{a}\right)\right) \]

       11. /-lowering-/.f64 96.3%

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(b, a\right)\right) \]

   14. Applied egg-rr 96.3%

       \[\leadsto \color{blue}{-\frac{b}{a}} \]

   if -2.55e154 < b < 2.00000000000000007e139

   1. Initial program 89.6%

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

   3. Simplified 89.6%

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

   if 2.00000000000000007e139 < b

   1. Initial program 40.0%

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

   3. Simplified 42.2%

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

      1. flip-+ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\frac{c \cdot -2}{\frac{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{\color{blue}{b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. associate-/r/ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\frac{c \cdot -2}{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}} \cdot \color{blue}{\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\left(\frac{c \cdot -2}{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}\right), \color{blue}{\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in b around -inf

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

      1. if-same N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b\right)}{\color{blue}{a}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right)\right)}{a} \]

      4. sub-neg N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b\right)}{a} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b\right)\right), \color{blue}{a}\right) \]

   9. Simplified 2.2%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)} - b\right)}{a}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

       4. /-lowering-/.f64 100.0%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   12. Simplified 100.0%

       \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   13. Step-by-step derivation

       1. sub0-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

       2. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{c}{b}\right)\right) \]

       3. /-lowering-/.f64 100.0%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(c, b\right)\right) \]

   14. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{-\frac{c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.55 \cdot 10^{+154}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.55 \cdot 10^{+154}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{+139}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.55e+154)
   (- 0.0 (/ b a))
   (if (<= b 3.6e+139)
     (if (>= b 0.0)
       (* c (/ -2.0 (+ b (sqrt (+ (* b b) (* c (* a -4.0)))))))
       (/ (- (sqrt (+ (* b b) (* a (* c -4.0)))) b) (* a 2.0)))
     (/ c (- 0.0 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.55e+154) {
		tmp = 0.0 - (b / a);
	} else if (b <= 3.6e+139) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = c * (-2.0 / (b + sqrt(((b * b) + (c * (a * -4.0))))));
		} else {
			tmp_1 = (sqrt(((b * b) + (a * (c * -4.0)))) - b) / (a * 2.0);
		}
		tmp = tmp_1;
	} else {
		tmp = c / (0.0 - b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    real(8) :: tmp_1
    if (b <= (-2.55d+154)) then
        tmp = 0.0d0 - (b / a)
    else if (b <= 3.6d+139) then
        if (b >= 0.0d0) then
            tmp_1 = c * ((-2.0d0) / (b + sqrt(((b * b) + (c * (a * (-4.0d0)))))))
        else
            tmp_1 = (sqrt(((b * b) + (a * (c * (-4.0d0))))) - b) / (a * 2.0d0)
        end if
        tmp = tmp_1
    else
        tmp = c / (0.0d0 - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.55e+154) {
		tmp = 0.0 - (b / a);
	} else if (b <= 3.6e+139) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = c * (-2.0 / (b + Math.sqrt(((b * b) + (c * (a * -4.0))))));
		} else {
			tmp_1 = (Math.sqrt(((b * b) + (a * (c * -4.0)))) - b) / (a * 2.0);
		}
		tmp = tmp_1;
	} else {
		tmp = c / (0.0 - b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.55e+154:
		tmp = 0.0 - (b / a)
	elif b <= 3.6e+139:
		tmp_1 = 0
		if b >= 0.0:
			tmp_1 = c * (-2.0 / (b + math.sqrt(((b * b) + (c * (a * -4.0))))))
		else:
			tmp_1 = (math.sqrt(((b * b) + (a * (c * -4.0)))) - b) / (a * 2.0)
		tmp = tmp_1
	else:
		tmp = c / (0.0 - b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.55e+154)
		tmp = Float64(0.0 - Float64(b / a));
	elseif (b <= 3.6e+139)
		tmp_1 = 0.0
		if (b >= 0.0)
			tmp_1 = Float64(c * Float64(-2.0 / Float64(b + sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0)))))));
		else
			tmp_1 = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0)))) - b) / Float64(a * 2.0));
		end
		tmp = tmp_1;
	else
		tmp = Float64(c / Float64(0.0 - b));
	end
	return tmp
end

MATLAB

function tmp_3 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.55e+154)
		tmp = 0.0 - (b / a);
	elseif (b <= 3.6e+139)
		tmp_2 = 0.0;
		if (b >= 0.0)
			tmp_2 = c * (-2.0 / (b + sqrt(((b * b) + (c * (a * -4.0))))));
		else
			tmp_2 = (sqrt(((b * b) + (a * (c * -4.0)))) - b) / (a * 2.0);
		end
		tmp = tmp_2;
	else
		tmp = c / (0.0 - b);
	end
	tmp_3 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.55e+154], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.6e+139], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], N[(c / N[(0.0 - b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.55e154

   1. Initial program 32.3%

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

   3. Simplified 32.3%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(b\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. neg-sub0 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(0 - b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. --lowering--.f64 96.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   7. Simplified 96.3%

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

   8. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{neg}\left(\frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. neg-sub0 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;0 - \color{blue}{\frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. --lowering--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      4. /-lowering-/.f64 96.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   10. Simplified 96.3%

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

       1. associate-/r* N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(0 - b\right) - b}{2}}{a}\\ \end{array} \]

       2. div-inv N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0 - b\right) - b}{2} \cdot \frac{1}{a}\\ \end{array} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(\left(\frac{\left(0 - b\right) - b}{2}\right), \left(\frac{1}{a}\right)\right)\\ \end{array} \]

   12. Applied egg-rr 95.9%

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

       1. associate-/l* N/A

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

       2. metadata-eval N/A

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

       3. *-commutative N/A

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

       4. neg-mul-1 N/A

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

       5. distribute-lft-neg-in N/A

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

       6. div-inv N/A

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

       7. sub0-neg N/A

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

       8. if-same N/A

          \[\leadsto 0 - \color{blue}{\frac{b}{a}} \]

       9. sub0-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

       10. neg-lowering-neg.f64 N/A

           \[\leadsto \mathsf{neg.f64}\left(\left(\frac{b}{a}\right)\right) \]

       11. /-lowering-/.f64 96.3%

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(b, a\right)\right) \]

