jeff quadratic root 1

Percentage Accurate: 72.2% → 89.8%

Time: 5.0s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{if}\;b \leq -9.8 \cdot 10^{+153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{e}{\left(e \cdot b\right) \cdot -2} \cdot \left(c + c\right)\\ \end{array}\\ \mathbf{elif}\;b \leq -3.6 \cdot 10^{-308}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \left(\frac{b}{a} - \sqrt{\frac{c}{a} \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{+41}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(b + \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* -2.0 b) (* 2.0 a))))
   (if (<= b -9.8e+153)
     (if (>= b 0.0) t_0 (* (/ E (* (* E b) -2.0)) (+ c c)))
     (if (<= b -3.6e-308)
       (if (>= b 0.0)
         (* -0.5 (- (/ b a) (sqrt (* (/ c a) -4.0))))
         (/ (+ c c) (- (sqrt (fma (* a c) -4.0 (* b b))) b)))
       (if (<= b 3.4e+41)
         (if (>= b 0.0)
           (* (+ b (sqrt (fma (* c -4.0) a (* b b)))) (/ -0.5 a))
           (/ 2.0 (sqrt (* -4.0 (/ a c)))))
         (if (>= b 0.0)
           t_0
           (/ (* 2.0 c) (+ (- b) (sqrt (* -4.0 (* a c)))))))))))

C

double code(double a, double b, double c) {
	double t_0 = (-2.0 * b) / (2.0 * a);
	double tmp_1;
	if (b <= -9.8e+153) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = (((double) M_E) / ((((double) M_E) * b) * -2.0)) * (c + c);
		}
		tmp_1 = tmp_2;
	} else if (b <= -3.6e-308) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -0.5 * ((b / a) - sqrt(((c / a) * -4.0)));
		} else {
			tmp_3 = (c + c) / (sqrt(fma((a * c), -4.0, (b * b))) - b);
		}
		tmp_1 = tmp_3;
	} else if (b <= 3.4e+41) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (b + sqrt(fma((c * -4.0), a, (b * b)))) * (-0.5 / a);
		} else {
			tmp_4 = 2.0 / sqrt((-4.0 * (a / c)));
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = (2.0 * c) / (-b + sqrt((-4.0 * (a * c))));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-2.0 * b) / Float64(2.0 * a))
	tmp_1 = 0.0
	if (b <= -9.8e+153)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(Float64(exp(1) / Float64(Float64(exp(1) * b) * -2.0)) * Float64(c + c));
		end
		tmp_1 = tmp_2;
	elseif (b <= -3.6e-308)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(-0.5 * Float64(Float64(b / a) - sqrt(Float64(Float64(c / a) * -4.0))));
		else
			tmp_3 = Float64(Float64(c + c) / Float64(sqrt(fma(Float64(a * c), -4.0, Float64(b * b))) - b));
		end
		tmp_1 = tmp_3;
	elseif (b <= 3.4e+41)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(b + sqrt(fma(Float64(c * -4.0), a, Float64(b * b)))) * Float64(-0.5 / a));
		else
			tmp_4 = Float64(2.0 / sqrt(Float64(-4.0 * Float64(a / c))));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = t_0;
	else
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + sqrt(Float64(-4.0 * Float64(a * c)))));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-2.0 * b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -9.8e+153], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(E / N[(N[(E * b), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] * N[(c + c), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -3.6e-308], If[GreaterEqual[b, 0.0], N[(-0.5 * N[(N[(b / a), $MachinePrecision] - N[Sqrt[N[(N[(c / a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 3.4e+41], If[GreaterEqual[b, 0.0], N[(N[(b + N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -9.80000000000000003e153

   1. Initial program 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. 1-exp N/A

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

      11. metadata-eval N/A

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

      12. exp-diff N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-E.f64 70.1

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

   6. Applied rewrites 70.1%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 67.9

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

   9. Applied rewrites 67.9%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

   11. Applied rewrites 67.9%

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

   if -9.80000000000000003e153 < b < -3.5999999999999999e-308

   1. Initial program 72.2%

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

   2. Taylor expanded in a around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 48.8

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

   4. Applied rewrites 48.8%

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

   6. Applied rewrites 48.8%

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

   if -3.5999999999999999e-308 < b < 3.39999999999999998e41

   1. Initial program 72.2%

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 44.5

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

   6. Applied rewrites 44.5%

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

   if 3.39999999999999998e41 < b

   1. Initial program 72.2%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.6

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

   4. Applied rewrites 56.6%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 40.7

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

   7. Applied rewrites 40.7%

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

   8. Taylor expanded in b around inf

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

      1. lower-*.f64 54.0

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

   10. Applied rewrites 54.0%

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

Alternative 2: 89.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-2 \cdot b}{2 \cdot a}\\ t_1 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;b \leq -9.8 \cdot 10^{+153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{e}{\left(e \cdot b\right) \cdot -2} \cdot \left(c + c\right)\\ \end{array}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{+41}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t\_1}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{a \cdot \left(e \cdot \sqrt{-4 \cdot \frac{c}{a}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* -2.0 b) (* 2.0 a)))
        (t_1 (sqrt (- (* b b) (* (* 4.0 a) c)))))
   (if (<= b -9.8e+153)
     (if (>= b 0.0) t_0 (* (/ E (* (* E b) -2.0)) (+ c c)))
     (if (<= b 3.4e+41)
       (if (>= b 0.0) (/ (- (- b) t_1) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) t_1)))
       (if (>= b 0.0)
         t_0
         (/ (* (+ c c) E) (* a (* E (sqrt (* -4.0 (/ c a)))))))))))

C

double code(double a, double b, double c) {
	double t_0 = (-2.0 * b) / (2.0 * a);
	double t_1 = sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp_1;
	if (b <= -9.8e+153) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = (((double) M_E) / ((((double) M_E) * b) * -2.0)) * (c + c);
		}
		tmp_1 = tmp_2;
	} else if (b <= 3.4e+41) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_1) / (2.0 * a);
		} else {
			tmp_3 = (2.0 * c) / (-b + t_1);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = ((c + c) * ((double) M_E)) / (a * (((double) M_E) * sqrt((-4.0 * (c / a)))));
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = (-2.0 * b) / (2.0 * a);
	double t_1 = Math.sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp_1;
	if (b <= -9.8e+153) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = (Math.E / ((Math.E * b) * -2.0)) * (c + c);
		}
		tmp_1 = tmp_2;
	} else if (b <= 3.4e+41) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_1) / (2.0 * a);
		} else {
			tmp_3 = (2.0 * c) / (-b + t_1);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = ((c + c) * Math.E) / (a * (Math.E * Math.sqrt((-4.0 * (c / a)))));
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (-2.0 * b) / (2.0 * a)
	t_1 = math.sqrt(((b * b) - ((4.0 * a) * c)))
	tmp_1 = 0
	if b <= -9.8e+153:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_0
		else:
			tmp_2 = (math.e / ((math.e * b) * -2.0)) * (c + c)
		tmp_1 = tmp_2
	elif b <= 3.4e+41:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (-b - t_1) / (2.0 * a)
		else:
			tmp_3 = (2.0 * c) / (-b + t_1)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = t_0
	else:
		tmp_1 = ((c + c) * math.e) / (a * (math.e * math.sqrt((-4.0 * (c / a)))))
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-2.0 * b) / Float64(2.0 * a))
	t_1 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	tmp_1 = 0.0
	if (b <= -9.8e+153)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(Float64(exp(1) / Float64(Float64(exp(1) * b) * -2.0)) * Float64(c + c));
		end
		tmp_1 = tmp_2;
	elseif (b <= 3.4e+41)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(-b) - t_1) / Float64(2.0 * a));
		else
			tmp_3 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_1));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = t_0;
	else
		tmp_1 = Float64(Float64(Float64(c + c) * exp(1)) / Float64(a * Float64(exp(1) * sqrt(Float64(-4.0 * Float64(c / a))))));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = (-2.0 * b) / (2.0 * a);
	t_1 = sqrt(((b * b) - ((4.0 * a) * c)));
	tmp_2 = 0.0;
	if (b <= -9.8e+153)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_0;
		else
			tmp_3 = (2.71828182845904523536 / ((2.71828182845904523536 * b) * -2.0)) * (c + c);
		end
		tmp_2 = tmp_3;
	elseif (b <= 3.4e+41)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (-b - t_1) / (2.0 * a);
		else
			tmp_4 = (2.0 * c) / (-b + t_1);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = t_0;
	else
		tmp_2 = ((c + c) * 2.71828182845904523536) / (a * (2.71828182845904523536 * sqrt((-4.0 * (c / a)))));
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-2.0 * b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -9.8e+153], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(E / N[(N[(E * b), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] * N[(c + c), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 3.4e+41], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$1), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$1), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(N[(c + c), $MachinePrecision] * E), $MachinePrecision] / N[(a * N[(E * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.80000000000000003e153

   1. Initial program 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. 1-exp N/A

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

      11. metadata-eval N/A

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

      12. exp-diff N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-E.f64 70.1

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

   6. Applied rewrites 70.1%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 67.9

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

   9. Applied rewrites 67.9%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

   11. Applied rewrites 67.9%

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

   if -9.80000000000000003e153 < b < 3.39999999999999998e41

   1. Initial program 72.2%

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

   if 3.39999999999999998e41 < b

   1. Initial program 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. 1-exp N/A

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

      11. metadata-eval N/A

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

      12. exp-diff N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-E.f64 70.1

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

   6. Applied rewrites 70.1%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 67.9

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

   9. Applied rewrites 67.9%

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

   10. Taylor expanded in a around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-E.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 41.0

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

   12. Applied rewrites 41.0%

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

Alternative 3: 89.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-2 \cdot b}{2 \cdot a}\\ t_1 := \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}\\ \mathbf{if}\;b \leq -9.8 \cdot 10^{+153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{e}{\left(e \cdot b\right) \cdot -2} \cdot \left(c + c\right)\\ \end{array}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{+41}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(b + t\_1\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_1 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* -2.0 b) (* 2.0 a))) (t_1 (sqrt (fma (* c -4.0) a (* b b)))))
   (if (<= b -9.8e+153)
     (if (>= b 0.0) t_0 (* (/ E (* (* E b) -2.0)) (+ c c)))
     (if (<= b 3.4e+41)
       (if (>= b 0.0) (* (+ b t_1) (/ -0.5 a)) (/ (+ c c) (- t_1 b)))
       (if (>= b 0.0) t_0 (/ (* 2.0 c) (+ (- b) (sqrt (* -4.0 (* a c))))))))))

C

double code(double a, double b, double c) {
	double t_0 = (-2.0 * b) / (2.0 * a);
	double t_1 = sqrt(fma((c * -4.0), a, (b * b)));
	double tmp_1;
	if (b <= -9.8e+153) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = (((double) M_E) / ((((double) M_E) * b) * -2.0)) * (c + c);
		}
		tmp_1 = tmp_2;
	} else if (b <= 3.4e+41) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (b + t_1) * (-0.5 / a);
		} else {
			tmp_3 = (c + c) / (t_1 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = (2.0 * c) / (-b + sqrt((-4.0 * (a * c))));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-2.0 * b) / Float64(2.0 * a))
	t_1 = sqrt(fma(Float64(c * -4.0), a, Float64(b * b)))
	tmp_1 = 0.0
	if (b <= -9.8e+153)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(Float64(exp(1) / Float64(Float64(exp(1) * b) * -2.0)) * Float64(c + c));
		end
		tmp_1 = tmp_2;
	elseif (b <= 3.4e+41)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(b + t_1) * Float64(-0.5 / a));
		else
			tmp_3 = Float64(Float64(c + c) / Float64(t_1 - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = t_0;
	else
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + sqrt(Float64(-4.0 * Float64(a * c)))));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-2.0 * b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -9.8e+153], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(E / N[(N[(E * b), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] * N[(c + c), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 3.4e+41], If[GreaterEqual[b, 0.0], N[(N[(b + t$95$1), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(t$95$1 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.80000000000000003e153

