jeff quadratic root 2

Percentage Accurate: 72.5% → 90.6%

Time: 5.2s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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 = (2.0d0 * c) / (-b - t_0)
    else
        tmp = (-b + t_0) / (2.0d0 * a)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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 = (2.0d0 * c) / (-b - t_0)
    else
        tmp = (-b + t_0) / (2.0d0 * a)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;b \leq -1.8 \cdot 10^{+117}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{\sqrt{-4 \cdot \frac{a}{c}}} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|b\right| - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq -2.15 \cdot 10^{-266}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1 \cdot \left(c + \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+56}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{t\_0 + b} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot \left(\sqrt{-4 \cdot \frac{c}{a}} + \frac{b}{a}\right)\right) \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b + \left|b\right|} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b} - b}{a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 a) c (* b b)))))
   (if (<= b -1.8e+117)
     (if (>= b 0.0)
       (* (/ -1.0 (sqrt (* -4.0 (/ a c)))) -2.0)
       (* (/ (- (fabs b) b) a) 0.5))
     (if (<= b -2.15e-266)
       (if (>= b 0.0)
         (/ (* -1.0 (+ c (/ (* a (* c c)) (* b b)))) b)
         (* (/ (- t_0 b) a) 0.5))
       (if (<= b 2e+56)
         (if (>= b 0.0)
           (* (/ c (+ t_0 b)) -2.0)
           (* (* -1.0 (+ (sqrt (* -4.0 (/ c a))) (/ b a))) 0.5))
         (if (>= b 0.0)
           (* (/ c (+ b (fabs b))) -2.0)
           (* (/ (- (sqrt (* b b)) b) a) 0.5)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-4.0 * a), c, (b * b)));
	double tmp_1;
	if (b <= -1.8e+117) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-1.0 / sqrt((-4.0 * (a / c)))) * -2.0;
		} else {
			tmp_2 = ((fabs(b) - b) / a) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= -2.15e-266) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-1.0 * (c + ((a * (c * c)) / (b * b)))) / b;
		} else {
			tmp_3 = ((t_0 - b) / a) * 0.5;
		}
		tmp_1 = tmp_3;
	} else if (b <= 2e+56) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c / (t_0 + b)) * -2.0;
		} else {
			tmp_4 = (-1.0 * (sqrt((-4.0 * (c / a))) + (b / a))) * 0.5;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / (b + fabs(b))) * -2.0;
	} else {
		tmp_1 = ((sqrt((b * b)) - b) / a) * 0.5;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))
	tmp_1 = 0.0
	if (b <= -1.8e+117)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-1.0 / sqrt(Float64(-4.0 * Float64(a / c)))) * -2.0);
		else
			tmp_2 = Float64(Float64(Float64(abs(b) - b) / a) * 0.5);
		end
		tmp_1 = tmp_2;
	elseif (b <= -2.15e-266)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-1.0 * Float64(c + Float64(Float64(a * Float64(c * c)) / Float64(b * b)))) / b);
		else
			tmp_3 = Float64(Float64(Float64(t_0 - b) / a) * 0.5);
		end
		tmp_1 = tmp_3;
	elseif (b <= 2e+56)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(c / Float64(t_0 + b)) * -2.0);
		else
			tmp_4 = Float64(Float64(-1.0 * Float64(sqrt(Float64(-4.0 * Float64(c / a))) + Float64(b / a))) * 0.5);
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / Float64(b + abs(b))) * -2.0);
	else
		tmp_1 = Float64(Float64(Float64(sqrt(Float64(b * b)) - b) / a) * 0.5);
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.8e+117], If[GreaterEqual[b, 0.0], N[(N[(-1.0 / N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(N[Abs[b], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, -2.15e-266], If[GreaterEqual[b, 0.0], N[(N[(-1.0 * N[(c + N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], N[(N[(N[(t$95$0 - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, 2e+56], If[GreaterEqual[b, 0.0], N[(N[(c / N[(t$95$0 + b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(-1.0 * N[(N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c / N[(b + N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.80000000000000006e117

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-sqrt.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 32.7

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

   10. Applied rewrites 32.7%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. rem-sqrt-square N/A

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

       4. lower-fabs.f64 42.5

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

   12. Applied rewrites 42.5%

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

   if -1.80000000000000006e117 < b < -2.15000000000000014e-266

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 63.4

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

   7. Applied rewrites 63.4%

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

   if -2.15000000000000014e-266 < b < 2.00000000000000018e56

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in a around -inf

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lower-/.f64 48.7

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

   7. Applied rewrites 48.7%

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

   if 2.00000000000000018e56 < b

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-sqrt.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 32.7