   14. Applied egg-rr 96.3%

       \[\leadsto \color{blue}{-\frac{b}{a}} \]

   if -2.55e154 < b < 3.59999999999999985e139

   1. Initial program 89.6%

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

   3. Simplified 89.6%

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

      1. associate-/l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;c \cdot \color{blue}{\frac{-2}{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\frac{-2}{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}} \cdot \color{blue}{c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\left(\frac{-2}{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}\right), \color{blue}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   6. Applied egg-rr 89.5%

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

   if 3.59999999999999985e139 < b

   1. Initial program 40.0%

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

   3. Simplified 42.2%

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

      1. flip-+ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\frac{c \cdot -2}{\frac{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{\color{blue}{b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. associate-/r/ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\frac{c \cdot -2}{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}} \cdot \color{blue}{\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\left(\frac{c \cdot -2}{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}\right), \color{blue}{\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in b around -inf

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

      1. if-same N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b\right)}{\color{blue}{a}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right)\right)}{a} \]

      4. sub-neg N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b\right)}{a} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b\right)\right), \color{blue}{a}\right) \]

   9. Simplified 2.2%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)} - b\right)}{a}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

       4. /-lowering-/.f64 100.0%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   12. Simplified 100.0%

       \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   13. Step-by-step derivation

       1. sub0-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

       2. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{c}{b}\right)\right) \]

       3. /-lowering-/.f64 100.0%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(c, b\right)\right) \]

   14. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{-\frac{c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.55 \cdot 10^{+154}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{+139}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.55 \cdot 10^{+154}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-91}:\\ \;\;\;\;\frac{0.5 \cdot \left(\sqrt{b \cdot b + -4 \cdot \left(a \cdot c\right)} - b\right)}{a}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{-2 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0 - b\right) - b}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.55e+154)
   (- 0.0 (/ b a))
   (if (<= b 3.9e-91)
     (/ (* 0.5 (- (sqrt (+ (* b b) (* -4.0 (* a c)))) b)) a)
     (if (>= b 0.0)
       (/ (* c -2.0) (+ (* -2.0 (* c (/ a b))) (* b 2.0)))
       (/ (- (- 0.0 b) b) (* a 2.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.55e+154) {
		tmp = 0.0 - (b / a);
	} else if (b <= 3.9e-91) {
		tmp = (0.5 * (sqrt(((b * b) + (-4.0 * (a * c)))) - b)) / a;
	} else if (b >= 0.0) {
		tmp = (c * -2.0) / ((-2.0 * (c * (a / b))) + (b * 2.0));
	} else {
		tmp = ((0.0 - b) - b) / (a * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.55d+154)) then
        tmp = 0.0d0 - (b / a)
    else if (b <= 3.9d-91) then
        tmp = (0.5d0 * (sqrt(((b * b) + ((-4.0d0) * (a * c)))) - b)) / a
    else if (b >= 0.0d0) then
        tmp = (c * (-2.0d0)) / (((-2.0d0) * (c * (a / b))) + (b * 2.0d0))
    else
        tmp = ((0.0d0 - b) - b) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.55e+154) {
		tmp = 0.0 - (b / a);
	} else if (b <= 3.9e-91) {
		tmp = (0.5 * (Math.sqrt(((b * b) + (-4.0 * (a * c)))) - b)) / a;
	} else if (b >= 0.0) {
		tmp = (c * -2.0) / ((-2.0 * (c * (a / b))) + (b * 2.0));
	} else {
		tmp = ((0.0 - b) - b) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.55e+154:
		tmp = 0.0 - (b / a)
	elif b <= 3.9e-91:
		tmp = (0.5 * (math.sqrt(((b * b) + (-4.0 * (a * c)))) - b)) / a
	elif b >= 0.0:
		tmp = (c * -2.0) / ((-2.0 * (c * (a / b))) + (b * 2.0))
	else:
		tmp = ((0.0 - b) - b) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.55e+154)
		tmp = Float64(0.0 - Float64(b / a));
	elseif (b <= 3.9e-91)
		tmp = Float64(Float64(0.5 * Float64(sqrt(Float64(Float64(b * b) + Float64(-4.0 * Float64(a * c)))) - b)) / a);
	elseif (b >= 0.0)
		tmp = Float64(Float64(c * -2.0) / Float64(Float64(-2.0 * Float64(c * Float64(a / b))) + Float64(b * 2.0)));
	else
		tmp = Float64(Float64(Float64(0.0 - b) - b) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.55e+154)
		tmp = 0.0 - (b / a);
	elseif (b <= 3.9e-91)
		tmp = (0.5 * (sqrt(((b * b) + (-4.0 * (a * c)))) - b)) / a;
	elseif (b >= 0.0)
		tmp = (c * -2.0) / ((-2.0 * (c * (a / b))) + (b * 2.0));
	else
		tmp = ((0.0 - b) - b) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.55e+154], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.9e-91], N[(N[(0.5 * N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[GreaterEqual[b, 0.0], N[(N[(c * -2.0), $MachinePrecision] / N[(N[(-2.0 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.0 - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.55e154

   1. Initial program 32.3%

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

   3. Simplified 32.3%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(b\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. neg-sub0 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(0 - b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. --lowering--.f64 96.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   7. Simplified 96.3%

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

   8. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{neg}\left(\frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. neg-sub0 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;0 - \color{blue}{\frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. --lowering--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      4. /-lowering-/.f64 96.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   10. Simplified 96.3%

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

       1. associate-/r* N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(0 - b\right) - b}{2}}{a}\\ \end{array} \]

       2. div-inv N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0 - b\right) - b}{2} \cdot \frac{1}{a}\\ \end{array} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(\left(\frac{\left(0 - b\right) - b}{2}\right), \left(\frac{1}{a}\right)\right)\\ \end{array} \]

   12. Applied egg-rr 95.9%

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

       1. associate-/l* N/A

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

       2. metadata-eval N/A

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

       3. *-commutative N/A

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

       4. neg-mul-1 N/A

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

       5. distribute-lft-neg-in N/A

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

       6. div-inv N/A

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

       7. sub0-neg N/A

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

       8. if-same N/A

          \[\leadsto 0 - \color{blue}{\frac{b}{a}} \]

       9. sub0-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

       10. neg-lowering-neg.f64 N/A

           \[\leadsto \mathsf{neg.f64}\left(\left(\frac{b}{a}\right)\right) \]

       11. /-lowering-/.f64 96.3%

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(b, a\right)\right) \]