   1. Initial program 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. 1-exp N/A

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

      11. metadata-eval N/A

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

      12. exp-diff N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-E.f64 70.1

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

   6. Applied rewrites 70.1%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 67.9

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

   9. Applied rewrites 67.9%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

   11. Applied rewrites 67.9%

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

   if -9.80000000000000003e153 < b < 3.39999999999999998e41

   1. Initial program 72.2%

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

   3. Applied rewrites 72.1%

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

   if 3.39999999999999998e41 < b

   1. Initial program 72.2%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.6

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

   4. Applied rewrites 56.6%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 40.7

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

   7. Applied rewrites 40.7%

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

   8. Taylor expanded in b around inf

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

      1. lower-*.f64 54.0

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

   10. Applied rewrites 54.0%

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

Alternative 4: 89.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-2 \cdot b}{2 \cdot a}\\ t_1 := \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}\\ \mathbf{if}\;b \leq -9.8 \cdot 10^{+153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{e}{\left(e \cdot b\right) \cdot -2} \cdot \left(c + c\right)\\ \end{array}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{+41}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(b + t\_1\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_1 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{a \cdot \left(e \cdot \sqrt{-4 \cdot \frac{c}{a}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* -2.0 b) (* 2.0 a))) (t_1 (sqrt (fma (* c -4.0) a (* b b)))))
   (if (<= b -9.8e+153)
     (if (>= b 0.0) t_0 (* (/ E (* (* E b) -2.0)) (+ c c)))
     (if (<= b 3.4e+41)
       (if (>= b 0.0) (* (+ b t_1) (/ -0.5 a)) (/ (+ c c) (- t_1 b)))
       (if (>= b 0.0)
         t_0
         (/ (* (+ c c) E) (* a (* E (sqrt (* -4.0 (/ c a)))))))))))

C

double code(double a, double b, double c) {
	double t_0 = (-2.0 * b) / (2.0 * a);
	double t_1 = sqrt(fma((c * -4.0), a, (b * b)));
	double tmp_1;
	if (b <= -9.8e+153) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = (((double) M_E) / ((((double) M_E) * b) * -2.0)) * (c + c);
		}
		tmp_1 = tmp_2;
	} else if (b <= 3.4e+41) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (b + t_1) * (-0.5 / a);
		} else {
			tmp_3 = (c + c) / (t_1 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = ((c + c) * ((double) M_E)) / (a * (((double) M_E) * sqrt((-4.0 * (c / a)))));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-2.0 * b) / Float64(2.0 * a))
	t_1 = sqrt(fma(Float64(c * -4.0), a, Float64(b * b)))
	tmp_1 = 0.0
	if (b <= -9.8e+153)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(Float64(exp(1) / Float64(Float64(exp(1) * b) * -2.0)) * Float64(c + c));
		end
		tmp_1 = tmp_2;
	elseif (b <= 3.4e+41)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(b + t_1) * Float64(-0.5 / a));
		else
			tmp_3 = Float64(Float64(c + c) / Float64(t_1 - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = t_0;
	else
		tmp_1 = Float64(Float64(Float64(c + c) * exp(1)) / Float64(a * Float64(exp(1) * sqrt(Float64(-4.0 * Float64(c / a))))));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-2.0 * b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -9.8e+153], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(E / N[(N[(E * b), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] * N[(c + c), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 3.4e+41], If[GreaterEqual[b, 0.0], N[(N[(b + t$95$1), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(t$95$1 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(N[(c + c), $MachinePrecision] * E), $MachinePrecision] / N[(a * N[(E * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.80000000000000003e153

   1. Initial program 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. 1-exp N/A

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

      11. metadata-eval N/A

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

      12. exp-diff N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-E.f64 70.1

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

   6. Applied rewrites 70.1%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 67.9

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

   9. Applied rewrites 67.9%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

   11. Applied rewrites 67.9%

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

   if -9.80000000000000003e153 < b < 3.39999999999999998e41

   1. Initial program 72.2%

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

   3. Applied rewrites 72.1%

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

   if 3.39999999999999998e41 < b

   1. Initial program 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. 1-exp N/A

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

      11. metadata-eval N/A

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

      12. exp-diff N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-E.f64 70.1

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

   6. Applied rewrites 70.1%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 67.9

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

   9. Applied rewrites 67.9%

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

   10. Taylor expanded in a around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-E.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 41.0

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

   12. Applied rewrites 41.0%

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

Alternative 5: 85.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-4 \cdot \frac{a}{c}}\\ t_1 := \frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{if}\;b \leq -9.8 \cdot 10^{+153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{e}{\left(e \cdot b\right) \cdot -2} \cdot \left(c + c\right)\\ \end{array}\\ \mathbf{elif}\;b \leq -3.6 \cdot 10^{-308}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \left(\frac{b}{a} - \sqrt{\frac{c}{a} \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.28 \cdot 10^{-105}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot c\right) \cdot a} + b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 \cdot t\_0}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* -4.0 (/ a c)))) (t_1 (/ (* -2.0 b) (* 2.0 a))))
   (if (<= b -9.8e+153)
     (if (>= b 0.0) t_1 (* (/ E (* (* E b) -2.0)) (+ c c)))
     (if (<= b -3.6e-308)
       (if (>= b 0.0)
         (* -0.5 (- (/ b a) (sqrt (* (/ c a) -4.0))))
         (/ (+ c c) (- (sqrt (fma (* a c) -4.0 (* b b))) b)))
       (if (<= b 1.28e-105)
         (if (>= b 0.0)
           (/ (+ (sqrt (* (* -4.0 c) a)) b) (* -2.0 a))
           (/ 1.0 (* 0.5 t_0)))
         (if (>= b 0.0) t_1 (/ 2.0 t_0)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((-4.0 * (a / c)));
	double t_1 = (-2.0 * b) / (2.0 * a);
	double tmp_1;
	if (b <= -9.8e+153) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = (((double) M_E) / ((((double) M_E) * b) * -2.0)) * (c + c);
		}
		tmp_1 = tmp_2;
	} else if (b <= -3.6e-308) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -0.5 * ((b / a) - sqrt(((c / a) * -4.0)));
		} else {
			tmp_3 = (c + c) / (sqrt(fma((a * c), -4.0, (b * b))) - b);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.28e-105) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (sqrt(((-4.0 * c) * a)) + b) / (-2.0 * a);
		} else {
			tmp_4 = 1.0 / (0.5 * t_0);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = t_1;
	} else {
		tmp_1 = 2.0 / t_0;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(-4.0 * Float64(a / c)))
	t_1 = Float64(Float64(-2.0 * b) / Float64(2.0 * a))
	tmp_1 = 0.0
	if (b <= -9.8e+153)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(Float64(exp(1) / Float64(Float64(exp(1) * b) * -2.0)) * Float64(c + c));
		end
		tmp_1 = tmp_2;
	elseif (b <= -3.6e-308)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(-0.5 * Float64(Float64(b / a) - sqrt(Float64(Float64(c / a) * -4.0))));
		else
			tmp_3 = Float64(Float64(c + c) / Float64(sqrt(fma(Float64(a * c), -4.0, Float64(b * b))) - b));
		end
		tmp_1 = tmp_3;
	elseif (b <= 1.28e-105)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(sqrt(Float64(Float64(-4.0 * c) * a)) + b) / Float64(-2.0 * a));
		else
			tmp_4 = Float64(1.0 / Float64(0.5 * t_0));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = t_1;
	else
		tmp_1 = Float64(2.0 / t_0);
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -9.8e+153], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(E / N[(N[(E * b), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] * N[(c + c), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -3.6e-308], If[GreaterEqual[b, 0.0], N[(-0.5 * N[(N[(b / a), $MachinePrecision] - N[Sqrt[N[(N[(c / a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.28e-105], If[GreaterEqual[b, 0.0], N[(N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$1, N[(2.0 / t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -9.80000000000000003e153