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

   10. Applied rewrites 32.7%

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

   11. Taylor expanded in a around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. pow2 N/A

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

       4. rem-sqrt-square N/A

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

       5. lower-fabs.f64 58.7

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

   13. Applied rewrites 58.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b + \left|b\right|} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b} - b}{a} \cdot 0.5\\ \end{array} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 90.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;b \leq -1.8 \cdot 10^{+117}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{\sqrt{-4 \cdot \frac{a}{c}}} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|b\right| - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+56}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{t\_0 + b} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t\_0}{a} - \frac{b}{a}\right) \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b + \left|b\right|} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b} - b}{a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 a) c (* b b)))))
   (if (<= b -1.8e+117)
     (if (>= b 0.0)
       (* (/ -1.0 (sqrt (* -4.0 (/ a c)))) -2.0)
       (* (/ (- (fabs b) b) a) 0.5))
     (if (<= b 2e+56)
       (if (>= b 0.0) (* (/ c (+ t_0 b)) -2.0) (* (- (/ t_0 a) (/ b a)) 0.5))
       (if (>= b 0.0)
         (* (/ c (+ b (fabs b))) -2.0)
         (* (/ (- (sqrt (* b b)) b) a) 0.5))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-4.0 * a), c, (b * b)));
	double tmp_1;
	if (b <= -1.8e+117) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-1.0 / sqrt((-4.0 * (a / c)))) * -2.0;
		} else {
			tmp_2 = ((fabs(b) - b) / a) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= 2e+56) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c / (t_0 + b)) * -2.0;
		} else {
			tmp_3 = ((t_0 / a) - (b / a)) * 0.5;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c / (b + fabs(b))) * -2.0;
	} else {
		tmp_1 = ((sqrt((b * b)) - b) / a) * 0.5;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))
	tmp_1 = 0.0
	if (b <= -1.8e+117)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-1.0 / sqrt(Float64(-4.0 * Float64(a / c)))) * -2.0);
		else
			tmp_2 = Float64(Float64(Float64(abs(b) - b) / a) * 0.5);
		end
		tmp_1 = tmp_2;
	elseif (b <= 2e+56)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(c / Float64(t_0 + b)) * -2.0);
		else
			tmp_3 = Float64(Float64(Float64(t_0 / a) - Float64(b / a)) * 0.5);
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / Float64(b + abs(b))) * -2.0);
	else
		tmp_1 = Float64(Float64(Float64(sqrt(Float64(b * b)) - b) / a) * 0.5);
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.8e+117], If[GreaterEqual[b, 0.0], N[(N[(-1.0 / N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(N[Abs[b], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, 2e+56], If[GreaterEqual[b, 0.0], N[(N[(c / N[(t$95$0 + b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(t$95$0 / a), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c / N[(b + N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.80000000000000006e117

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-sqrt.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 32.7

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

   10. Applied rewrites 32.7%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. rem-sqrt-square N/A

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

       4. lower-fabs.f64 42.5

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

   12. Applied rewrites 42.5%

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

   if -1.80000000000000006e117 < b < 2.00000000000000018e56

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. div-sub N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lower-/.f64 72.6

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

   6. Applied rewrites 72.6%

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

   if 2.00000000000000018e56 < b

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-sqrt.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 32.7

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

   10. Applied rewrites 32.7%

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

   11. Taylor expanded in a around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. pow2 N/A

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

       4. rem-sqrt-square N/A

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

       5. lower-fabs.f64 58.7

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

   13. Applied rewrites 58.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b + \left|b\right|} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b} - b}{a} \cdot 0.5\\ \end{array} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 90.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;b \leq -1.8 \cdot 10^{+117}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{\sqrt{-4 \cdot \frac{a}{c}}} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|b\right| - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+56}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{t\_0 + b} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b + \left|b\right|} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b} - b}{a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 a) c (* b b)))))
   (if (<= b -1.8e+117)
     (if (>= b 0.0)
       (* (/ -1.0 (sqrt (* -4.0 (/ a c)))) -2.0)
       (* (/ (- (fabs b) b) a) 0.5))
     (if (<= b 2e+56)
       (if (>= b 0.0) (* (/ c (+ t_0 b)) -2.0) (* (/ (- t_0 b) a) 0.5))
       (if (>= b 0.0)
         (* (/ c (+ b (fabs b))) -2.0)
         (* (/ (- (sqrt (* b b)) b) a) 0.5))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-4.0 * a), c, (b * b)));
	double tmp_1;
	if (b <= -1.8e+117) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-1.0 / sqrt((-4.0 * (a / c)))) * -2.0;
		} else {
			tmp_2 = ((fabs(b) - b) / a) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= 2e+56) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c / (t_0 + b)) * -2.0;
		} else {
			tmp_3 = ((t_0 - b) / a) * 0.5;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c / (b + fabs(b))) * -2.0;
	} else {
		tmp_1 = ((sqrt((b * b)) - b) / a) * 0.5;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))
	tmp_1 = 0.0
	if (b <= -1.8e+117)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-1.0 / sqrt(Float64(-4.0 * Float64(a / c)))) * -2.0);
		else
			tmp_2 = Float64(Float64(Float64(abs(b) - b) / a) * 0.5);
		end
		tmp_1 = tmp_2;
	elseif (b <= 2e+56)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(c / Float64(t_0 + b)) * -2.0);
		else
			tmp_3 = Float64(Float64(Float64(t_0 - b) / a) * 0.5);
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / Float64(b + abs(b))) * -2.0);
	else
		tmp_1 = Float64(Float64(Float64(sqrt(Float64(b * b)) - b) / a) * 0.5);
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.8e+117], If[GreaterEqual[b, 0.0], N[(N[(-1.0 / N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(N[Abs[b], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, 2e+56], If[GreaterEqual[b, 0.0], N[(N[(c / N[(t$95$0 + b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(t$95$0 - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c / N[(b + N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.80000000000000006e117