   14. Applied egg-rr 96.3%

       \[\leadsto \color{blue}{-\frac{b}{a}} \]

   if -2.55e154 < b < 3.89999999999999994e-91

   1. Initial program 89.0%

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

   3. Simplified 89.0%

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

      1. flip-+ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\frac{c \cdot -2}{\frac{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{\color{blue}{b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. associate-/r/ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\frac{c \cdot -2}{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}} \cdot \color{blue}{\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\left(\frac{c \cdot -2}{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}\right), \color{blue}{\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   6. Applied egg-rr 86.9%

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

   7. Taylor expanded in b around -inf

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

      1. if-same N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b\right)}{\color{blue}{a}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right)\right)}{a} \]

      4. sub-neg N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b\right)}{a} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b\right)\right), \color{blue}{a}\right) \]

   9. Simplified 87.2%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)} - b\right)}{a}} \]

   if 3.89999999999999994e-91 < b

   1. Initial program 62.3%

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

   3. Simplified 63.5%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(b\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. neg-sub0 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(0 - b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. --lowering--.f64 63.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   7. Simplified 63.5%

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

   8. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \color{blue}{\left(-2 \cdot \frac{a \cdot c}{b} + 2 \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\left(-2 \cdot \frac{a \cdot c}{b}\right), \color{blue}{\left(2 \cdot b\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. associate-*r/ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\left(\frac{-2 \cdot \left(a \cdot c\right)}{b}\right), \left(\color{blue}{2} \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \left(a \cdot c\right)\right), b\right), \left(\color{blue}{2} \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(a \cdot c\right)\right), b\right), \left(2 \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      5. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(c \cdot a\right)\right), b\right), \left(2 \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), b\right), \left(2 \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      7. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), b\right), \left(b \cdot \color{blue}{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      8. *-lowering-*.f64 86.1%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), b\right), \mathsf{*.f64}\left(b, \color{blue}{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   10. Simplified 86.1%

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

       1. associate-/l* N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\left(-2 \cdot \frac{c \cdot a}{b}\right), \mathsf{*.f64}\left(\color{blue}{b}, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       2. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\left(\frac{c \cdot a}{b} \cdot -2\right), \mathsf{*.f64}\left(\color{blue}{b}, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{c \cdot a}{b}\right), -2\right), \mathsf{*.f64}\left(\color{blue}{b}, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       4. associate-/l* N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(c \cdot \frac{a}{b}\right), -2\right), \mathsf{*.f64}\left(b, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \left(\frac{a}{b}\right)\right), -2\right), \mathsf{*.f64}\left(b, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       6. /-lowering-/.f64 88.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(a, b\right)\right), -2\right), \mathsf{*.f64}\left(b, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   12. Applied egg-rr 88.5%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.55 \cdot 10^{+154}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-91}:\\ \;\;\;\;\frac{0.5 \cdot \left(\sqrt{b \cdot b + -4 \cdot \left(a \cdot c\right)} - b\right)}{a}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{-2 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0 - b\right) - b}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{b \cdot \left(a \cdot \frac{c}{b \cdot b} + -1\right)}{a}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-91}:\\ \;\;\;\;\frac{0.5 \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right)} - b\right)}{a}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{-2 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0 - b\right) - b}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.5e-39)
   (/ (* b (+ (* a (/ c (* b b))) -1.0)) a)
   (if (<= b 4.4e-91)
     (/ (* 0.5 (- (sqrt (* -4.0 (* a c))) b)) a)
     (if (>= b 0.0)
       (/ (* c -2.0) (+ (* -2.0 (* c (/ a b))) (* b 2.0)))
       (/ (- (- 0.0 b) b) (* a 2.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.5e-39) {
		tmp = (b * ((a * (c / (b * b))) + -1.0)) / a;
	} else if (b <= 4.4e-91) {
		tmp = (0.5 * (sqrt((-4.0 * (a * c))) - b)) / a;
	} else if (b >= 0.0) {
		tmp = (c * -2.0) / ((-2.0 * (c * (a / b))) + (b * 2.0));
	} else {
		tmp = ((0.0 - b) - b) / (a * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.5d-39)) then
        tmp = (b * ((a * (c / (b * b))) + (-1.0d0))) / a
    else if (b <= 4.4d-91) then
        tmp = (0.5d0 * (sqrt(((-4.0d0) * (a * c))) - b)) / a
    else if (b >= 0.0d0) then
        tmp = (c * (-2.0d0)) / (((-2.0d0) * (c * (a / b))) + (b * 2.0d0))
    else
        tmp = ((0.0d0 - b) - b) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.5e-39) {
		tmp = (b * ((a * (c / (b * b))) + -1.0)) / a;
	} else if (b <= 4.4e-91) {
		tmp = (0.5 * (Math.sqrt((-4.0 * (a * c))) - b)) / a;
	} else if (b >= 0.0) {
		tmp = (c * -2.0) / ((-2.0 * (c * (a / b))) + (b * 2.0));
	} else {
		tmp = ((0.0 - b) - b) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.5e-39:
		tmp = (b * ((a * (c / (b * b))) + -1.0)) / a
	elif b <= 4.4e-91:
		tmp = (0.5 * (math.sqrt((-4.0 * (a * c))) - b)) / a
	elif b >= 0.0:
		tmp = (c * -2.0) / ((-2.0 * (c * (a / b))) + (b * 2.0))
	else:
		tmp = ((0.0 - b) - b) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.5e-39)
		tmp = Float64(Float64(b * Float64(Float64(a * Float64(c / Float64(b * b))) + -1.0)) / a);
	elseif (b <= 4.4e-91)
		tmp = Float64(Float64(0.5 * Float64(sqrt(Float64(-4.0 * Float64(a * c))) - b)) / a);
	elseif (b >= 0.0)
		tmp = Float64(Float64(c * -2.0) / Float64(Float64(-2.0 * Float64(c * Float64(a / b))) + Float64(b * 2.0)));
	else
		tmp = Float64(Float64(Float64(0.0 - b) - b) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.5e-39)
		tmp = (b * ((a * (c / (b * b))) + -1.0)) / a;
	elseif (b <= 4.4e-91)
		tmp = (0.5 * (sqrt((-4.0 * (a * c))) - b)) / a;
	elseif (b >= 0.0)
		tmp = (c * -2.0) / ((-2.0 * (c * (a / b))) + (b * 2.0));
	else
		tmp = ((0.0 - b) - b) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.5e-39], N[(N[(b * N[(N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 4.4e-91], N[(N[(0.5 * N[(N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[GreaterEqual[b, 0.0], N[(N[(c * -2.0), $MachinePrecision] / N[(N[(-2.0 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.0 - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.4999999999999999e-39