   1. Initial program 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. 1-exp N/A

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

      11. metadata-eval N/A

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

      12. exp-diff N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-E.f64 70.1

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

   6. Applied rewrites 70.1%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 67.9

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

   9. Applied rewrites 67.9%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

   11. Applied rewrites 67.9%

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

   if -9.80000000000000003e153 < b < -3.5999999999999999e-308

   1. Initial program 72.2%

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

   2. Taylor expanded in a around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 48.8

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

   4. Applied rewrites 48.8%

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

   6. Applied rewrites 48.8%

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

   if -3.5999999999999999e-308 < b < 1.28e-105

   1. Initial program 72.2%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.6

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

   4. Applied rewrites 56.6%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 40.7

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

   7. Applied rewrites 40.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 40.7

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

   9. Applied rewrites 40.7%

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

   10. Taylor expanded in c around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 29.0

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

   12. Applied rewrites 29.0%

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

       1. lift-/.f64 N/A

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

       2. frac-2neg N/A

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

       3. lower-/.f64 N/A

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

   14. Applied rewrites 29.0%

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

   if 1.28e-105 < b

   1. Initial program 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. 1-exp N/A

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

      11. metadata-eval N/A

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

      12. exp-diff N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-E.f64 70.1

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

   6. Applied rewrites 70.1%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 67.9

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

   9. Applied rewrites 67.9%

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

   10. Taylor expanded in c around inf

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

       1. lower-/.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 42.4

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

   12. Applied rewrites 42.4%

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

Alternative 6: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left(a \cdot -4\right) \cdot c}\\ \mathbf{if}\;b \leq -1.7 \cdot 10^{-113}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;0.5 \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.28 \cdot 10^{-105}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_0}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t\_0}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* (* a -4.0) c))))
   (if (<= b -1.7e-113)
     (if (>= b 0.0)
       (* 0.5 (* c (sqrt (/ -4.0 (* a c)))))
       (/ (* (+ c c) E) (* -2.0 (* b E))))
     (if (<= b 1.28e-105)
       (if (>= b 0.0) (/ (- (- b) t_0) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) t_0)))
       (if (>= b 0.0)
         (/ (* -2.0 b) (* 2.0 a))
         (/ 2.0 (sqrt (* -4.0 (/ a c)))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((a * -4.0) * c));
	double tmp_1;
	if (b <= -1.7e-113) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 0.5 * (c * sqrt((-4.0 / (a * c))));
		} else {
			tmp_2 = ((c + c) * ((double) M_E)) / (-2.0 * (b * ((double) M_E)));
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.28e-105) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_0) / (2.0 * a);
		} else {
			tmp_3 = (2.0 * c) / (-b + t_0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * b) / (2.0 * a);
	} else {
		tmp_1 = 2.0 / sqrt((-4.0 * (a / c)));
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((a * -4.0) * c));
	double tmp_1;
	if (b <= -1.7e-113) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 0.5 * (c * Math.sqrt((-4.0 / (a * c))));
		} else {
			tmp_2 = ((c + c) * Math.E) / (-2.0 * (b * Math.E));
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.28e-105) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_0) / (2.0 * a);
		} else {
			tmp_3 = (2.0 * c) / (-b + t_0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * b) / (2.0 * a);
	} else {
		tmp_1 = 2.0 / Math.sqrt((-4.0 * (a / c)));
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((a * -4.0) * c))
	tmp_1 = 0
	if b <= -1.7e-113:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = 0.5 * (c * math.sqrt((-4.0 / (a * c))))
		else:
			tmp_2 = ((c + c) * math.e) / (-2.0 * (b * math.e))
		tmp_1 = tmp_2
	elif b <= 1.28e-105:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (-b - t_0) / (2.0 * a)
		else:
			tmp_3 = (2.0 * c) / (-b + t_0)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = (-2.0 * b) / (2.0 * a)
	else:
		tmp_1 = 2.0 / math.sqrt((-4.0 * (a / c)))
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(a * -4.0) * c))
	tmp_1 = 0.0
	if (b <= -1.7e-113)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(0.5 * Float64(c * sqrt(Float64(-4.0 / Float64(a * c)))));
		else
			tmp_2 = Float64(Float64(Float64(c + c) * exp(1)) / Float64(-2.0 * Float64(b * exp(1))));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.28e-105)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(-b) - t_0) / Float64(2.0 * a));
		else
			tmp_3 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_0));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-2.0 * b) / Float64(2.0 * a));
	else
		tmp_1 = Float64(2.0 / sqrt(Float64(-4.0 * Float64(a / c))));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((a * -4.0) * c));
	tmp_2 = 0.0;
	if (b <= -1.7e-113)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = 0.5 * (c * sqrt((-4.0 / (a * c))));
		else
			tmp_3 = ((c + c) * 2.71828182845904523536) / (-2.0 * (b * 2.71828182845904523536));
		end
		tmp_2 = tmp_3;
	elseif (b <= 1.28e-105)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (-b - t_0) / (2.0 * a);
		else
			tmp_4 = (2.0 * c) / (-b + t_0);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = (-2.0 * b) / (2.0 * a);
	else
		tmp_2 = 2.0 / sqrt((-4.0 * (a / c)));
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(a * -4.0), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.7e-113], If[GreaterEqual[b, 0.0], N[(0.5 * N[(c * N[Sqrt[N[(-4.0 / N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c + c), $MachinePrecision] * E), $MachinePrecision] / N[(-2.0 * N[(b * E), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.28e-105], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.7000000000000001e-113

   1. Initial program 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. 1-exp N/A

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

      11. metadata-eval N/A

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

      12. exp-diff N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-E.f64 70.1

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

   6. Applied rewrites 70.1%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 41.8

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

   9. Applied rewrites 41.8%

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

   10. Taylor expanded in c around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 48.7

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

   12. Applied rewrites 48.7%

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

   if -1.7000000000000001e-113 < b < 1.28e-105

   1. Initial program 72.2%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.6

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

   4. Applied rewrites 56.6%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 40.7

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

   7. Applied rewrites 40.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 40.7

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

   9. Applied rewrites 40.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 40.7

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

   11. Applied rewrites 40.7%

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

   if 1.28e-105 < b

   1. Initial program 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. 1-exp N/A

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

      11. metadata-eval N/A

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

      12. exp-diff N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-E.f64 70.1

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

   6. Applied rewrites 70.1%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 67.9

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

   9. Applied rewrites 67.9%

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

   10. Taylor expanded in c around inf

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

       1. lower-/.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 42.4

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

   12. Applied rewrites 42.4%

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

Alternative 7: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b \leq -1.7 \cdot 10^{-113}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;0.5 \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.28 \cdot 10^{-105}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_0}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t\_0}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* -4.0 (* a c)))))
   (if (<= b -1.7e-113)
     (if (>= b 0.0)
       (* 0.5 (* c (sqrt (/ -4.0 (* a c)))))
       (/ (* (+ c c) E) (* -2.0 (* b E))))
     (if (<= b 1.28e-105)
       (if (>= b 0.0) (/ (- (- b) t_0) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) t_0)))
       (if (>= b 0.0)
         (/ (* -2.0 b) (* 2.0 a))
         (/ 2.0 (sqrt (* -4.0 (/ a c)))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((-4.0 * (a * c)));
	double tmp_1;
	if (b <= -1.7e-113) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 0.5 * (c * sqrt((-4.0 / (a * c))));
		} else {
			tmp_2 = ((c + c) * ((double) M_E)) / (-2.0 * (b * ((double) M_E)));
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.28e-105) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_0) / (2.0 * a);
		} else {
			tmp_3 = (2.0 * c) / (-b + t_0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * b) / (2.0 * a);
	} else {
		tmp_1 = 2.0 / sqrt((-4.0 * (a / c)));
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt((-4.0 * (a * c)));
	double tmp_1;
	if (b <= -1.7e-113) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 0.5 * (c * Math.sqrt((-4.0 / (a * c))));
		} else {
			tmp_2 = ((c + c) * Math.E) / (-2.0 * (b * Math.E));
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.28e-105) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_0) / (2.0 * a);
		} else {
			tmp_3 = (2.0 * c) / (-b + t_0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * b) / (2.0 * a);
	} else {
		tmp_1 = 2.0 / Math.sqrt((-4.0 * (a / c)));
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt((-4.0 * (a * c)))
	tmp_1 = 0
	if b <= -1.7e-113:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = 0.5 * (c * math.sqrt((-4.0 / (a * c))))
		else:
			tmp_2 = ((c + c) * math.e) / (-2.0 * (b * math.e))
		tmp_1 = tmp_2
	elif b <= 1.28e-105:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (-b - t_0) / (2.0 * a)
		else:
			tmp_3 = (2.0 * c) / (-b + t_0)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = (-2.0 * b) / (2.0 * a)
	else:
		tmp_1 = 2.0 / math.sqrt((-4.0 * (a / c)))
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(-4.0 * Float64(a * c)))
	tmp_1 = 0.0
	if (b <= -1.7e-113)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(0.5 * Float64(c * sqrt(Float64(-4.0 / Float64(a * c)))));
		else
			tmp_2 = Float64(Float64(Float64(c + c) * exp(1)) / Float64(-2.0 * Float64(b * exp(1))));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.28e-105)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(-b) - t_0) / Float64(2.0 * a));
		else
			tmp_3 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_0));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-2.0 * b) / Float64(2.0 * a));
	else
		tmp_1 = Float64(2.0 / sqrt(Float64(-4.0 * Float64(a / c))));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt((-4.0 * (a * c)));
	tmp_2 = 0.0;
	if (b <= -1.7e-113)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = 0.5 * (c * sqrt((-4.0 / (a * c))));
		else
			tmp_3 = ((c + c) * 2.71828182845904523536) / (-2.0 * (b * 2.71828182845904523536));
		end
		tmp_2 = tmp_3;
	elseif (b <= 1.28e-105)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (-b - t_0) / (2.0 * a);
		else
			tmp_4 = (2.0 * c) / (-b + t_0);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = (-2.0 * b) / (2.0 * a);
	else
		tmp_2 = 2.0 / sqrt((-4.0 * (a / c)));
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.7e-113], If[GreaterEqual[b, 0.0], N[(0.5 * N[(c * N[Sqrt[N[(-4.0 / N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c + c), $MachinePrecision] * E), $MachinePrecision] / N[(-2.0 * N[(b * E), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.28e-105], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.7000000000000001e-113

   1. Initial program 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. 1-exp N/A

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

      11. metadata-eval N/A

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

      12. exp-diff N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-E.f64 70.1

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

   6. Applied rewrites 70.1%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 41.8

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

   9. Applied rewrites 41.8%

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

   10. Taylor expanded in c around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 48.7

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

   12. Applied rewrites 48.7%

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

   if -1.7000000000000001e-113 < b < 1.28e-105

   1. Initial program 72.2%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.6

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

   4. Applied rewrites 56.6%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 40.7

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

   7. Applied rewrites 40.7%

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

   if 1.28e-105 < b

   1. Initial program 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. 1-exp N/A

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

      11. metadata-eval N/A

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

      12. exp-diff N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-E.f64 70.1

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

   6. Applied rewrites 70.1%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 67.9

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

   9. Applied rewrites 67.9%

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

   10. Taylor expanded in c around inf

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

       1. lower-/.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 42.4

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

   12. Applied rewrites 42.4%

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

Alternative 8: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left(a \cdot c\right) \cdot -4}\\ \mathbf{if}\;b \leq -1.7 \cdot 10^{-113}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;0.5 \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.28 \cdot 10^{-105}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(t\_0 + b\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* (* a c) -4.0))))
   (if (<= b -1.7e-113)
     (if (>= b 0.0)
       (* 0.5 (* c (sqrt (/ -4.0 (* a c)))))
       (/ (* (+ c c) E) (* -2.0 (* b E))))
     (if (<= b 1.28e-105)
       (if (>= b 0.0) (* (+ t_0 b) (/ -0.5 a)) (/ (+ c c) (- t_0 b)))
       (if (>= b 0.0)
         (/ (* -2.0 b) (* 2.0 a))
         (/ 2.0 (sqrt (* -4.0 (/ a c)))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((a * c) * -4.0));
	double tmp_1;
	if (b <= -1.7e-113) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 0.5 * (c * sqrt((-4.0 / (a * c))));
		} else {
			tmp_2 = ((c + c) * ((double) M_E)) / (-2.0 * (b * ((double) M_E)));
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.28e-105) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (t_0 + b) * (-0.5 / a);
		} else {
			tmp_3 = (c + c) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * b) / (2.0 * a);
	} else {
		tmp_1 = 2.0 / sqrt((-4.0 * (a / c)));
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((a * c) * -4.0));
	double tmp_1;
	if (b <= -1.7e-113) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 0.5 * (c * Math.sqrt((-4.0 / (a * c))));
		} else {
			tmp_2 = ((c + c) * Math.E) / (-2.0 * (b * Math.E));
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.28e-105) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (t_0 + b) * (-0.5 / a);
		} else {
			tmp_3 = (c + c) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * b) / (2.0 * a);
	} else {
		tmp_1 = 2.0 / Math.sqrt((-4.0 * (a / c)));
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((a * c) * -4.0))
	tmp_1 = 0
	if b <= -1.7e-113:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = 0.5 * (c * math.sqrt((-4.0 / (a * c))))
		else:
			tmp_2 = ((c + c) * math.e) / (-2.0 * (b * math.e))
		tmp_1 = tmp_2
	elif b <= 1.28e-105:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (t_0 + b) * (-0.5 / a)
		else:
			tmp_3 = (c + c) / (t_0 - b)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = (-2.0 * b) / (2.0 * a)
	else:
		tmp_1 = 2.0 / math.sqrt((-4.0 * (a / c)))
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(a * c) * -4.0))
	tmp_1 = 0.0
	if (b <= -1.7e-113)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(0.5 * Float64(c * sqrt(Float64(-4.0 / Float64(a * c)))));
		else
			tmp_2 = Float64(Float64(Float64(c + c) * exp(1)) / Float64(-2.0 * Float64(b * exp(1))));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.28e-105)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(t_0 + b) * Float64(-0.5 / a));
		else
			tmp_3 = Float64(Float64(c + c) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-2.0 * b) / Float64(2.0 * a));
	else
		tmp_1 = Float64(2.0 / sqrt(Float64(-4.0 * Float64(a / c))));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((a * c) * -4.0));
	tmp_2 = 0.0;
	if (b <= -1.7e-113)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = 0.5 * (c * sqrt((-4.0 / (a * c))));
		else
			tmp_3 = ((c + c) * 2.71828182845904523536) / (-2.0 * (b * 2.71828182845904523536));
		end
		tmp_2 = tmp_3;
	elseif (b <= 1.28e-105)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (t_0 + b) * (-0.5 / a);
		else
			tmp_4 = (c + c) / (t_0 - b);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = (-2.0 * b) / (2.0 * a);
	else
		tmp_2 = 2.0 / sqrt((-4.0 * (a / c)));
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.7e-113], If[GreaterEqual[b, 0.0], N[(0.5 * N[(c * N[Sqrt[N[(-4.0 / N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c + c), $MachinePrecision] * E), $MachinePrecision] / N[(-2.0 * N[(b * E), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.28e-105], If[GreaterEqual[b, 0.0], N[(N[(t$95$0 + b), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.7000000000000001e-113