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-sqrt.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 32.7

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

   10. Applied rewrites 32.7%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. rem-sqrt-square N/A

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

       4. lower-fabs.f64 42.5

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

   12. Applied rewrites 42.5%

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

   if -1.80000000000000006e117 < b < 2.00000000000000018e56

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   if 2.00000000000000018e56 < b

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-sqrt.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 32.7

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

   10. Applied rewrites 32.7%

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

   11. Taylor expanded in a around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. pow2 N/A

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

       4. rem-sqrt-square N/A

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

       5. lower-fabs.f64 58.7

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

   13. Applied rewrites 58.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b + \left|b\right|} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b} - b}{a} \cdot 0.5\\ \end{array} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 90.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;b \leq -1.8 \cdot 10^{+117}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{\sqrt{-4 \cdot \frac{a}{c}}} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|b\right| - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+56}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + t\_0}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b + \left|b\right|} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b} - b}{a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* (* 4.0 a) c)))))
   (if (<= b -1.8e+117)
     (if (>= b 0.0)
       (* (/ -1.0 (sqrt (* -4.0 (/ a c)))) -2.0)
       (* (/ (- (fabs b) b) a) 0.5))
     (if (<= b 2e+56)
       (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_0)) (/ (+ (- b) t_0) (* 2.0 a)))
       (if (>= b 0.0)
         (* (/ c (+ b (fabs b))) -2.0)
         (* (/ (- (sqrt (* b b)) b) a) 0.5))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp_1;
	if (b <= -1.8e+117) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-1.0 / sqrt((-4.0 * (a / c)))) * -2.0;
		} else {
			tmp_2 = ((fabs(b) - b) / a) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= 2e+56) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_3 = (-b + t_0) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c / (b + fabs(b))) * -2.0;
	} else {
		tmp_1 = ((sqrt((b * b)) - b) / a) * 0.5;
	}
	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
    real(8) :: tmp_3
    t_0 = sqrt(((b * b) - ((4.0d0 * a) * c)))
    if (b <= (-1.8d+117)) then
        if (b >= 0.0d0) then
            tmp_2 = ((-1.0d0) / sqrt(((-4.0d0) * (a / c)))) * (-2.0d0)
        else
            tmp_2 = ((abs(b) - b) / a) * 0.5d0
        end if
        tmp_1 = tmp_2
    else if (b <= 2d+56) then
        if (b >= 0.0d0) then
            tmp_3 = (2.0d0 * c) / (-b - t_0)
        else
            tmp_3 = (-b + t_0) / (2.0d0 * a)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = (c / (b + abs(b))) * (-2.0d0)
    else
        tmp_1 = ((sqrt((b * b)) - b) / a) * 0.5d0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp_1;
	if (b <= -1.8e+117) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-1.0 / Math.sqrt((-4.0 * (a / c)))) * -2.0;
		} else {
			tmp_2 = ((Math.abs(b) - b) / a) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= 2e+56) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_3 = (-b + t_0) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c / (b + Math.abs(b))) * -2.0;
	} else {
		tmp_1 = ((Math.sqrt((b * b)) - b) / a) * 0.5;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - ((4.0 * a) * c)))
	tmp_1 = 0
	if b <= -1.8e+117:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (-1.0 / math.sqrt((-4.0 * (a / c)))) * -2.0
		else:
			tmp_2 = ((math.fabs(b) - b) / a) * 0.5
		tmp_1 = tmp_2
	elif b <= 2e+56:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (2.0 * c) / (-b - t_0)
		else:
			tmp_3 = (-b + t_0) / (2.0 * a)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = (c / (b + math.fabs(b))) * -2.0
	else:
		tmp_1 = ((math.sqrt((b * b)) - b) / a) * 0.5
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	tmp_1 = 0.0
	if (b <= -1.8e+117)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-1.0 / sqrt(Float64(-4.0 * Float64(a / c)))) * -2.0);
		else
			tmp_2 = Float64(Float64(Float64(abs(b) - b) / a) * 0.5);
		end
		tmp_1 = tmp_2;
	elseif (b <= 2e+56)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0));
		else
			tmp_3 = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / Float64(b + abs(b))) * -2.0);
	else
		tmp_1 = Float64(Float64(Float64(sqrt(Float64(b * b)) - b) / a) * 0.5);
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	tmp_2 = 0.0;
	if (b <= -1.8e+117)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (-1.0 / sqrt((-4.0 * (a / c)))) * -2.0;
		else
			tmp_3 = ((abs(b) - b) / a) * 0.5;
		end
		tmp_2 = tmp_3;
	elseif (b <= 2e+56)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (2.0 * c) / (-b - t_0);
		else
			tmp_4 = (-b + t_0) / (2.0 * a);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = (c / (b + abs(b))) * -2.0;
	else
		tmp_2 = ((sqrt((b * b)) - b) / a) * 0.5;
	end
	tmp_5 = tmp_2;
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[LessEqual[b, -1.8e+117], If[GreaterEqual[b, 0.0], N[(N[(-1.0 / N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(N[Abs[b], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, 2e+56], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c / N[(b + N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.80000000000000006e117