   1. Initial program 62.8%

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

   3. Simplified 62.8%

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

      1. flip-+ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\frac{c \cdot -2}{\frac{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{\color{blue}{b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. associate-/r/ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\frac{c \cdot -2}{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}} \cdot \color{blue}{\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\left(\frac{c \cdot -2}{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}\right), \color{blue}{\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   6. Applied egg-rr 62.8%

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

   7. Taylor expanded in b around -inf

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

      1. if-same N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b\right)}{\color{blue}{a}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right)\right)}{a} \]

      4. sub-neg N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b\right)}{a} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b\right)\right), \color{blue}{a}\right) \]

   9. Simplified 62.8%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)} - b\right)}{a}} \]

   10. Taylor expanded in b around -inf

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \left(b \cdot \left(1 + -1 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)}, a\right) \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(-1 \cdot b\right) \cdot \left(1 + -1 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right), a\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(-1 \cdot b\right), \left(1 + -1 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right), a\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{neg}\left(b\right)\right), \left(1 + -1 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right), a\right) \]

       4. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(0 - b\right), \left(1 + -1 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right), a\right) \]

       5. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \left(1 + -1 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right), a\right) \]

       6. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \left(1 + \left(\mathsf{neg}\left(\frac{a \cdot c}{{b}^{2}}\right)\right)\right)\right), a\right) \]

       7. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \left(1 - \frac{a \cdot c}{{b}^{2}}\right)\right), a\right) \]

       8. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{\_.f64}\left(1, \left(\frac{a \cdot c}{{b}^{2}}\right)\right)\right), a\right) \]

       9. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{\_.f64}\left(1, \left(a \cdot \frac{c}{{b}^{2}}\right)\right)\right), a\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{c}{{b}^{2}}\right)\right)\right)\right), a\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(c, \left({b}^{2}\right)\right)\right)\right)\right), a\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(c, \left(b \cdot b\right)\right)\right)\right)\right), a\right) \]

       13. *-lowering-*.f64 91.8%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), a\right) \]

   12. Simplified 91.8%

       \[\leadsto \frac{\color{blue}{\left(0 - b\right) \cdot \left(1 - a \cdot \frac{c}{b \cdot b}\right)}}{a} \]

   if -2.4999999999999999e-39 < b < 4.4000000000000002e-91

   1. Initial program 84.8%

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

   3. Simplified 84.8%

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

      1. flip-+ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\frac{c \cdot -2}{\frac{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{\color{blue}{b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. associate-/r/ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\frac{c \cdot -2}{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}} \cdot \color{blue}{\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\left(\frac{c \cdot -2}{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}\right), \color{blue}{\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   6. Applied egg-rr 81.6%

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

   7. Taylor expanded in b around -inf

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

      1. if-same N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b\right)}{\color{blue}{a}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right)\right)}{a} \]

      4. sub-neg N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b\right)}{a} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b\right)\right), \color{blue}{a}\right) \]

   9. Simplified 82.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)} - b\right)}{a}} \]

   10. Taylor expanded in b around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right)}\right), b\right)\right), a\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \left(a \cdot c\right)\right)\right), b\right)\right), a\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \left(c \cdot a\right)\right)\right), b\right)\right), a\right) \]

       3. *-lowering-*.f64 75.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(c, a\right)\right)\right), b\right)\right), a\right) \]

   12. Simplified 75.8%

       \[\leadsto \frac{0.5 \cdot \left(\sqrt{\color{blue}{-4 \cdot \left(c \cdot a\right)}} - b\right)}{a} \]

   if 4.4000000000000002e-91 < b

   1. Initial program 62.3%

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

   3. Simplified 63.5%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(b\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. neg-sub0 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(0 - b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. --lowering--.f64 63.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   7. Simplified 63.5%

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

   8. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \color{blue}{\left(-2 \cdot \frac{a \cdot c}{b} + 2 \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\left(-2 \cdot \frac{a \cdot c}{b}\right), \color{blue}{\left(2 \cdot b\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. associate-*r/ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\left(\frac{-2 \cdot \left(a \cdot c\right)}{b}\right), \left(\color{blue}{2} \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \left(a \cdot c\right)\right), b\right), \left(\color{blue}{2} \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(a \cdot c\right)\right), b\right), \left(2 \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      5. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(c \cdot a\right)\right), b\right), \left(2 \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), b\right), \left(2 \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      7. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), b\right), \left(b \cdot \color{blue}{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      8. *-lowering-*.f64 86.1%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), b\right), \mathsf{*.f64}\left(b, \color{blue}{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   10. Simplified 86.1%