   1. Initial program 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. 1-exp N/A

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

      11. metadata-eval N/A

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

      12. exp-diff N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-E.f64 70.1

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

   6. Applied rewrites 70.1%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 41.8

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

   9. Applied rewrites 41.8%

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

   10. Taylor expanded in c around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 48.7

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

   12. Applied rewrites 48.7%

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

   if -1.7000000000000001e-113 < b < 1.28e-105

   1. Initial program 72.2%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.6

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

   4. Applied rewrites 56.6%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 40.7

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

   7. Applied rewrites 40.7%

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

   9. Applied rewrites 40.7%

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

   if 1.28e-105 < b

   1. Initial program 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. 1-exp N/A

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

      11. metadata-eval N/A

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

      12. exp-diff N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-E.f64 70.1

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

   6. Applied rewrites 70.1%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 67.9

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

   9. Applied rewrites 67.9%

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

   10. Taylor expanded in c around inf

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

       1. lower-/.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 42.4

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

   12. Applied rewrites 42.4%

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

Alternative 9: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{-113}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;0.5 \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.28 \cdot 10^{-105}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(\sqrt{\left(a \cdot c\right) \cdot -4} + b\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-2}{b - \sqrt{\left(-4 \cdot c\right) \cdot a}}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.7e-113)
   (if (>= b 0.0)
     (* 0.5 (* c (sqrt (/ -4.0 (* a c)))))
     (/ (* (+ c c) E) (* -2.0 (* b E))))
   (if (<= b 1.28e-105)
     (if (>= b 0.0)
       (* (+ (sqrt (* (* a c) -4.0)) b) (/ -0.5 a))
       (* c (/ -2.0 (- b (sqrt (* (* -4.0 c) a))))))
     (if (>= b 0.0)
       (/ (* -2.0 b) (* 2.0 a))
       (/ 2.0 (sqrt (* -4.0 (/ a c))))))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1.7e-113) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 0.5 * (c * sqrt((-4.0 / (a * c))));
		} else {
			tmp_2 = ((c + c) * ((double) M_E)) / (-2.0 * (b * ((double) M_E)));
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.28e-105) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (sqrt(((a * c) * -4.0)) + b) * (-0.5 / a);
		} else {
			tmp_3 = c * (-2.0 / (b - sqrt(((-4.0 * c) * a))));
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * b) / (2.0 * a);
	} else {
		tmp_1 = 2.0 / sqrt((-4.0 * (a / c)));
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1.7e-113) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 0.5 * (c * Math.sqrt((-4.0 / (a * c))));
		} else {
			tmp_2 = ((c + c) * Math.E) / (-2.0 * (b * Math.E));
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.28e-105) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (Math.sqrt(((a * c) * -4.0)) + b) * (-0.5 / a);
		} else {
			tmp_3 = c * (-2.0 / (b - Math.sqrt(((-4.0 * c) * a))));
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * b) / (2.0 * a);
	} else {
		tmp_1 = 2.0 / Math.sqrt((-4.0 * (a / c)));
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if b <= -1.7e-113:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = 0.5 * (c * math.sqrt((-4.0 / (a * c))))
		else:
			tmp_2 = ((c + c) * math.e) / (-2.0 * (b * math.e))
		tmp_1 = tmp_2
	elif b <= 1.28e-105:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (math.sqrt(((a * c) * -4.0)) + b) * (-0.5 / a)
		else:
			tmp_3 = c * (-2.0 / (b - math.sqrt(((-4.0 * c) * a))))
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = (-2.0 * b) / (2.0 * a)
	else:
		tmp_1 = 2.0 / math.sqrt((-4.0 * (a / c)))
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -1.7e-113)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(0.5 * Float64(c * sqrt(Float64(-4.0 / Float64(a * c)))));
		else
			tmp_2 = Float64(Float64(Float64(c + c) * exp(1)) / Float64(-2.0 * Float64(b * exp(1))));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.28e-105)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(sqrt(Float64(Float64(a * c) * -4.0)) + b) * Float64(-0.5 / a));
		else
			tmp_3 = Float64(c * Float64(-2.0 / Float64(b - sqrt(Float64(Float64(-4.0 * c) * a)))));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-2.0 * b) / Float64(2.0 * a));
	else
		tmp_1 = Float64(2.0 / sqrt(Float64(-4.0 * Float64(a / c))));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	tmp_2 = 0.0;
	if (b <= -1.7e-113)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = 0.5 * (c * sqrt((-4.0 / (a * c))));
		else
			tmp_3 = ((c + c) * 2.71828182845904523536) / (-2.0 * (b * 2.71828182845904523536));
		end
		tmp_2 = tmp_3;
	elseif (b <= 1.28e-105)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (sqrt(((a * c) * -4.0)) + b) * (-0.5 / a);
		else
			tmp_4 = c * (-2.0 / (b - sqrt(((-4.0 * c) * a))));
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = (-2.0 * b) / (2.0 * a);
	else
		tmp_2 = 2.0 / sqrt((-4.0 * (a / c)));
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.7e-113], If[GreaterEqual[b, 0.0], N[(0.5 * N[(c * N[Sqrt[N[(-4.0 / N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c + c), $MachinePrecision] * E), $MachinePrecision] / N[(-2.0 * N[(b * E), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.28e-105], If[GreaterEqual[b, 0.0], N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-2.0 / N[(b - N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.7000000000000001e-113

   1. Initial program 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. 1-exp N/A

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

      11. metadata-eval N/A

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

      12. exp-diff N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-E.f64 70.1

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

   6. Applied rewrites 70.1%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 41.8

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

   9. Applied rewrites 41.8%

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

   10. Taylor expanded in c around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 48.7

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

   12. Applied rewrites 48.7%

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

   if -1.7000000000000001e-113 < b < 1.28e-105

   1. Initial program 72.2%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.6

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

   4. Applied rewrites 56.6%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 40.7

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

   7. Applied rewrites 40.7%

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

   9. Applied rewrites 40.7%

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

       1. lift-/.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. count-2-rev N/A

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

       4. *-commutative N/A

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

       5. associate-/l* N/A

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

       6. lower-*.f64 N/A

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

       7. frac-2neg N/A

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

       8. metadata-eval N/A

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

       9. lower-/.f64 N/A

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

       10. lift--.f64 N/A

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

       11. sub-negate-rev N/A

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

       12. lower--.f64 40.7

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

       13. lift-*.f64 N/A

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

   11. Applied rewrites 40.7%

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

   if 1.28e-105 < b

   1. Initial program 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. 1-exp N/A

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

      11. metadata-eval N/A

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

      12. exp-diff N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-E.f64 70.1

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

   6. Applied rewrites 70.1%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 67.9

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

   9. Applied rewrites 67.9%

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

   10. Taylor expanded in c around inf

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

       1. lower-/.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 42.4

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

   12. Applied rewrites 42.4%

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

Alternative 10: 80.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \mathbf{if}\;b \leq -1.7 \cdot 10^{-113}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;0.5 \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \leq -3.6 \cdot 10^{-308}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(2 \cdot \frac{a \cdot c}{b}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(a \cdot c\right) \cdot -4} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.28 \cdot 10^{-105}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* (+ c c) E) (* -2.0 (* b E)))))
   (if (<= b -1.7e-113)
     (if (>= b 0.0) (* 0.5 (* c (sqrt (/ -4.0 (* a c))))) t_0)
     (if (<= b -3.6e-308)
       (if (>= b 0.0)
         (* (* 2.0 (/ (* a c) b)) (/ -0.5 a))
         (/ (+ c c) (- (sqrt (* (* a c) -4.0)) b)))
       (if (<= b 1.28e-105)
         (if (>= b 0.0) (* -0.5 (/ (sqrt (* -4.0 (* a c))) a)) t_0)
         (if (>= b 0.0)
           (/ (* -2.0 b) (* 2.0 a))
           (/ 2.0 (sqrt (* -4.0 (/ a c))))))))))

C

double code(double a, double b, double c) {
	double t_0 = ((c + c) * ((double) M_E)) / (-2.0 * (b * ((double) M_E)));
	double tmp_1;
	if (b <= -1.7e-113) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 0.5 * (c * sqrt((-4.0 / (a * c))));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -3.6e-308) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * ((a * c) / b)) * (-0.5 / a);
		} else {
			tmp_3 = (c + c) / (sqrt(((a * c) * -4.0)) - b);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.28e-105) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = -0.5 * (sqrt((-4.0 * (a * c))) / a);
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * b) / (2.0 * a);
	} else {
		tmp_1 = 2.0 / sqrt((-4.0 * (a / c)));
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = ((c + c) * Math.E) / (-2.0 * (b * Math.E));
	double tmp_1;
	if (b <= -1.7e-113) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 0.5 * (c * Math.sqrt((-4.0 / (a * c))));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -3.6e-308) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * ((a * c) / b)) * (-0.5 / a);
		} else {
			tmp_3 = (c + c) / (Math.sqrt(((a * c) * -4.0)) - b);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.28e-105) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = -0.5 * (Math.sqrt((-4.0 * (a * c))) / a);
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * b) / (2.0 * a);
	} else {
		tmp_1 = 2.0 / Math.sqrt((-4.0 * (a / c)));
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = ((c + c) * math.e) / (-2.0 * (b * math.e))
	tmp_1 = 0
	if b <= -1.7e-113:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = 0.5 * (c * math.sqrt((-4.0 / (a * c))))
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b <= -3.6e-308:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (2.0 * ((a * c) / b)) * (-0.5 / a)
		else:
			tmp_3 = (c + c) / (math.sqrt(((a * c) * -4.0)) - b)
		tmp_1 = tmp_3
	elif b <= 1.28e-105:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = -0.5 * (math.sqrt((-4.0 * (a * c))) / a)
		else:
			tmp_4 = t_0
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = (-2.0 * b) / (2.0 * a)
	else:
		tmp_1 = 2.0 / math.sqrt((-4.0 * (a / c)))
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(c + c) * exp(1)) / Float64(-2.0 * Float64(b * exp(1))))
	tmp_1 = 0.0
	if (b <= -1.7e-113)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(0.5 * Float64(c * sqrt(Float64(-4.0 / Float64(a * c)))));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b <= -3.6e-308)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(2.0 * Float64(Float64(a * c) / b)) * Float64(-0.5 / a));
		else
			tmp_3 = Float64(Float64(c + c) / Float64(sqrt(Float64(Float64(a * c) * -4.0)) - b));
		end
		tmp_1 = tmp_3;
	elseif (b <= 1.28e-105)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(-0.5 * Float64(sqrt(Float64(-4.0 * Float64(a * c))) / a));
		else
			tmp_4 = t_0;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-2.0 * b) / Float64(2.0 * a));
	else
		tmp_1 = Float64(2.0 / sqrt(Float64(-4.0 * Float64(a / c))));
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = ((c + c) * 2.71828182845904523536) / (-2.0 * (b * 2.71828182845904523536));
	tmp_2 = 0.0;
	if (b <= -1.7e-113)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = 0.5 * (c * sqrt((-4.0 / (a * c))));
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b <= -3.6e-308)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (2.0 * ((a * c) / b)) * (-0.5 / a);
		else
			tmp_4 = (c + c) / (sqrt(((a * c) * -4.0)) - b);
		end
		tmp_2 = tmp_4;
	elseif (b <= 1.28e-105)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = -0.5 * (sqrt((-4.0 * (a * c))) / a);
		else
			tmp_5 = t_0;
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = (-2.0 * b) / (2.0 * a);
	else
		tmp_2 = 2.0 / sqrt((-4.0 * (a / c)));
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[(c + c), $MachinePrecision] * E), $MachinePrecision] / N[(-2.0 * N[(b * E), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.7e-113], If[GreaterEqual[b, 0.0], N[(0.5 * N[(c * N[Sqrt[N[(-4.0 / N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0], If[LessEqual[b, -3.6e-308], If[GreaterEqual[b, 0.0], N[(N[(2.0 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.28e-105], If[GreaterEqual[b, 0.0], N[(-0.5 * N[(N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.7000000000000001e-113