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-sqrt.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 32.7

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

   10. Applied rewrites 32.7%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. rem-sqrt-square N/A

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

       4. lower-fabs.f64 42.5

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

   12. Applied rewrites 42.5%

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

   if -1.80000000000000006e117 < b < 2.00000000000000018e56

   1. Initial program 72.5%

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

   if 2.00000000000000018e56 < b

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-sqrt.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 32.7

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

   10. Applied rewrites 32.7%

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

   11. Taylor expanded in a around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. pow2 N/A

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

       4. rem-sqrt-square N/A

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

       5. lower-fabs.f64 58.7

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

   13. Applied rewrites 58.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b + \left|b\right|} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b} - b}{a} \cdot 0.5\\ \end{array} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 89.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;b \leq -1.8 \cdot 10^{+117}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{\sqrt{-4 \cdot \frac{a}{c}}} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|b\right| - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1 \cdot \left(c + \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+56}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{t\_0 + b} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{b}{a}\right) \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b + \left|b\right|} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b} - b}{a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 a) c (* b b)))))
   (if (<= b -1.8e+117)
     (if (>= b 0.0)
       (* (/ -1.0 (sqrt (* -4.0 (/ a c)))) -2.0)
       (* (/ (- (fabs b) b) a) 0.5))
     (if (<= b -5e-310)
       (if (>= b 0.0)
         (/ (* -1.0 (+ c (/ (* a (* c c)) (* b b)))) b)
         (* (/ (- t_0 b) a) 0.5))
       (if (<= b 2e+56)
         (if (>= b 0.0) (* (/ c (+ t_0 b)) -2.0) (* (* -2.0 (/ b a)) 0.5))
         (if (>= b 0.0)
           (* (/ c (+ b (fabs b))) -2.0)
           (* (/ (- (sqrt (* b b)) b) a) 0.5)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-4.0 * a), c, (b * b)));
	double tmp_1;
	if (b <= -1.8e+117) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-1.0 / sqrt((-4.0 * (a / c)))) * -2.0;
		} else {
			tmp_2 = ((fabs(b) - b) / a) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= -5e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-1.0 * (c + ((a * (c * c)) / (b * b)))) / b;
		} else {
			tmp_3 = ((t_0 - b) / a) * 0.5;
		}
		tmp_1 = tmp_3;
	} else if (b <= 2e+56) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c / (t_0 + b)) * -2.0;
		} else {
			tmp_4 = (-2.0 * (b / a)) * 0.5;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / (b + fabs(b))) * -2.0;
	} else {
		tmp_1 = ((sqrt((b * b)) - b) / a) * 0.5;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))
	tmp_1 = 0.0
	if (b <= -1.8e+117)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-1.0 / sqrt(Float64(-4.0 * Float64(a / c)))) * -2.0);
		else
			tmp_2 = Float64(Float64(Float64(abs(b) - b) / a) * 0.5);
		end
		tmp_1 = tmp_2;
	elseif (b <= -5e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-1.0 * Float64(c + Float64(Float64(a * Float64(c * c)) / Float64(b * b)))) / b);
		else
			tmp_3 = Float64(Float64(Float64(t_0 - b) / a) * 0.5);
		end
		tmp_1 = tmp_3;
	elseif (b <= 2e+56)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(c / Float64(t_0 + b)) * -2.0);
		else
			tmp_4 = Float64(Float64(-2.0 * Float64(b / a)) * 0.5);
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / Float64(b + abs(b))) * -2.0);
	else
		tmp_1 = Float64(Float64(Float64(sqrt(Float64(b * b)) - b) / a) * 0.5);
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.8e+117], If[GreaterEqual[b, 0.0], N[(N[(-1.0 / N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(N[Abs[b], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, -5e-310], If[GreaterEqual[b, 0.0], N[(N[(-1.0 * N[(c + N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], N[(N[(N[(t$95$0 - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, 2e+56], If[GreaterEqual[b, 0.0], N[(N[(c / N[(t$95$0 + b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(-2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c / N[(b + N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.80000000000000006e117

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-sqrt.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 32.7

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

   10. Applied rewrites 32.7%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. rem-sqrt-square N/A

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

       4. lower-fabs.f64 42.5

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

   12. Applied rewrites 42.5%

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

   if -1.80000000000000006e117 < b < -4.999999999999985e-310

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 63.4

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

   7. Applied rewrites 63.4%

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

   if -4.999999999999985e-310 < b < 2.00000000000000018e56

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 70.6

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

   7. Applied rewrites 70.6%

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

   if 2.00000000000000018e56 < b

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-sqrt.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 32.7