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

       1. associate-/l* N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\left(-2 \cdot \frac{c \cdot a}{b}\right), \mathsf{*.f64}\left(\color{blue}{b}, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       2. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\left(\frac{c \cdot a}{b} \cdot -2\right), \mathsf{*.f64}\left(\color{blue}{b}, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{c \cdot a}{b}\right), -2\right), \mathsf{*.f64}\left(\color{blue}{b}, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       4. associate-/l* N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(c \cdot \frac{a}{b}\right), -2\right), \mathsf{*.f64}\left(b, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \left(\frac{a}{b}\right)\right), -2\right), \mathsf{*.f64}\left(b, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       6. /-lowering-/.f64 88.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(a, b\right)\right), -2\right), \mathsf{*.f64}\left(b, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   12. Applied egg-rr 88.5%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{b \cdot \left(a \cdot \frac{c}{b \cdot b} + -1\right)}{a}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-91}:\\ \;\;\;\;\frac{0.5 \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right)} - b\right)}{a}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{-2 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0 - b\right) - b}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.9 \cdot 10^{-39}:\\ \;\;\;\;\frac{b \cdot \left(a \cdot \frac{c}{b \cdot b} + -1\right)}{a}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-91}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{-2 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0 - b\right) - b}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.9e-39)
   (/ (* b (+ (* a (/ c (* b b))) -1.0)) a)
   (if (<= b 6e-91)
     (* (/ 0.5 a) (+ b (sqrt (* c (* a -4.0)))))
     (if (>= b 0.0)
       (/ (* c -2.0) (+ (* -2.0 (* c (/ a b))) (* b 2.0)))
       (/ (- (- 0.0 b) b) (* a 2.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.9e-39) {
		tmp = (b * ((a * (c / (b * b))) + -1.0)) / a;
	} else if (b <= 6e-91) {
		tmp = (0.5 / a) * (b + sqrt((c * (a * -4.0))));
	} else if (b >= 0.0) {
		tmp = (c * -2.0) / ((-2.0 * (c * (a / b))) + (b * 2.0));
	} else {
		tmp = ((0.0 - b) - b) / (a * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.9d-39)) then
        tmp = (b * ((a * (c / (b * b))) + (-1.0d0))) / a
    else if (b <= 6d-91) then
        tmp = (0.5d0 / a) * (b + sqrt((c * (a * (-4.0d0)))))
    else if (b >= 0.0d0) then
        tmp = (c * (-2.0d0)) / (((-2.0d0) * (c * (a / b))) + (b * 2.0d0))
    else
        tmp = ((0.0d0 - b) - b) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.9e-39) {
		tmp = (b * ((a * (c / (b * b))) + -1.0)) / a;
	} else if (b <= 6e-91) {
		tmp = (0.5 / a) * (b + Math.sqrt((c * (a * -4.0))));
	} else if (b >= 0.0) {
		tmp = (c * -2.0) / ((-2.0 * (c * (a / b))) + (b * 2.0));
	} else {
		tmp = ((0.0 - b) - b) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.9e-39:
		tmp = (b * ((a * (c / (b * b))) + -1.0)) / a
	elif b <= 6e-91:
		tmp = (0.5 / a) * (b + math.sqrt((c * (a * -4.0))))
	elif b >= 0.0:
		tmp = (c * -2.0) / ((-2.0 * (c * (a / b))) + (b * 2.0))
	else:
		tmp = ((0.0 - b) - b) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.9e-39)
		tmp = Float64(Float64(b * Float64(Float64(a * Float64(c / Float64(b * b))) + -1.0)) / a);
	elseif (b <= 6e-91)
		tmp = Float64(Float64(0.5 / a) * Float64(b + sqrt(Float64(c * Float64(a * -4.0)))));
	elseif (b >= 0.0)
		tmp = Float64(Float64(c * -2.0) / Float64(Float64(-2.0 * Float64(c * Float64(a / b))) + Float64(b * 2.0)));
	else
		tmp = Float64(Float64(Float64(0.0 - b) - b) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.9e-39)
		tmp = (b * ((a * (c / (b * b))) + -1.0)) / a;
	elseif (b <= 6e-91)
		tmp = (0.5 / a) * (b + sqrt((c * (a * -4.0))));
	elseif (b >= 0.0)
		tmp = (c * -2.0) / ((-2.0 * (c * (a / b))) + (b * 2.0));
	else
		tmp = ((0.0 - b) - b) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.9e-39], N[(N[(b * N[(N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 6e-91], N[(N[(0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[GreaterEqual[b, 0.0], N[(N[(c * -2.0), $MachinePrecision] / N[(N[(-2.0 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.0 - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.9000000000000003e-39

   1. Initial program 62.8%

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

   3. Simplified 62.8%

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

      1. flip-+ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\frac{c \cdot -2}{\frac{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{\color{blue}{b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. associate-/r/ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\frac{c \cdot -2}{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}} \cdot \color{blue}{\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\left(\frac{c \cdot -2}{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}\right), \color{blue}{\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   6. Applied egg-rr 62.8%

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

   7. Taylor expanded in b around -inf

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

      1. if-same N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b\right)}{\color{blue}{a}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right)\right)}{a} \]

      4. sub-neg N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b\right)}{a} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b\right)\right), \color{blue}{a}\right) \]

   9. Simplified 62.8%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)} - b\right)}{a}} \]

   10. Taylor expanded in b around -inf

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \left(b \cdot \left(1 + -1 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)}, a\right) \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(-1 \cdot b\right) \cdot \left(1 + -1 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right), a\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(-1 \cdot b\right), \left(1 + -1 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right), a\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{neg}\left(b\right)\right), \left(1 + -1 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right), a\right) \]

       4. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(0 - b\right), \left(1 + -1 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right), a\right) \]

       5. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \left(1 + -1 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right), a\right) \]

       6. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \left(1 + \left(\mathsf{neg}\left(\frac{a \cdot c}{{b}^{2}}\right)\right)\right)\right), a\right) \]

       7. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \left(1 - \frac{a \cdot c}{{b}^{2}}\right)\right), a\right) \]

       8. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{\_.f64}\left(1, \left(\frac{a \cdot c}{{b}^{2}}\right)\right)\right), a\right) \]

       9. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{\_.f64}\left(1, \left(a \cdot \frac{c}{{b}^{2}}\right)\right)\right), a\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, \left(\frac{c}{{b}^{2}}\right)\right)\right)\right), a\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(c, \left({b}^{2}\right)\right)\right)\right)\right), a\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(c, \left(b \cdot b\right)\right)\right)\right)\right), a\right) \]

       13. *-lowering-*.f64 91.8%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, b\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), a\right) \]

   12. Simplified 91.8%

       \[\leadsto \frac{\color{blue}{\left(0 - b\right) \cdot \left(1 - a \cdot \frac{c}{b \cdot b}\right)}}{a} \]

   if -3.9000000000000003e-39 < b < 6.0000000000000004e-91

   1. Initial program 84.8%

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

   3. Simplified 84.8%

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

      1. flip-+ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\frac{c \cdot -2}{\frac{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{\color{blue}{b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. associate-/r/ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\frac{c \cdot -2}{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}} \cdot \color{blue}{\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\left(\frac{c \cdot -2}{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}\right), \color{blue}{\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   6. Applied egg-rr 81.6%

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

   7. Taylor expanded in b around -inf

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

      1. if-same N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b\right)}{\color{blue}{a}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right)\right)}{a} \]

      4. sub-neg N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b\right)}{a} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b\right)\right), \color{blue}{a}\right) \]

   9. Simplified 82.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)} - b\right)}{a}} \]

   11. Applied egg-rr 74.6%

       \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right)} \]

   12. Taylor expanded in b around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right)}\right)\right)\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\left(c \cdot a\right) \cdot -4\right)\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(c \cdot \left(a \cdot -4\right)\right)\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(c \cdot \left(-4 \cdot a\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(-4 \cdot a\right)\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(a \cdot -4\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f64 74.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right) \]