   1. Initial program 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. 1-exp N/A

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

      11. metadata-eval N/A

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

      12. exp-diff N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-E.f64 70.1

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

   6. Applied rewrites 70.1%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 41.8

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

   9. Applied rewrites 41.8%

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

   10. Taylor expanded in c around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 48.7

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

   12. Applied rewrites 48.7%

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

   if -1.7000000000000001e-113 < b < -3.5999999999999999e-308

   1. Initial program 72.2%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.6

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

   4. Applied rewrites 56.6%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 40.7

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

   7. Applied rewrites 40.7%

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

   9. Applied rewrites 40.7%

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

   10. Taylor expanded in b around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 21.3

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

   12. Applied rewrites 21.3%

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

   if -3.5999999999999999e-308 < b < 1.28e-105

   1. Initial program 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. 1-exp N/A

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

      11. metadata-eval N/A

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

      12. exp-diff N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-E.f64 70.1

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

   6. Applied rewrites 70.1%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 41.8

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

   9. Applied rewrites 41.8%

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

   10. Taylor expanded in a around 0

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

       1. lower-/.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 47.2

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

   12. Applied rewrites 47.2%

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

   if 1.28e-105 < b

   1. Initial program 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. 1-exp N/A

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

      11. metadata-eval N/A

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

      12. exp-diff N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-E.f64 70.1

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

   6. Applied rewrites 70.1%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 67.9

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

   9. Applied rewrites 67.9%

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

   10. Taylor expanded in c around inf

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

       1. lower-/.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 42.4

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

   12. Applied rewrites 42.4%

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

Alternative 11: 80.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ t_1 := \sqrt{-4 \cdot \left(a \cdot c\right)}\\ t_2 := \frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{if}\;b \leq -1.7 \cdot 10^{-113}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;0.5 \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \leq -3.6 \cdot 10^{-308}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{t\_1}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.28 \cdot 10^{-105}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{t\_1}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* (+ c c) E) (* -2.0 (* b E))))
        (t_1 (sqrt (* -4.0 (* a c))))
        (t_2 (/ (* -2.0 b) (* 2.0 a))))
   (if (<= b -1.7e-113)
     (if (>= b 0.0) (* 0.5 (* c (sqrt (/ -4.0 (* a c))))) t_0)
     (if (<= b -3.6e-308)
       (if (>= b 0.0) t_2 (* 2.0 (/ c t_1)))
       (if (<= b 1.28e-105)
         (if (>= b 0.0) (* -0.5 (/ t_1 a)) t_0)
         (if (>= b 0.0) t_2 (/ 2.0 (sqrt (* -4.0 (/ a c))))))))))

C

double code(double a, double b, double c) {
	double t_0 = ((c + c) * ((double) M_E)) / (-2.0 * (b * ((double) M_E)));
	double t_1 = sqrt((-4.0 * (a * c)));
	double t_2 = (-2.0 * b) / (2.0 * a);
	double tmp_1;
	if (b <= -1.7e-113) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 0.5 * (c * sqrt((-4.0 / (a * c))));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -3.6e-308) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_2;
		} else {
			tmp_3 = 2.0 * (c / t_1);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.28e-105) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = -0.5 * (t_1 / a);
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = t_2;
	} else {
		tmp_1 = 2.0 / sqrt((-4.0 * (a / c)));
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = ((c + c) * Math.E) / (-2.0 * (b * Math.E));
	double t_1 = Math.sqrt((-4.0 * (a * c)));
	double t_2 = (-2.0 * b) / (2.0 * a);
	double tmp_1;
	if (b <= -1.7e-113) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 0.5 * (c * Math.sqrt((-4.0 / (a * c))));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -3.6e-308) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_2;
		} else {
			tmp_3 = 2.0 * (c / t_1);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.28e-105) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = -0.5 * (t_1 / a);
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = t_2;
	} else {
		tmp_1 = 2.0 / Math.sqrt((-4.0 * (a / c)));
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = ((c + c) * math.e) / (-2.0 * (b * math.e))
	t_1 = math.sqrt((-4.0 * (a * c)))
	t_2 = (-2.0 * b) / (2.0 * a)
	tmp_1 = 0
	if b <= -1.7e-113:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = 0.5 * (c * math.sqrt((-4.0 / (a * c))))
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b <= -3.6e-308:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = t_2
		else:
			tmp_3 = 2.0 * (c / t_1)
		tmp_1 = tmp_3
	elif b <= 1.28e-105:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = -0.5 * (t_1 / a)
		else:
			tmp_4 = t_0
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = t_2
	else:
		tmp_1 = 2.0 / math.sqrt((-4.0 * (a / c)))
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(c + c) * exp(1)) / Float64(-2.0 * Float64(b * exp(1))))
	t_1 = sqrt(Float64(-4.0 * Float64(a * c)))
	t_2 = Float64(Float64(-2.0 * b) / Float64(2.0 * a))
	tmp_1 = 0.0
	if (b <= -1.7e-113)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(0.5 * Float64(c * sqrt(Float64(-4.0 / Float64(a * c)))));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b <= -3.6e-308)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_2;
		else
			tmp_3 = Float64(2.0 * Float64(c / t_1));
		end
		tmp_1 = tmp_3;
	elseif (b <= 1.28e-105)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(-0.5 * Float64(t_1 / a));
		else
			tmp_4 = t_0;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = t_2;
	else
		tmp_1 = Float64(2.0 / sqrt(Float64(-4.0 * Float64(a / c))));
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = ((c + c) * 2.71828182845904523536) / (-2.0 * (b * 2.71828182845904523536));
	t_1 = sqrt((-4.0 * (a * c)));
	t_2 = (-2.0 * b) / (2.0 * a);
	tmp_2 = 0.0;
	if (b <= -1.7e-113)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = 0.5 * (c * sqrt((-4.0 / (a * c))));
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b <= -3.6e-308)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = t_2;
		else
			tmp_4 = 2.0 * (c / t_1);
		end
		tmp_2 = tmp_4;
	elseif (b <= 1.28e-105)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = -0.5 * (t_1 / a);
		else
			tmp_5 = t_0;
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = t_2;
	else
		tmp_2 = 2.0 / sqrt((-4.0 * (a / c)));
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[(c + c), $MachinePrecision] * E), $MachinePrecision] / N[(-2.0 * N[(b * E), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 * b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.7e-113], If[GreaterEqual[b, 0.0], N[(0.5 * N[(c * N[Sqrt[N[(-4.0 / N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0], If[LessEqual[b, -3.6e-308], If[GreaterEqual[b, 0.0], t$95$2, N[(2.0 * N[(c / t$95$1), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.28e-105], If[GreaterEqual[b, 0.0], N[(-0.5 * N[(t$95$1 / a), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], t$95$2, N[(2.0 / N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.7000000000000001e-113

   1. Initial program 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. 1-exp N/A

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

      11. metadata-eval N/A

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

      12. exp-diff N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-E.f64 70.1

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

   6. Applied rewrites 70.1%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

      3. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

      4. lower-/.f64 41.8

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   9. Applied rewrites 41.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   10. Taylor expanded in c around -inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{1}{2} \cdot \color{blue}{\left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{1}{2} \cdot \left(c \cdot \color{blue}{\sqrt{\frac{-4}{a \cdot c}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

       2. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{1}{2} \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{1}{2} \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

       4. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{1}{2} \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

       5. lower-*.f64 48.7

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;0.5 \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   12. Applied rewrites 48.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;0.5 \cdot \color{blue}{\left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   if -1.7000000000000001e-113 < b < -3.5999999999999999e-308

   1. Initial program 72.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      2. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      3. associate-/l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      4. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 2\\ \end{array} \]

      5. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \left(2 \cdot 1\right)}\\ \end{array} \]

      6. associate-*r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 2\right) \cdot 1\\ \end{array} \]

      7. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot 1\\ \end{array} \]

      8. associate-/l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 1\\ \end{array} \]

      9. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 1\\ \end{array} \]

      10. 1-exp N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot e^{0}}\\ \end{array} \]

      11. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}} \cdot e^{1 - 1}\\ \end{array} \]

      12. exp-diff N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \frac{e^{1}}{e^{1}}}\\ \end{array} \]

      13. frac-times N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot c\right) \cdot e^{1}}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot e^{1}}\\ \end{array} \]

      14. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot c\right) \cdot e^{1}}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot e^{1}}\\ \end{array} \]

   3. Applied rewrites 72.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{\left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b\right) \cdot e}\\ \end{array} \]

   4. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{\color{blue}{-2 \cdot \left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]

      2. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \color{blue}{\left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]

      3. lower-E.f64 70.1

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot \color{blue}{e}\right)}\\ \end{array} \]

   6. Applied rewrites 70.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}}\\ \end{array} \]

   7. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{-2 \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]
   8. Step-by-step derivation

      1. lower-*.f64 67.9

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot \color{blue}{b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   9. Applied rewrites 67.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{-2 \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   10. Taylor expanded in b around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}}\\ \end{array} \]

       2. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \color{blue}{\frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}}\\ \end{array} \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right)}}}\\ \end{array} \]

       4. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \end{array} \]

       5. lower-*.f64 46.5

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \end{array} \]

   12. Applied rewrites 46.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]

   if -3.5999999999999999e-308 < b < 1.28e-105

   1. Initial program 72.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      2. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      3. associate-/l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      4. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 2\\ \end{array} \]