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

   10. Applied rewrites 32.7%

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

   11. Taylor expanded in a around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. pow2 N/A

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

       4. rem-sqrt-square N/A

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

       5. lower-fabs.f64 58.7

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

   13. Applied rewrites 58.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b + \left|b\right|} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b} - b}{a} \cdot 0.5\\ \end{array} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 86.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b \leq -1.8 \cdot 10^{+117}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{\sqrt{-4 \cdot \frac{a}{c}}} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|b\right| - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq -6.5 \cdot 10^{-162}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1 \cdot \left(c + \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-75}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{t\_0 + b} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b + \left|b\right|} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b} - b}{a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* -4.0 (* a c)))))
   (if (<= b -1.8e+117)
     (if (>= b 0.0)
       (* (/ -1.0 (sqrt (* -4.0 (/ a c)))) -2.0)
       (* (/ (- (fabs b) b) a) 0.5))
     (if (<= b -6.5e-162)
       (if (>= b 0.0)
         (/ (* -1.0 (+ c (/ (* a (* c c)) (* b b)))) b)
         (* (/ (- (sqrt (fma (* -4.0 a) c (* b b))) b) a) 0.5))
       (if (<= b 2.5e-75)
         (if (>= b 0.0) (* (/ c (+ t_0 b)) -2.0) (* (/ (- t_0 b) a) 0.5))
         (if (>= b 0.0)
           (* (/ c (+ b (fabs b))) -2.0)
           (* (/ (- (sqrt (* b b)) b) a) 0.5)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((-4.0 * (a * c)));
	double tmp_1;
	if (b <= -1.8e+117) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-1.0 / sqrt((-4.0 * (a / c)))) * -2.0;
		} else {
			tmp_2 = ((fabs(b) - b) / a) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= -6.5e-162) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-1.0 * (c + ((a * (c * c)) / (b * b)))) / b;
		} else {
			tmp_3 = ((sqrt(fma((-4.0 * a), c, (b * b))) - b) / a) * 0.5;
		}
		tmp_1 = tmp_3;
	} else if (b <= 2.5e-75) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c / (t_0 + b)) * -2.0;
		} else {
			tmp_4 = ((t_0 - b) / a) * 0.5;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / (b + fabs(b))) * -2.0;
	} else {
		tmp_1 = ((sqrt((b * b)) - b) / a) * 0.5;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(-4.0 * Float64(a * c)))
	tmp_1 = 0.0
	if (b <= -1.8e+117)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-1.0 / sqrt(Float64(-4.0 * Float64(a / c)))) * -2.0);
		else
			tmp_2 = Float64(Float64(Float64(abs(b) - b) / a) * 0.5);
		end
		tmp_1 = tmp_2;
	elseif (b <= -6.5e-162)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-1.0 * Float64(c + Float64(Float64(a * Float64(c * c)) / Float64(b * b)))) / b);
		else
			tmp_3 = Float64(Float64(Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) - b) / a) * 0.5);
		end
		tmp_1 = tmp_3;
	elseif (b <= 2.5e-75)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(c / Float64(t_0 + b)) * -2.0);
		else
			tmp_4 = Float64(Float64(Float64(t_0 - b) / a) * 0.5);
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / Float64(b + abs(b))) * -2.0);
	else
		tmp_1 = Float64(Float64(Float64(sqrt(Float64(b * b)) - b) / a) * 0.5);
	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]}, If[LessEqual[b, -1.8e+117], If[GreaterEqual[b, 0.0], N[(N[(-1.0 / N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(N[Abs[b], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, -6.5e-162], If[GreaterEqual[b, 0.0], N[(N[(-1.0 * N[(c + N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, 2.5e-75], If[GreaterEqual[b, 0.0], N[(N[(c / N[(t$95$0 + b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(t$95$0 - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c / N[(b + N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.80000000000000006e117

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-sqrt.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 32.7

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

   10. Applied rewrites 32.7%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. rem-sqrt-square N/A

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

       4. lower-fabs.f64 42.5

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

   12. Applied rewrites 42.5%

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

   if -1.80000000000000006e117 < b < -6.49999999999999989e-162

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 63.4

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

   7. Applied rewrites 63.4%

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

   if -6.49999999999999989e-162 < b < 2.49999999999999989e-75

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.9

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

   7. Applied rewrites 56.9%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 40.7

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

   10. Applied rewrites 40.7%

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

   if 2.49999999999999989e-75 < b

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-sqrt.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 32.7