   14. Simplified 74.4%

       \[\leadsto \frac{0.5}{a} \cdot \left(b + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}\right) \]

   if 6.0000000000000004e-91 < b

   1. Initial program 62.3%

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

   3. Simplified 63.5%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(b\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. neg-sub0 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(0 - b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. --lowering--.f64 63.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   7. Simplified 63.5%

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

   8. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \color{blue}{\left(-2 \cdot \frac{a \cdot c}{b} + 2 \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\left(-2 \cdot \frac{a \cdot c}{b}\right), \color{blue}{\left(2 \cdot b\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. associate-*r/ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\left(\frac{-2 \cdot \left(a \cdot c\right)}{b}\right), \left(\color{blue}{2} \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \left(a \cdot c\right)\right), b\right), \left(\color{blue}{2} \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(a \cdot c\right)\right), b\right), \left(2 \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      5. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(c \cdot a\right)\right), b\right), \left(2 \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), b\right), \left(2 \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      7. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), b\right), \left(b \cdot \color{blue}{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      8. *-lowering-*.f64 86.1%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), b\right), \mathsf{*.f64}\left(b, \color{blue}{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   10. Simplified 86.1%

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

       1. associate-/l* N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\left(-2 \cdot \frac{c \cdot a}{b}\right), \mathsf{*.f64}\left(\color{blue}{b}, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       2. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\left(\frac{c \cdot a}{b} \cdot -2\right), \mathsf{*.f64}\left(\color{blue}{b}, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{c \cdot a}{b}\right), -2\right), \mathsf{*.f64}\left(\color{blue}{b}, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       4. associate-/l* N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(c \cdot \frac{a}{b}\right), -2\right), \mathsf{*.f64}\left(b, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \left(\frac{a}{b}\right)\right), -2\right), \mathsf{*.f64}\left(b, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       6. /-lowering-/.f64 88.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(a, b\right)\right), -2\right), \mathsf{*.f64}\left(b, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   12. Applied egg-rr 88.5%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.9 \cdot 10^{-39}:\\ \;\;\;\;\frac{b \cdot \left(a \cdot \frac{c}{b \cdot b} + -1\right)}{a}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-91}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{-2 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0 - b\right) - b}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.4% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.9%

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

3. Simplified 70.3%

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

5. Taylor expanded in b around -inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(b\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   2. neg-sub0 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(0 - b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   3. --lowering--.f64 68.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

7. Simplified 68.5%

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

8. Taylor expanded in a around 0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \color{blue}{\left(-2 \cdot \frac{a \cdot c}{b} + 2 \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
9. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\left(-2 \cdot \frac{a \cdot c}{b}\right), \color{blue}{\left(2 \cdot b\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   2. associate-*r/ N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\left(\frac{-2 \cdot \left(a \cdot c\right)}{b}\right), \left(\color{blue}{2} \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \left(a \cdot c\right)\right), b\right), \left(\color{blue}{2} \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(a \cdot c\right)\right), b\right), \left(2 \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   5. *-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(c \cdot a\right)\right), b\right), \left(2 \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), b\right), \left(2 \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   7. *-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), b\right), \left(b \cdot \color{blue}{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   8. *-lowering-*.f64 65.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, a\right)\right), b\right), \mathsf{*.f64}\left(b, \color{blue}{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

10. Simplified 65.6%

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

    1. associate-/l* N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\left(-2 \cdot \frac{c \cdot a}{b}\right), \mathsf{*.f64}\left(\color{blue}{b}, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

    2. *-commutative N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\left(\frac{c \cdot a}{b} \cdot -2\right), \mathsf{*.f64}\left(\color{blue}{b}, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{c \cdot a}{b}\right), -2\right), \mathsf{*.f64}\left(\color{blue}{b}, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

    4. associate-/l* N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(c \cdot \frac{a}{b}\right), -2\right), \mathsf{*.f64}\left(b, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \left(\frac{a}{b}\right)\right), -2\right), \mathsf{*.f64}\left(b, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

    6. /-lowering-/.f64 66.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(a, b\right)\right), -2\right), \mathsf{*.f64}\left(b, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

12. Applied egg-rr 66.4%

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

13. Final simplification 66.4%

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

Alternative 7: 67.3% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.7 \cdot 10^{-282}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;b \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0 - b\right) - b}{a \cdot 2}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.7e-282)
   (if (>= b 0.0) (* b (/ 1.0 a)) (/ (- (- 0.0 b) b) (* a 2.0)))
   (/ c (- 0.0 b))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= 1.7e-282) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b * (1.0 / a);
		} else {
			tmp_2 = ((0.0 - b) - b) / (a * 2.0);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = c / (0.0 - b);
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    if (b <= 1.7d-282) then
        if (b >= 0.0d0) then
            tmp_2 = b * (1.0d0 / a)
        else
            tmp_2 = ((0.0d0 - b) - b) / (a * 2.0d0)
        end if
        tmp_1 = tmp_2
    else
        tmp_1 = c / (0.0d0 - b)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if (b <= 1.7e-282) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = b * (1.0 / a);
		} else {
			tmp_2 = ((0.0 - b) - b) / (a * 2.0);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = c / (0.0 - b);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if b <= 1.7e-282:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = b * (1.0 / a)
		else:
			tmp_2 = ((0.0 - b) - b) / (a * 2.0)
		tmp_1 = tmp_2
	else:
		tmp_1 = c / (0.0 - b)
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= 1.7e-282)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(b * Float64(1.0 / a));
		else
			tmp_2 = Float64(Float64(Float64(0.0 - b) - b) / Float64(a * 2.0));
		end
		tmp_1 = tmp_2;
	else
		tmp_1 = Float64(c / Float64(0.0 - b));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp_2 = 0.0;
	if (b <= 1.7e-282)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = b * (1.0 / a);
		else
			tmp_3 = ((0.0 - b) - b) / (a * 2.0);
		end
		tmp_2 = tmp_3;
	else
		tmp_2 = c / (0.0 - b);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.7e-282], If[GreaterEqual[b, 0.0], N[(b * N[(1.0 / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.0 - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], N[(c / N[(0.0 - b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.69999999999999999e-282

   1. Initial program 70.9%

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

   3. Simplified 70.9%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(b\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. neg-sub0 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(0 - b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. --lowering--.f64 67.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   7. Simplified 67.6%