      5. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \left(2 \cdot 1\right)}\\ \end{array} \]

      6. associate-*r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 2\right) \cdot 1\\ \end{array} \]

      7. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot 1\\ \end{array} \]

      8. associate-/l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 1\\ \end{array} \]

      9. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 1\\ \end{array} \]

      10. 1-exp N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot e^{0}}\\ \end{array} \]

      11. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}} \cdot e^{1 - 1}\\ \end{array} \]

      12. exp-diff N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \frac{e^{1}}{e^{1}}}\\ \end{array} \]

      13. frac-times N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot c\right) \cdot e^{1}}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot e^{1}}\\ \end{array} \]

      14. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot c\right) \cdot e^{1}}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot e^{1}}\\ \end{array} \]

   3. Applied rewrites 72.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{\left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b\right) \cdot e}\\ \end{array} \]

   4. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{\color{blue}{-2 \cdot \left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]

      2. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \color{blue}{\left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]

      3. lower-E.f64 70.1

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot \color{blue}{e}\right)}\\ \end{array} \]

   6. Applied rewrites 70.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}}\\ \end{array} \]

   7. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \color{blue}{\sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

      3. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

      4. lower-/.f64 41.8

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   9. Applied rewrites 41.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   10. Taylor expanded in a around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{\color{blue}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

       3. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

       4. lower-*.f64 47.2

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   12. Applied rewrites 47.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{\color{blue}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   if 1.28e-105 < b

   1. Initial program 72.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      2. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      3. associate-/l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      4. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 2\\ \end{array} \]

      5. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \left(2 \cdot 1\right)}\\ \end{array} \]

      6. associate-*r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 2\right) \cdot 1\\ \end{array} \]

      7. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot 1\\ \end{array} \]

      8. associate-/l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 1\\ \end{array} \]

      9. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 1\\ \end{array} \]

      10. 1-exp N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot e^{0}}\\ \end{array} \]

      11. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}} \cdot e^{1 - 1}\\ \end{array} \]

      12. exp-diff N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \frac{e^{1}}{e^{1}}}\\ \end{array} \]

      13. frac-times N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot c\right) \cdot e^{1}}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot e^{1}}\\ \end{array} \]

      14. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot c\right) \cdot e^{1}}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot e^{1}}\\ \end{array} \]

   3. Applied rewrites 72.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{\left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b\right) \cdot e}\\ \end{array} \]

   4. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{\color{blue}{-2 \cdot \left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]

      2. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \color{blue}{\left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]

      3. lower-E.f64 70.1

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot \color{blue}{e}\right)}\\ \end{array} \]

   6. Applied rewrites 70.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}}\\ \end{array} \]

   7. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{-2 \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]
   8. Step-by-step derivation

      1. lower-*.f64 67.9

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot \color{blue}{b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   9. Applied rewrites 67.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{-2 \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   10. Taylor expanded in c around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}}\\ \end{array} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\color{blue}{\sqrt{-4 \cdot \frac{a}{c}}}}\\ \end{array} \]

       3. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{\color{blue}{-4 \cdot \frac{a}{c}}}}\\ \end{array} \]

       4. lower-/.f64 42.4

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{-4 \cdot \color{blue}{\frac{a}{c}}}}\\ \end{array} \]

   12. Applied rewrites 42.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 80.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;0.5 \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array}\\ t_1 := \frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{if}\;b \leq -1.7 \cdot 10^{-113}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq -3.6 \cdot 10^{-308}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.28 \cdot 10^{-105}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0
         (if (>= b 0.0)
           (* 0.5 (* c (sqrt (/ -4.0 (* a c)))))
           (/ (* (+ c c) E) (* -2.0 (* b E)))))
        (t_1 (/ (* -2.0 b) (* 2.0 a))))
   (if (<= b -1.7e-113)
     t_0
     (if (<= b -3.6e-308)
       (if (>= b 0.0) t_1 (* 2.0 (/ c (sqrt (* -4.0 (* a c))))))
       (if (<= b 1.28e-105)
         t_0
         (if (>= b 0.0) t_1 (/ 2.0 (sqrt (* -4.0 (/ a c))))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = 0.5 * (c * sqrt((-4.0 / (a * c))));
	} else {
		tmp = ((c + c) * ((double) M_E)) / (-2.0 * (b * ((double) M_E)));
	}
	double t_0 = tmp;
	double t_1 = (-2.0 * b) / (2.0 * a);
	double tmp_1;
	if (b <= -1.7e-113) {
		tmp_1 = t_0;
	} else if (b <= -3.6e-308) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = 2.0 * (c / sqrt((-4.0 * (a * c))));
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.28e-105) {
		tmp_1 = t_0;
	} else if (b >= 0.0) {
		tmp_1 = t_1;
	} else {
		tmp_1 = 2.0 / sqrt((-4.0 * (a / c)));
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = 0.5 * (c * Math.sqrt((-4.0 / (a * c))));
	} else {
		tmp = ((c + c) * Math.E) / (-2.0 * (b * Math.E));
	}
	double t_0 = tmp;
	double t_1 = (-2.0 * b) / (2.0 * a);
	double tmp_1;
	if (b <= -1.7e-113) {
		tmp_1 = t_0;
	} else if (b <= -3.6e-308) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = 2.0 * (c / Math.sqrt((-4.0 * (a * c))));
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.28e-105) {
		tmp_1 = t_0;
	} else if (b >= 0.0) {
		tmp_1 = t_1;
	} else {
		tmp_1 = 2.0 / Math.sqrt((-4.0 * (a / c)));
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = 0.5 * (c * math.sqrt((-4.0 / (a * c))))
	else:
		tmp = ((c + c) * math.e) / (-2.0 * (b * math.e))
	t_0 = tmp
	t_1 = (-2.0 * b) / (2.0 * a)
	tmp_1 = 0
	if b <= -1.7e-113:
		tmp_1 = t_0
	elif b <= -3.6e-308:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_1
		else:
			tmp_2 = 2.0 * (c / math.sqrt((-4.0 * (a * c))))
		tmp_1 = tmp_2
	elif b <= 1.28e-105:
		tmp_1 = t_0
	elif b >= 0.0:
		tmp_1 = t_1
	else:
		tmp_1 = 2.0 / math.sqrt((-4.0 * (a / c)))
	return tmp_1

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(0.5 * Float64(c * sqrt(Float64(-4.0 / Float64(a * c)))));
	else
		tmp = Float64(Float64(Float64(c + c) * exp(1)) / Float64(-2.0 * Float64(b * exp(1))));
	end
	t_0 = tmp
	t_1 = Float64(Float64(-2.0 * b) / Float64(2.0 * a))
	tmp_1 = 0.0
	if (b <= -1.7e-113)
		tmp_1 = t_0;
	elseif (b <= -3.6e-308)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(2.0 * Float64(c / sqrt(Float64(-4.0 * Float64(a * c)))));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.28e-105)
		tmp_1 = t_0;
	elseif (b >= 0.0)
		tmp_1 = t_1;
	else
		tmp_1 = Float64(2.0 / sqrt(Float64(-4.0 * Float64(a / c))));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = 0.5 * (c * sqrt((-4.0 / (a * c))));
	else
		tmp = ((c + c) * 2.71828182845904523536) / (-2.0 * (b * 2.71828182845904523536));
	end
	t_0 = tmp;
	t_1 = (-2.0 * b) / (2.0 * a);
	tmp_2 = 0.0;
	if (b <= -1.7e-113)
		tmp_2 = t_0;
	elseif (b <= -3.6e-308)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = 2.0 * (c / sqrt((-4.0 * (a * c))));
		end
		tmp_2 = tmp_3;
	elseif (b <= 1.28e-105)
		tmp_2 = t_0;
	elseif (b >= 0.0)
		tmp_2 = t_1;
	else
		tmp_2 = 2.0 / sqrt((-4.0 * (a / c)));
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = If[GreaterEqual[b, 0.0], N[(0.5 * N[(c * N[Sqrt[N[(-4.0 / N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c + c), $MachinePrecision] * E), $MachinePrecision] / N[(-2.0 * N[(b * E), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]}, Block[{t$95$1 = N[(N[(-2.0 * b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.7e-113], t$95$0, If[LessEqual[b, -3.6e-308], If[GreaterEqual[b, 0.0], t$95$1, N[(2.0 * N[(c / N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.28e-105], t$95$0, If[GreaterEqual[b, 0.0], t$95$1, N[(2.0 / N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.7000000000000001e-113 or -3.5999999999999999e-308 < b < 1.28e-105

   1. Initial program 72.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      2. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      3. associate-/l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      4. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 2\\ \end{array} \]

      5. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \left(2 \cdot 1\right)}\\ \end{array} \]

      6. associate-*r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 2\right) \cdot 1\\ \end{array} \]

      7. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot 1\\ \end{array} \]

      8. associate-/l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 1\\ \end{array} \]

      9. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 1\\ \end{array} \]

      10. 1-exp N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot e^{0}}\\ \end{array} \]

      11. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}} \cdot e^{1 - 1}\\ \end{array} \]

      12. exp-diff N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \frac{e^{1}}{e^{1}}}\\ \end{array} \]

      13. frac-times N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot c\right) \cdot e^{1}}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot e^{1}}\\ \end{array} \]

      14. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot c\right) \cdot e^{1}}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot e^{1}}\\ \end{array} \]

   3. Applied rewrites 72.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{\left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b\right) \cdot e}\\ \end{array} \]

   4. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{\color{blue}{-2 \cdot \left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]

      2. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \color{blue}{\left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]

      3. lower-E.f64 70.1

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot \color{blue}{e}\right)}\\ \end{array} \]

   6. Applied rewrites 70.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}}\\ \end{array} \]

   7. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \color{blue}{\sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

      3. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

      4. lower-/.f64 41.8

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   9. Applied rewrites 41.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   10. Taylor expanded in c around -inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{1}{2} \cdot \color{blue}{\left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{1}{2} \cdot \left(c \cdot \color{blue}{\sqrt{\frac{-4}{a \cdot c}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

       2. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{1}{2} \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{1}{2} \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

       4. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{1}{2} \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

       5. lower-*.f64 48.7

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;0.5 \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   12. Applied rewrites 48.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;0.5 \cdot \color{blue}{\left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   if -1.7000000000000001e-113 < b < -3.5999999999999999e-308

   1. Initial program 72.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      2. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      3. associate-/l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      4. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 2\\ \end{array} \]

      5. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \left(2 \cdot 1\right)}\\ \end{array} \]

      6. associate-*r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 2\right) \cdot 1\\ \end{array} \]

      7. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot 1\\ \end{array} \]

      8. associate-/l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 1\\ \end{array} \]

      9. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 1\\ \end{array} \]

      10. 1-exp N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot e^{0}}\\ \end{array} \]

      11. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}} \cdot e^{1 - 1}\\ \end{array} \]

      12. exp-diff N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \frac{e^{1}}{e^{1}}}\\ \end{array} \]

      13. frac-times N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot c\right) \cdot e^{1}}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot e^{1}}\\ \end{array} \]

      14. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot c\right) \cdot e^{1}}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot e^{1}}\\ \end{array} \]

   3. Applied rewrites 72.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{\left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b\right) \cdot e}\\ \end{array} \]

   4. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{\color{blue}{-2 \cdot \left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]