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

   10. Applied rewrites 32.7%

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

   11. Taylor expanded in a around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. pow2 N/A

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

       4. rem-sqrt-square N/A

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

       5. lower-fabs.f64 58.7

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

   13. Applied rewrites 58.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b + \left|b\right|} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b} - b}{a} \cdot 0.5\\ \end{array} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 86.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{+117}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{\sqrt{-4 \cdot \frac{a}{c}}} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|b\right| - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 1.06 \cdot 10^{-75}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{a \cdot \sqrt{\frac{-4}{a \cdot c}}} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b + \left|b\right|} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b} - b}{a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.8e+117)
   (if (>= b 0.0)
     (* (/ -1.0 (sqrt (* -4.0 (/ a c)))) -2.0)
     (* (/ (- (fabs b) b) a) 0.5))
   (if (<= b 1.06e-75)
     (if (>= b 0.0)
       (* (/ -1.0 (* a (sqrt (/ -4.0 (* a c))))) -2.0)
       (* (/ (- (sqrt (fma (* -4.0 a) c (* b b))) b) a) 0.5))
     (if (>= b 0.0)
       (* (/ c (+ b (fabs b))) -2.0)
       (* (/ (- (sqrt (* b b)) b) a) 0.5)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1.8e+117) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-1.0 / sqrt((-4.0 * (a / c)))) * -2.0;
		} else {
			tmp_2 = ((fabs(b) - b) / a) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.06e-75) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-1.0 / (a * sqrt((-4.0 / (a * c))))) * -2.0;
		} else {
			tmp_3 = ((sqrt(fma((-4.0 * a), c, (b * b))) - b) / a) * 0.5;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c / (b + fabs(b))) * -2.0;
	} else {
		tmp_1 = ((sqrt((b * b)) - b) / a) * 0.5;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -1.8e+117)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-1.0 / sqrt(Float64(-4.0 * Float64(a / c)))) * -2.0);
		else
			tmp_2 = Float64(Float64(Float64(abs(b) - b) / a) * 0.5);
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.06e-75)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-1.0 / Float64(a * sqrt(Float64(-4.0 / Float64(a * c))))) * -2.0);
		else
			tmp_3 = Float64(Float64(Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) - b) / a) * 0.5);
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / Float64(b + abs(b))) * -2.0);
	else
		tmp_1 = Float64(Float64(Float64(sqrt(Float64(b * b)) - b) / a) * 0.5);
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.8e+117], If[GreaterEqual[b, 0.0], N[(N[(-1.0 / N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(N[Abs[b], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, 1.06e-75], If[GreaterEqual[b, 0.0], N[(N[(-1.0 / N[(a * N[Sqrt[N[(-4.0 / N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c / N[(b + N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.80000000000000006e117

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-sqrt.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 32.7

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

   10. Applied rewrites 32.7%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. rem-sqrt-square N/A

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

       4. lower-fabs.f64 42.5

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

   12. Applied rewrites 42.5%

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

   if -1.80000000000000006e117 < b < 1.06e-75

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 50.8

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

   10. Applied rewrites 50.8%

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

   if 1.06e-75 < b

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-sqrt.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 32.7

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

   10. Applied rewrites 32.7%

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

   11. Taylor expanded in a around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. pow2 N/A

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

       4. rem-sqrt-square N/A

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

       5. lower-fabs.f64 58.7

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

   13. Applied rewrites 58.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b + \left|b\right|} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b} - b}{a} \cdot 0.5\\ \end{array} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b \leq -2.9 \cdot 10^{-28}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{\sqrt{-4 \cdot \frac{a}{c}}} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|b\right| - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-75}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{t\_0 + b} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b + \left|b\right|} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b} - b}{a} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* -4.0 (* a c)))))
   (if (<= b -2.9e-28)
     (if (>= b 0.0)
       (* (/ -1.0 (sqrt (* -4.0 (/ a c)))) -2.0)
       (* (/ (- (fabs b) b) a) 0.5))
     (if (<= b 2.5e-75)
       (if (>= b 0.0) (* (/ c (+ t_0 b)) -2.0) (* (/ (- t_0 b) a) 0.5))
       (if (>= b 0.0)
         (* (/ c (+ b (fabs b))) -2.0)
         (* (/ (- (sqrt (* b b)) b) a) 0.5))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((-4.0 * (a * c)));
	double tmp_1;
	if (b <= -2.9e-28) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-1.0 / sqrt((-4.0 * (a / c)))) * -2.0;
		} else {
			tmp_2 = ((fabs(b) - b) / a) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= 2.5e-75) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c / (t_0 + b)) * -2.0;
		} else {
			tmp_3 = ((t_0 - b) / a) * 0.5;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c / (b + fabs(b))) * -2.0;
	} else {
		tmp_1 = ((sqrt((b * b)) - b) / a) * 0.5;
	}
	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
    real(8) :: tmp_3
    t_0 = sqrt(((-4.0d0) * (a * c)))
    if (b <= (-2.9d-28)) then
        if (b >= 0.0d0) then
            tmp_2 = ((-1.0d0) / sqrt(((-4.0d0) * (a / c)))) * (-2.0d0)
        else
            tmp_2 = ((abs(b) - b) / a) * 0.5d0
        end if
        tmp_1 = tmp_2
    else if (b <= 2.5d-75) then
        if (b >= 0.0d0) then
            tmp_3 = (c / (t_0 + b)) * (-2.0d0)
        else
            tmp_3 = ((t_0 - b) / a) * 0.5d0
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = (c / (b + abs(b))) * (-2.0d0)
    else
        tmp_1 = ((sqrt((b * b)) - b) / a) * 0.5d0
    end if
    code = tmp_1
end function