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

   8. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{neg}\left(\frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. neg-sub0 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;0 - \color{blue}{\frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. --lowering--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      4. /-lowering-/.f64 65.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   10. Simplified 65.6%

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

       1. sub0-neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{neg}\left(\frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       2. distribute-neg-frac N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\frac{\mathsf{neg}\left(b\right)}{\color{blue}{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

       3. sub0-neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\frac{0 - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   12. Applied egg-rr 65.6%

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

   if 1.69999999999999999e-282 < b

   1. Initial program 68.7%

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

   3. Simplified 69.6%

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

      1. flip-+ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\frac{c \cdot -2}{\frac{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{\color{blue}{b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. associate-/r/ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\frac{c \cdot -2}{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}} \cdot \color{blue}{\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\left(\frac{c \cdot -2}{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}\right), \color{blue}{\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   6. Applied egg-rr 29.1%

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

   7. Taylor expanded in b around -inf

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

      1. if-same N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b\right)}{\color{blue}{a}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right)\right)}{a} \]

      4. sub-neg N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b\right)}{a} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b\right)\right), \color{blue}{a}\right) \]

   9. Simplified 33.7%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)} - b\right)}{a}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

       4. /-lowering-/.f64 67.0%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   12. Simplified 67.0%

       \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   13. Step-by-step derivation

       1. sub0-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

       2. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{c}{b}\right)\right) \]

       3. /-lowering-/.f64 67.0%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(c, b\right)\right) \]

   14. Applied egg-rr 67.0%

       \[\leadsto \color{blue}{-\frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.7 \cdot 10^{-282}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;b \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0 - b\right) - b}{a \cdot 2}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.2% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.9%

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

3. Simplified 70.3%

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

5. Taylor expanded in b around -inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(b\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   2. neg-sub0 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(0 - b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   3. --lowering--.f64 68.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

7. Simplified 68.5%

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

8. Taylor expanded in b around inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \color{blue}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
9. Step-by-step derivation

10. Simplified 66.2%

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

11. Final simplification 66.2%

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

Alternative 9: 67.3% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{-283}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2e-283) (- 0.0 (/ b a)) (/ c (- 0.0 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2e-283) {
		tmp = 0.0 - (b / a);
	} else {
		tmp = c / (0.0 - b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2d-283) then
        tmp = 0.0d0 - (b / a)
    else
        tmp = c / (0.0d0 - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2e-283) {
		tmp = 0.0 - (b / a);
	} else {
		tmp = c / (0.0 - b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2e-283:
		tmp = 0.0 - (b / a)
	else:
		tmp = c / (0.0 - b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2e-283)
		tmp = Float64(0.0 - Float64(b / a));
	else
		tmp = Float64(c / Float64(0.0 - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2e-283)
		tmp = 0.0 - (b / a);
	else
		tmp = c / (0.0 - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2e-283], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(c / N[(0.0 - b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.99999999999999989e-283

   1. Initial program 70.9%

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

   3. Simplified 70.9%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(b\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. neg-sub0 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(0 - b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. --lowering--.f64 67.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   7. Simplified 67.6%

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

   8. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{neg}\left(\frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. neg-sub0 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;0 - \color{blue}{\frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. --lowering--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      4. /-lowering-/.f64 65.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   10. Simplified 65.6%

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

       1. associate-/r* N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(0 - b\right) - b}{2}}{a}\\ \end{array} \]

       2. div-inv N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0 - b\right) - b}{2} \cdot \frac{1}{a}\\ \end{array} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(\left(\frac{\left(0 - b\right) - b}{2}\right), \left(\frac{1}{a}\right)\right)\\ \end{array} \]

   12. Applied egg-rr 65.4%

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

       1. associate-/l* N/A

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

       2. metadata-eval N/A

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

       3. *-commutative N/A

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

       4. neg-mul-1 N/A

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

       5. distribute-lft-neg-in N/A

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

       6. div-inv N/A

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

       7. sub0-neg N/A

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

       8. if-same N/A

          \[\leadsto 0 - \color{blue}{\frac{b}{a}} \]

       9. sub0-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

       10. neg-lowering-neg.f64 N/A

           \[\leadsto \mathsf{neg.f64}\left(\left(\frac{b}{a}\right)\right) \]

       11. /-lowering-/.f64 65.6%

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(b, a\right)\right) \]

   14. Applied egg-rr 65.6%

       \[\leadsto \color{blue}{-\frac{b}{a}} \]

   if 1.99999999999999989e-283 < b

   1. Initial program 68.7%

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

   3. Simplified 69.6%

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

      1. flip-+ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\frac{c \cdot -2}{\frac{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{\color{blue}{b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. associate-/r/ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\frac{c \cdot -2}{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}} \cdot \color{blue}{\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\left(\frac{c \cdot -2}{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}\right), \color{blue}{\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   6. Applied egg-rr 29.1%

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

   7. Taylor expanded in b around -inf

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

      1. if-same N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b\right)}{\color{blue}{a}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right)\right)}{a} \]

      4. sub-neg N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b\right)}{a} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b\right)\right), \color{blue}{a}\right) \]

   9. Simplified 33.7%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)} - b\right)}{a}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

       4. /-lowering-/.f64 67.0%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   12. Simplified 67.0%

       \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
   13. Step-by-step derivation

       1. sub0-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

       2. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{c}{b}\right)\right) \]

       3. /-lowering-/.f64 67.0%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(c, b\right)\right) \]

   14. Applied egg-rr 67.0%

       \[\leadsto \color{blue}{-\frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{-283}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0 - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.4% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.8 \cdot 10^{+65}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6.8e+65) (- 0.0 (/ b a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.8e+65) {
		tmp = 0.0 - (b / a);
	} else {
		tmp = c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 6.8d+65) then
        tmp = 0.0d0 - (b / a)
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.8e+65) {
		tmp = 0.0 - (b / a);
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 6.8e+65:
		tmp = 0.0 - (b / a)
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.8e+65)
		tmp = Float64(0.0 - Float64(b / a));
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 6.8e+65)
		tmp = 0.0 - (b / a);
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.8e+65], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.7999999999999999e65

   1. Initial program 74.8%

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

   3. Simplified 74.8%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(b\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. neg-sub0 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(0 - b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. --lowering--.f64 72.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   7. Simplified 72.4%

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

   8. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{neg}\left(\frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. neg-sub0 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;0 - \color{blue}{\frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. --lowering--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      4. /-lowering-/.f64 49.1%