      2. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \color{blue}{\left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]

      3. lower-E.f64 70.1

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot \color{blue}{e}\right)}\\ \end{array} \]

   6. Applied rewrites 70.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}}\\ \end{array} \]

   7. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{-2 \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]
   8. Step-by-step derivation

      1. lower-*.f64 67.9

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot \color{blue}{b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   9. Applied rewrites 67.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{-2 \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   10. Taylor expanded in b around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}}\\ \end{array} \]

       2. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \color{blue}{\frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}}\\ \end{array} \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right)}}}\\ \end{array} \]

       4. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \end{array} \]

       5. lower-*.f64 46.5

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \end{array} \]

   12. Applied rewrites 46.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]

   if 1.28e-105 < b

   1. Initial program 72.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      2. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      3. associate-/l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      4. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 2\\ \end{array} \]

      5. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \left(2 \cdot 1\right)}\\ \end{array} \]

      6. associate-*r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 2\right) \cdot 1\\ \end{array} \]

      7. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot 1\\ \end{array} \]

      8. associate-/l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 1\\ \end{array} \]

      9. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 1\\ \end{array} \]

      10. 1-exp N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot e^{0}}\\ \end{array} \]

      11. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}} \cdot e^{1 - 1}\\ \end{array} \]

      12. exp-diff N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \frac{e^{1}}{e^{1}}}\\ \end{array} \]

      13. frac-times N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot c\right) \cdot e^{1}}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot e^{1}}\\ \end{array} \]

      14. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot c\right) \cdot e^{1}}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot e^{1}}\\ \end{array} \]

   3. Applied rewrites 72.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{\left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b\right) \cdot e}\\ \end{array} \]

   4. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{\color{blue}{-2 \cdot \left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]

      2. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \color{blue}{\left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]

      3. lower-E.f64 70.1

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot \color{blue}{e}\right)}\\ \end{array} \]

   6. Applied rewrites 70.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}}\\ \end{array} \]

   7. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{-2 \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]
   8. Step-by-step derivation

      1. lower-*.f64 67.9

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot \color{blue}{b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   9. Applied rewrites 67.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{-2 \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   10. Taylor expanded in c around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}}\\ \end{array} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\color{blue}{\sqrt{-4 \cdot \frac{a}{c}}}}\\ \end{array} \]

       3. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{\color{blue}{-4 \cdot \frac{a}{c}}}}\\ \end{array} \]

       4. lower-/.f64 42.4

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{-4 \cdot \color{blue}{\frac{a}{c}}}}\\ \end{array} \]

   12. Applied rewrites 42.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 50.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ t_1 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ t_2 := \frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{if}\;b \leq -3.8 \cdot 10^{+125}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -3.6 \cdot 10^{-308}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.36 \cdot 10^{-115}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ 2.0 (sqrt (* -4.0 (/ a c)))))
        (t_1 (if (>= b 0.0) (* -0.5 (sqrt (* -4.0 (/ c a)))) t_0))
        (t_2 (/ (* -2.0 b) (* 2.0 a))))
   (if (<= b -3.8e+125)
     t_1
     (if (<= b -3.6e-308)
       (if (>= b 0.0) t_2 (* 2.0 (/ c (sqrt (* -4.0 (* a c))))))
       (if (<= b 1.36e-115) t_1 (if (>= b 0.0) t_2 t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = 2.0 / sqrt((-4.0 * (a / c)));
	double tmp;
	if (b >= 0.0) {
		tmp = -0.5 * sqrt((-4.0 * (c / a)));
	} else {
		tmp = t_0;
	}
	double t_1 = tmp;
	double t_2 = (-2.0 * b) / (2.0 * a);
	double tmp_1;
	if (b <= -3.8e+125) {
		tmp_1 = t_1;
	} else if (b <= -3.6e-308) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_2;
		} else {
			tmp_2 = 2.0 * (c / sqrt((-4.0 * (a * c))));
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.36e-115) {
		tmp_1 = t_1;
	} else if (b >= 0.0) {
		tmp_1 = t_2;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = 2.0d0 / sqrt(((-4.0d0) * (a / c)))
    if (b >= 0.0d0) then
        tmp = (-0.5d0) * sqrt(((-4.0d0) * (c / a)))
    else
        tmp = t_0
    end if
    t_1 = tmp
    t_2 = ((-2.0d0) * b) / (2.0d0 * a)
    if (b <= (-3.8d+125)) then
        tmp_1 = t_1
    else if (b <= (-3.6d-308)) then
        if (b >= 0.0d0) then
            tmp_2 = t_2
        else
            tmp_2 = 2.0d0 * (c / sqrt(((-4.0d0) * (a * c))))
        end if
        tmp_1 = tmp_2
    else if (b <= 1.36d-115) then
        tmp_1 = t_1
    else if (b >= 0.0d0) then
        tmp_1 = t_2
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = 2.0 / Math.sqrt((-4.0 * (a / c)));
	double tmp;
	if (b >= 0.0) {
		tmp = -0.5 * Math.sqrt((-4.0 * (c / a)));
	} else {
		tmp = t_0;
	}
	double t_1 = tmp;
	double t_2 = (-2.0 * b) / (2.0 * a);
	double tmp_1;
	if (b <= -3.8e+125) {
		tmp_1 = t_1;
	} else if (b <= -3.6e-308) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_2;
		} else {
			tmp_2 = 2.0 * (c / Math.sqrt((-4.0 * (a * c))));
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.36e-115) {
		tmp_1 = t_1;
	} else if (b >= 0.0) {
		tmp_1 = t_2;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = 2.0 / math.sqrt((-4.0 * (a / c)))
	tmp = 0
	if b >= 0.0:
		tmp = -0.5 * math.sqrt((-4.0 * (c / a)))
	else:
		tmp = t_0
	t_1 = tmp
	t_2 = (-2.0 * b) / (2.0 * a)
	tmp_1 = 0
	if b <= -3.8e+125:
		tmp_1 = t_1
	elif b <= -3.6e-308:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_2
		else:
			tmp_2 = 2.0 * (c / math.sqrt((-4.0 * (a * c))))
		tmp_1 = tmp_2
	elif b <= 1.36e-115:
		tmp_1 = t_1
	elif b >= 0.0:
		tmp_1 = t_2
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(2.0 / sqrt(Float64(-4.0 * Float64(a / c))))
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(-0.5 * sqrt(Float64(-4.0 * Float64(c / a))));
	else
		tmp = t_0;
	end
	t_1 = tmp
	t_2 = Float64(Float64(-2.0 * b) / Float64(2.0 * a))
	tmp_1 = 0.0
	if (b <= -3.8e+125)
		tmp_1 = t_1;
	elseif (b <= -3.6e-308)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_2;
		else
			tmp_2 = Float64(2.0 * Float64(c / sqrt(Float64(-4.0 * Float64(a * c)))));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.36e-115)
		tmp_1 = t_1;
	elseif (b >= 0.0)
		tmp_1 = t_2;
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = 2.0 / sqrt((-4.0 * (a / c)));
	tmp = 0.0;
	if (b >= 0.0)
		tmp = -0.5 * sqrt((-4.0 * (c / a)));
	else
		tmp = t_0;
	end
	t_1 = tmp;
	t_2 = (-2.0 * b) / (2.0 * a);
	tmp_2 = 0.0;
	if (b <= -3.8e+125)
		tmp_2 = t_1;
	elseif (b <= -3.6e-308)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_2;
		else
			tmp_3 = 2.0 * (c / sqrt((-4.0 * (a * c))));
		end
		tmp_2 = tmp_3;
	elseif (b <= 1.36e-115)
		tmp_2 = t_1;
	elseif (b >= 0.0)
		tmp_2 = t_2;
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(2.0 / N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = If[GreaterEqual[b, 0.0], N[(-0.5 * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]}, Block[{t$95$2 = N[(N[(-2.0 * b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.8e+125], t$95$1, If[LessEqual[b, -3.6e-308], If[GreaterEqual[b, 0.0], t$95$2, N[(2.0 * N[(c / N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.36e-115], t$95$1, If[GreaterEqual[b, 0.0], t$95$2, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.80000000000000002e125 or -3.5999999999999999e-308 < b < 1.35999999999999997e-115

   1. Initial program 72.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      2. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      3. associate-/l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      4. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 2\\ \end{array} \]

      5. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \left(2 \cdot 1\right)}\\ \end{array} \]

      6. associate-*r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 2\right) \cdot 1\\ \end{array} \]

      7. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot 1\\ \end{array} \]

      8. associate-/l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 1\\ \end{array} \]

      9. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 1\\ \end{array} \]

      10. 1-exp N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot e^{0}}\\ \end{array} \]

      11. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}} \cdot e^{1 - 1}\\ \end{array} \]

      12. exp-diff N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \frac{e^{1}}{e^{1}}}\\ \end{array} \]

      13. frac-times N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot c\right) \cdot e^{1}}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot e^{1}}\\ \end{array} \]

      14. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot c\right) \cdot e^{1}}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot e^{1}}\\ \end{array} \]

   3. Applied rewrites 72.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{\left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b\right) \cdot e}\\ \end{array} \]

   4. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{\color{blue}{-2 \cdot \left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]

      2. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \color{blue}{\left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]

      3. lower-E.f64 70.1

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot \color{blue}{e}\right)}\\ \end{array} \]

   6. Applied rewrites 70.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}}\\ \end{array} \]

   7. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \color{blue}{\sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

      3. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

      4. lower-/.f64 41.8

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   9. Applied rewrites 41.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   10. Taylor expanded in c around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}}\\ \end{array} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\color{blue}{\sqrt{-4 \cdot \frac{a}{c}}}}\\ \end{array} \]

       3. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{\color{blue}{-4 \cdot \frac{a}{c}}}}\\ \end{array} \]

       4. lower-/.f64 16.3

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{-4 \cdot \color{blue}{\frac{a}{c}}}}\\ \end{array} \]

   12. Applied rewrites 16.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array} \]

   if -3.80000000000000002e125 < b < -3.5999999999999999e-308

   1. Initial program 72.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      2. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      3. associate-/l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      4. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 2\\ \end{array} \]

      5. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \left(2 \cdot 1\right)}\\ \end{array} \]

      6. associate-*r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 2\right) \cdot 1\\ \end{array} \]

      7. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot 1\\ \end{array} \]

      8. associate-/l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 1\\ \end{array} \]

      9. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 1\\ \end{array} \]

      10. 1-exp N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot e^{0}}\\ \end{array} \]

      11. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}} \cdot e^{1 - 1}\\ \end{array} \]

      12. exp-diff N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \frac{e^{1}}{e^{1}}}\\ \end{array} \]

      13. frac-times N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot c\right) \cdot e^{1}}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot e^{1}}\\ \end{array} \]

      14. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot c\right) \cdot e^{1}}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot e^{1}}\\ \end{array} \]

   3. Applied rewrites 72.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{\left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b\right) \cdot e}\\ \end{array} \]

   4. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{\color{blue}{-2 \cdot \left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]

      2. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \color{blue}{\left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]

      3. lower-E.f64 70.1

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot \color{blue}{e}\right)}\\ \end{array} \]

   6. Applied rewrites 70.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}}\\ \end{array} \]

   7. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{-2 \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]
   8. Step-by-step derivation