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 <= -2.9e-28) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-1.0 / Math.sqrt((-4.0 * (a / c)))) * -2.0;
		} else {
			tmp_2 = ((Math.abs(b) - b) / a) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= 2.5e-75) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c / (t_0 + b)) * -2.0;
		} else {
			tmp_3 = ((t_0 - b) / a) * 0.5;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c / (b + Math.abs(b))) * -2.0;
	} else {
		tmp_1 = ((Math.sqrt((b * b)) - b) / a) * 0.5;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt((-4.0 * (a * c)))
	tmp_1 = 0
	if b <= -2.9e-28:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (-1.0 / math.sqrt((-4.0 * (a / c)))) * -2.0
		else:
			tmp_2 = ((math.fabs(b) - b) / a) * 0.5
		tmp_1 = tmp_2
	elif b <= 2.5e-75:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (c / (t_0 + b)) * -2.0
		else:
			tmp_3 = ((t_0 - b) / a) * 0.5
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = (c / (b + math.fabs(b))) * -2.0
	else:
		tmp_1 = ((math.sqrt((b * b)) - b) / a) * 0.5
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(-4.0 * Float64(a * c)))
	tmp_1 = 0.0
	if (b <= -2.9e-28)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-1.0 / sqrt(Float64(-4.0 * Float64(a / c)))) * -2.0);
		else
			tmp_2 = Float64(Float64(Float64(abs(b) - b) / a) * 0.5);
		end
		tmp_1 = tmp_2;
	elseif (b <= 2.5e-75)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(c / Float64(t_0 + b)) * -2.0);
		else
			tmp_3 = Float64(Float64(Float64(t_0 - b) / a) * 0.5);
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / Float64(b + abs(b))) * -2.0);
	else
		tmp_1 = Float64(Float64(Float64(sqrt(Float64(b * b)) - b) / a) * 0.5);
	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 <= -2.9e-28)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (-1.0 / sqrt((-4.0 * (a / c)))) * -2.0;
		else
			tmp_3 = ((abs(b) - b) / a) * 0.5;
		end
		tmp_2 = tmp_3;
	elseif (b <= 2.5e-75)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (c / (t_0 + b)) * -2.0;
		else
			tmp_4 = ((t_0 - b) / a) * 0.5;
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = (c / (b + abs(b))) * -2.0;
	else
		tmp_2 = ((sqrt((b * b)) - b) / a) * 0.5;
	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, -2.9e-28], If[GreaterEqual[b, 0.0], N[(N[(-1.0 / N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(N[Abs[b], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, 2.5e-75], If[GreaterEqual[b, 0.0], N[(N[(c / N[(t$95$0 + b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(t$95$0 - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c / N[(b + N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.90000000000000013e-28

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-sqrt.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 32.7

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

   10. Applied rewrites 32.7%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. rem-sqrt-square N/A

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

       4. lower-fabs.f64 42.5

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

   12. Applied rewrites 42.5%

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

   if -2.90000000000000013e-28 < b < 2.49999999999999989e-75

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.9

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

   7. Applied rewrites 56.9%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 40.7

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

   10. Applied rewrites 40.7%

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

   if 2.49999999999999989e-75 < b

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-sqrt.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 32.7