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   10. Simplified 49.1%

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

       1. associate-/r* N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(0 - b\right) - b}{2}}{a}\\ \end{array} \]

       2. div-inv N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0 - b\right) - b}{2} \cdot \frac{1}{a}\\ \end{array} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(\left(\frac{\left(0 - b\right) - b}{2}\right), \left(\frac{1}{a}\right)\right)\\ \end{array} \]

   12. Applied egg-rr 48.9%

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

       1. associate-/l* N/A

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

       2. metadata-eval N/A

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

       3. *-commutative N/A

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

       4. neg-mul-1 N/A

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

       5. distribute-lft-neg-in N/A

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

       6. div-inv N/A

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

       7. sub0-neg N/A

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

       8. if-same N/A

          \[\leadsto 0 - \color{blue}{\frac{b}{a}} \]

       9. sub0-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

       10. neg-lowering-neg.f64 N/A

           \[\leadsto \mathsf{neg.f64}\left(\left(\frac{b}{a}\right)\right) \]

       11. /-lowering-/.f64 49.1%

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(b, a\right)\right) \]

   14. Applied egg-rr 49.1%

       \[\leadsto \color{blue}{-\frac{b}{a}} \]

   if 6.7999999999999999e65 < b

   1. Initial program 55.4%

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

   3. Simplified 56.9%

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

      1. flip-+ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\frac{c \cdot -2}{\frac{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{\color{blue}{b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      2. associate-/r/ N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\frac{c \cdot -2}{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}} \cdot \color{blue}{\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\left(\frac{c \cdot -2}{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}\right), \color{blue}{\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   6. Applied egg-rr 4.0%

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

   7. Taylor expanded in b around -inf

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

      1. if-same N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b\right)}{\color{blue}{a}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right)\right)}{a} \]

      4. sub-neg N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b\right)}{a} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b\right)\right), \color{blue}{a}\right) \]

   9. Simplified 11.1%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)} - b\right)}{a}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

       4. /-lowering-/.f64 95.1%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   12. Simplified 95.1%

       \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

   14. Applied egg-rr 30.4%

       \[\leadsto \color{blue}{\frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.8 \cdot 10^{+65}:\\ \;\;\;\;0 - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 10.2% accurate, 40.3× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ c b))

C

double code(double a, double b, double c) {
	return c / b;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c / b
end function

Java

public static double code(double a, double b, double c) {
	return c / b;
}

Python

def code(a, b, c):
	return c / b

Julia

function code(a, b, c)
	return Float64(c / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = c / b;
end

Wolfram

code[a_, b_, c_] := N[(c / b), $MachinePrecision]

Derivation

1. Initial program 69.9%

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

3. Simplified 70.3%

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

   1. flip-+ N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\frac{c \cdot -2}{\frac{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{\color{blue}{b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   2. associate-/r/ N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\frac{c \cdot -2}{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}} \cdot \color{blue}{\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{*.f64}\left(\left(\frac{c \cdot -2}{b \cdot b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}\right), \color{blue}{\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

6. Applied egg-rr 52.1%

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

7. Taylor expanded in b around -inf

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

   1. if-same N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b}{a}} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + -1 \cdot b\right)}{\color{blue}{a}} \]

   3. mul-1-neg N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + \left(\mathsf{neg}\left(b\right)\right)\right)}{a} \]

   4. sub-neg N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b\right)}{a} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b\right)\right), \color{blue}{a}\right) \]

9. Simplified 54.2%

   \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)} - b\right)}{a}} \]

10. Taylor expanded in b around inf

    \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
11. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

    2. neg-sub0 N/A

       \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

    3. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

    4. /-lowering-/.f64 31.3%

       \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

12. Simplified 31.3%

    \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

14. Applied egg-rr 9.9%

    \[\leadsto \color{blue}{\frac{c}{b}} \]
15. Add Preprocessing

Alternative 12: 2.6% accurate, 40.3× speedup

Math

\[\begin{array}{l} \\ \frac{b}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ b a))

C

double code(double a, double b, double c) {
	return b / a;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = b / a
end function

Java

public static double code(double a, double b, double c) {
	return b / a;
}

Python

def code(a, b, c):
	return b / a

Julia

function code(a, b, c)
	return Float64(b / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = b / a;
end

Wolfram

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

Derivation

1. Initial program 69.9%

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

3. Simplified 70.3%

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

5. Taylor expanded in b around -inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(b\right)\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   2. neg-sub0 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(0 - b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   3. --lowering--.f64 68.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, -2\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

7. Simplified 68.5%

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

8. Taylor expanded in b around -inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]
9. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{neg}\left(\frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   2. neg-sub0 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;0 - \color{blue}{\frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   3. --lowering--.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

   4. /-lowering-/.f64 37.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, b\right), b\right), \mathsf{*.f64}\left(2, a\right)\right)\\ \end{array} \]

10. Simplified 37.4%

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

    1. associate-/r* N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(0 - b\right) - b}{2}}{a}\\ \end{array} \]

    2. div-inv N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0 - b\right) - b}{2} \cdot \frac{1}{a}\\ \end{array} \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{>=.f64}\left(b, 0\right):\\ \;\;\;\;\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{*.f64}\left(\left(\frac{\left(0 - b\right) - b}{2}\right), \left(\frac{1}{a}\right)\right)\\ \end{array} \]

12. Applied egg-rr 37.2%

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

    1. associate-/l* N/A

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

    2. metadata-eval N/A

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

    3. *-commutative N/A

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

    4. neg-mul-1 N/A

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

    5. distribute-lft-neg-in N/A

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

    6. div-inv N/A

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

    7. sub0-neg N/A

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

    8. if-same N/A

       \[\leadsto 0 - \color{blue}{\frac{b}{a}} \]

    9. flip3-- N/A

       \[\leadsto \frac{{0}^{3} - {\left(\frac{b}{a}\right)}^{3}}{\color{blue}{0 \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)}} \]

    10. metadata-eval N/A

        \[\leadsto \frac{0 - {\left(\frac{b}{a}\right)}^{3}}{\color{blue}{0} \cdot 0 + \left(\frac{b}{a} \cdot \frac{b}{a} + 0 \cdot \frac{b}{a}\right)} \]

14. Applied egg-rr 2.6%

    \[\leadsto \color{blue}{\frac{b}{a}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(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))))