      1. lower-*.f64 67.9

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot \color{blue}{b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   9. Applied rewrites 67.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{-2 \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   10. Taylor expanded in b around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}}\\ \end{array} \]

       2. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \color{blue}{\frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}}\\ \end{array} \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right)}}}\\ \end{array} \]

       4. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \end{array} \]

       5. lower-*.f64 46.5

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \end{array} \]

   12. Applied rewrites 46.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]

   if 1.35999999999999997e-115 < b

   1. Initial program 72.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      2. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      3. associate-/l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      4. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 2\\ \end{array} \]

      5. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \left(2 \cdot 1\right)}\\ \end{array} \]

      6. associate-*r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 2\right) \cdot 1\\ \end{array} \]

      7. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot 1\\ \end{array} \]

      8. associate-/l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 1\\ \end{array} \]

      9. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 1\\ \end{array} \]

      10. 1-exp N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot e^{0}}\\ \end{array} \]

      11. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}} \cdot e^{1 - 1}\\ \end{array} \]

      12. exp-diff N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \frac{e^{1}}{e^{1}}}\\ \end{array} \]

      13. frac-times N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot c\right) \cdot e^{1}}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot e^{1}}\\ \end{array} \]

      14. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot c\right) \cdot e^{1}}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot e^{1}}\\ \end{array} \]

   3. Applied rewrites 72.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{\left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b\right) \cdot e}\\ \end{array} \]

   4. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{\color{blue}{-2 \cdot \left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]

      2. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \color{blue}{\left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]

      3. lower-E.f64 70.1

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot \color{blue}{e}\right)}\\ \end{array} \]

   6. Applied rewrites 70.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}}\\ \end{array} \]

   7. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{-2 \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]
   8. Step-by-step derivation

      1. lower-*.f64 67.9

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot \color{blue}{b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   9. Applied rewrites 67.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{-2 \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   10. Taylor expanded in c around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}}\\ \end{array} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\color{blue}{\sqrt{-4 \cdot \frac{a}{c}}}}\\ \end{array} \]

       3. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{\color{blue}{-4 \cdot \frac{a}{c}}}}\\ \end{array} \]

       4. lower-/.f64 42.4

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{-4 \cdot \color{blue}{\frac{a}{c}}}}\\ \end{array} \]

   12. Applied rewrites 42.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 44.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \mathbf{if}\;b \leq 1.36 \cdot 10^{-115}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ 2.0 (sqrt (* -4.0 (/ a c))))))
   (if (<= b 1.36e-115)
     (if (>= b 0.0) (* -0.5 (sqrt (* -4.0 (/ c a)))) t_0)
     (if (>= b 0.0) (/ (* -2.0 b) (* 2.0 a)) t_0))))

C

double code(double a, double b, double c) {
	double t_0 = 2.0 / sqrt((-4.0 * (a / c)));
	double tmp_1;
	if (b <= 1.36e-115) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -0.5 * sqrt((-4.0 * (c / a)));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * b) / (2.0 * a);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = 2.0d0 / sqrt(((-4.0d0) * (a / c)))
    if (b <= 1.36d-115) then
        if (b >= 0.0d0) then
            tmp_2 = (-0.5d0) * sqrt(((-4.0d0) * (c / a)))
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = ((-2.0d0) * b) / (2.0d0 * a)
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = 2.0 / Math.sqrt((-4.0 * (a / c)));
	double tmp_1;
	if (b <= 1.36e-115) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -0.5 * Math.sqrt((-4.0 * (c / a)));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * b) / (2.0 * a);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = 2.0 / math.sqrt((-4.0 * (a / c)))
	tmp_1 = 0
	if b <= 1.36e-115:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -0.5 * math.sqrt((-4.0 * (c / a)))
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = (-2.0 * b) / (2.0 * a)
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(2.0 / sqrt(Float64(-4.0 * Float64(a / c))))
	tmp_1 = 0.0
	if (b <= 1.36e-115)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-0.5 * sqrt(Float64(-4.0 * Float64(c / a))));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-2.0 * b) / Float64(2.0 * a));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = 2.0 / sqrt((-4.0 * (a / c)));
	tmp_2 = 0.0;
	if (b <= 1.36e-115)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -0.5 * sqrt((-4.0 * (c / a)));
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = (-2.0 * b) / (2.0 * a);
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(2.0 / N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.36e-115], If[GreaterEqual[b, 0.0], N[(-0.5 * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.35999999999999997e-115

   1. Initial program 72.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      2. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      3. associate-/l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      4. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 2\\ \end{array} \]

      5. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \left(2 \cdot 1\right)}\\ \end{array} \]

      6. associate-*r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 2\right) \cdot 1\\ \end{array} \]

      7. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot 1\\ \end{array} \]

      8. associate-/l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 1\\ \end{array} \]

      9. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 1\\ \end{array} \]

      10. 1-exp N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot e^{0}}\\ \end{array} \]

      11. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}} \cdot e^{1 - 1}\\ \end{array} \]

      12. exp-diff N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \frac{e^{1}}{e^{1}}}\\ \end{array} \]

      13. frac-times N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot c\right) \cdot e^{1}}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot e^{1}}\\ \end{array} \]

      14. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot c\right) \cdot e^{1}}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot e^{1}}\\ \end{array} \]

   3. Applied rewrites 72.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{\left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b\right) \cdot e}\\ \end{array} \]

   4. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{\color{blue}{-2 \cdot \left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]

      2. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \color{blue}{\left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]

      3. lower-E.f64 70.1

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot \color{blue}{e}\right)}\\ \end{array} \]

   6. Applied rewrites 70.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}}\\ \end{array} \]

   7. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \color{blue}{\sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

      3. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

      4. lower-/.f64 41.8

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   9. Applied rewrites 41.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   10. Taylor expanded in c around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}}\\ \end{array} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\color{blue}{\sqrt{-4 \cdot \frac{a}{c}}}}\\ \end{array} \]

       3. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{\color{blue}{-4 \cdot \frac{a}{c}}}}\\ \end{array} \]

       4. lower-/.f64 16.3

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{-4 \cdot \color{blue}{\frac{a}{c}}}}\\ \end{array} \]

   12. Applied rewrites 16.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array} \]

   if 1.35999999999999997e-115 < b

   1. Initial program 72.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      2. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      3. associate-/l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      4. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 2\\ \end{array} \]

      5. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \left(2 \cdot 1\right)}\\ \end{array} \]

      6. associate-*r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 2\right) \cdot 1\\ \end{array} \]

      7. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot 1\\ \end{array} \]

      8. associate-/l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 1\\ \end{array} \]

      9. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 1\\ \end{array} \]

      10. 1-exp N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot e^{0}}\\ \end{array} \]

      11. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}} \cdot e^{1 - 1}\\ \end{array} \]

      12. exp-diff N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \frac{e^{1}}{e^{1}}}\\ \end{array} \]

      13. frac-times N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot c\right) \cdot e^{1}}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot e^{1}}\\ \end{array} \]

      14. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot c\right) \cdot e^{1}}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot e^{1}}\\ \end{array} \]

   3. Applied rewrites 72.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{\left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b\right) \cdot e}\\ \end{array} \]

   4. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{\color{blue}{-2 \cdot \left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]

      2. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \color{blue}{\left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]

      3. lower-E.f64 70.1

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot \color{blue}{e}\right)}\\ \end{array} \]

   6. Applied rewrites 70.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}}\\ \end{array} \]

   7. Taylor expanded in b around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{-2 \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]
   8. Step-by-step derivation

      1. lower-*.f64 67.9

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot \color{blue}{b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   9. Applied rewrites 67.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{-2 \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   10. Taylor expanded in c around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}}\\ \end{array} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\color{blue}{\sqrt{-4 \cdot \frac{a}{c}}}}\\ \end{array} \]

       3. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{\color{blue}{-4 \cdot \frac{a}{c}}}}\\ \end{array} \]

       4. lower-/.f64 42.4

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{-4 \cdot \color{blue}{\frac{a}{c}}}}\\ \end{array} \]

   12. Applied rewrites 42.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 16.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0)
   (* -0.5 (sqrt (* -4.0 (/ c a))))
   (/ 2.0 (sqrt (* -4.0 (/ a c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -0.5 * sqrt((-4.0 * (c / a)));
	} else {
		tmp = 2.0 / sqrt((-4.0 * (a / c)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b >= 0.0d0) then
        tmp = (-0.5d0) * sqrt(((-4.0d0) * (c / a)))
    else
        tmp = 2.0d0 / sqrt(((-4.0d0) * (a / c)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -0.5 * Math.sqrt((-4.0 * (c / a)));
	} else {
		tmp = 2.0 / Math.sqrt((-4.0 * (a / c)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = -0.5 * math.sqrt((-4.0 * (c / a)))
	else:
		tmp = 2.0 / math.sqrt((-4.0 * (a / c)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(-0.5 * sqrt(Float64(-4.0 * Float64(c / a))));
	else
		tmp = Float64(2.0 / sqrt(Float64(-4.0 * Float64(a / c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = -0.5 * sqrt((-4.0 * (c / a)));
	else
		tmp = 2.0 / sqrt((-4.0 * (a / c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(-0.5 * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 72.2%

   \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. lift-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. associate-/l* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   4. *-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 2\\ \end{array} \]

   5. metadata-eval N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \left(2 \cdot 1\right)}\\ \end{array} \]

   6. associate-*r* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 2\right) \cdot 1\\ \end{array} \]

   7. *-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot 1\\ \end{array} \]

   8. associate-/l* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 1\\ \end{array} \]

   9. lift-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 1\\ \end{array} \]

   10. 1-exp N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot e^{0}}\\ \end{array} \]

   11. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}} \cdot e^{1 - 1}\\ \end{array} \]

   12. exp-diff N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \frac{e^{1}}{e^{1}}}\\ \end{array} \]

   13. frac-times N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot c\right) \cdot e^{1}}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot e^{1}}\\ \end{array} \]

   14. lower-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot c\right) \cdot e^{1}}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot e^{1}}\\ \end{array} \]

3. Applied rewrites 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{\left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b\right) \cdot e}\\ \end{array} \]

4. Taylor expanded in b around -inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{\color{blue}{-2 \cdot \left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]

   2. lower-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \color{blue}{\left(b \cdot \mathsf{E}\left(\right)\right)}}\\ \end{array} \]

   3. lower-E.f64 70.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot \color{blue}{e}\right)}\\ \end{array} \]

6. Applied rewrites 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}}\\ \end{array} \]

7. Taylor expanded in a around inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \color{blue}{\sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   3. lower-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

   4. lower-/.f64 41.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

9. Applied rewrites 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c + c\right) \cdot e}{-2 \cdot \left(b \cdot e\right)}\\ \end{array} \]

10. Taylor expanded in c around inf

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array} \]
11. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}}\\ \end{array} \]

    2. lower-sqrt.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\color{blue}{\sqrt{-4 \cdot \frac{a}{c}}}}\\ \end{array} \]

    3. lower-*.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{\color{blue}{-4 \cdot \frac{a}{c}}}}\\ \end{array} \]

    4. lower-/.f64 16.3

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{-4 \cdot \color{blue}{\frac{a}{c}}}}\\ \end{array} \]

12. Applied rewrites 16.3%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025162 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  :precision binary64
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))