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

   10. Applied rewrites 32.7%

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

   11. Taylor expanded in a around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. pow2 N/A

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

       4. rem-sqrt-square N/A

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

       5. lower-fabs.f64 58.7

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

   13. Applied rewrites 58.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b + \left|b\right|} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b} - b}{a} \cdot 0.5\\ \end{array} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 80.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b} - b}{a} \cdot 0.5\\ t_1 := \frac{-1}{\sqrt{-4 \cdot \frac{a}{c}}} \cdot -2\\ t_2 := \sqrt{-4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b \leq -2.05 \cdot 10^{-29}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|b\right| - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 1.06 \cdot 10^{-75}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{t\_2} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b + \left|b\right|} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (/ (- (sqrt (* b b)) b) a) 0.5))
        (t_1 (* (/ -1.0 (sqrt (* -4.0 (/ a c)))) -2.0))
        (t_2 (sqrt (* -4.0 (* a c)))))
   (if (<= b -2.05e-29)
     (if (>= b 0.0) t_1 (* (/ (- (fabs b) b) a) 0.5))
     (if (<= b -5e-310)
       (if (>= b 0.0) t_1 (* (/ t_2 a) 0.5))
       (if (<= b 1.06e-75)
         (if (>= b 0.0) (* (/ c t_2) -2.0) t_0)
         (if (>= b 0.0) (* (/ c (+ b (fabs b))) -2.0) t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = ((sqrt((b * b)) - b) / a) * 0.5;
	double t_1 = (-1.0 / sqrt((-4.0 * (a / c)))) * -2.0;
	double t_2 = sqrt((-4.0 * (a * c)));
	double tmp_1;
	if (b <= -2.05e-29) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = ((fabs(b) - b) / a) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= -5e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = (t_2 / a) * 0.5;
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.06e-75) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c / t_2) * -2.0;
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / (b + fabs(b))) * -2.0;
	} 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
    real(8) :: tmp_3
    real(8) :: tmp_4
    t_0 = ((sqrt((b * b)) - b) / a) * 0.5d0
    t_1 = ((-1.0d0) / sqrt(((-4.0d0) * (a / c)))) * (-2.0d0)
    t_2 = sqrt(((-4.0d0) * (a * c)))
    if (b <= (-2.05d-29)) then
        if (b >= 0.0d0) then
            tmp_2 = t_1
        else
            tmp_2 = ((abs(b) - b) / a) * 0.5d0
        end if
        tmp_1 = tmp_2
    else if (b <= (-5d-310)) then
        if (b >= 0.0d0) then
            tmp_3 = t_1
        else
            tmp_3 = (t_2 / a) * 0.5d0
        end if
        tmp_1 = tmp_3
    else if (b <= 1.06d-75) then
        if (b >= 0.0d0) then
            tmp_4 = (c / t_2) * (-2.0d0)
        else
            tmp_4 = t_0
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = (c / (b + abs(b))) * (-2.0d0)
    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 = ((Math.sqrt((b * b)) - b) / a) * 0.5;
	double t_1 = (-1.0 / Math.sqrt((-4.0 * (a / c)))) * -2.0;
	double t_2 = Math.sqrt((-4.0 * (a * c)));
	double tmp_1;
	if (b <= -2.05e-29) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = ((Math.abs(b) - b) / a) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= -5e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = (t_2 / a) * 0.5;
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.06e-75) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c / t_2) * -2.0;
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / (b + Math.abs(b))) * -2.0;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = ((math.sqrt((b * b)) - b) / a) * 0.5
	t_1 = (-1.0 / math.sqrt((-4.0 * (a / c)))) * -2.0
	t_2 = math.sqrt((-4.0 * (a * c)))
	tmp_1 = 0
	if b <= -2.05e-29:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_1
		else:
			tmp_2 = ((math.fabs(b) - b) / a) * 0.5
		tmp_1 = tmp_2
	elif b <= -5e-310:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = t_1
		else:
			tmp_3 = (t_2 / a) * 0.5
		tmp_1 = tmp_3
	elif b <= 1.06e-75:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = (c / t_2) * -2.0
		else:
			tmp_4 = t_0
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = (c / (b + math.fabs(b))) * -2.0
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(sqrt(Float64(b * b)) - b) / a) * 0.5)
	t_1 = Float64(Float64(-1.0 / sqrt(Float64(-4.0 * Float64(a / c)))) * -2.0)
	t_2 = sqrt(Float64(-4.0 * Float64(a * c)))
	tmp_1 = 0.0
	if (b <= -2.05e-29)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(Float64(Float64(abs(b) - b) / a) * 0.5);
		end
		tmp_1 = tmp_2;
	elseif (b <= -5e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = Float64(Float64(t_2 / a) * 0.5);
		end
		tmp_1 = tmp_3;
	elseif (b <= 1.06e-75)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(c / t_2) * -2.0);
		else
			tmp_4 = t_0;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / Float64(b + abs(b))) * -2.0);
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = ((sqrt((b * b)) - b) / a) * 0.5;
	t_1 = (-1.0 / sqrt((-4.0 * (a / c)))) * -2.0;
	t_2 = sqrt((-4.0 * (a * c)));
	tmp_2 = 0.0;
	if (b <= -2.05e-29)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = ((abs(b) - b) / a) * 0.5;
		end
		tmp_2 = tmp_3;
	elseif (b <= -5e-310)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = t_1;
		else
			tmp_4 = (t_2 / a) * 0.5;
		end
		tmp_2 = tmp_4;
	elseif (b <= 1.06e-75)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = (c / t_2) * -2.0;
		else
			tmp_5 = t_0;
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = (c / (b + abs(b))) * -2.0;
	else
		tmp_2 = t_0;
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-1.0 / N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2.05e-29], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(N[(N[Abs[b], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, -5e-310], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(t$95$2 / a), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, 1.06e-75], If[GreaterEqual[b, 0.0], N[(N[(c / t$95$2), $MachinePrecision] * -2.0), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(c / N[(b + N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.0499999999999999e-29

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-sqrt.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 32.7

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

   10. Applied rewrites 32.7%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. rem-sqrt-square N/A

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

       4. lower-fabs.f64 42.5

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

   12. Applied rewrites 42.5%

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

   if -2.0499999999999999e-29 < b < -4.999999999999985e-310

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-sqrt.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 32.7

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

   10. Applied rewrites 32.7%

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

   11. Taylor expanded in b around 0

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

       1. lower-/.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 20.9

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

   13. Applied rewrites 20.9%

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

   if -4.999999999999985e-310 < b < 1.06e-75

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-sqrt.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 32.7

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

   10. Applied rewrites 32.7%

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

   11. Taylor expanded in b around 0

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

       1. lower-/.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 37.6

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

   13. Applied rewrites 37.6%

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

   if 1.06e-75 < b

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-sqrt.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 32.7

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

   10. Applied rewrites 32.7%

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

   11. Taylor expanded in a around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. pow2 N/A

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

       4. rem-sqrt-square N/A

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

       5. lower-fabs.f64 58.7

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

   13. Applied rewrites 58.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b + \left|b\right|} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b} - b}{a} \cdot 0.5\\ \end{array} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 10: 71.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b} - b}{a} \cdot 0.5\\ t_1 := \sqrt{-4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b \leq -2.05 \cdot 10^{-29}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(0.5 \cdot \frac{b}{a}\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{\sqrt{-4 \cdot \frac{a}{c}}} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 1.06 \cdot 10^{-75}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{t\_1} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b + \left|b\right|} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (/ (- (sqrt (* b b)) b) a) 0.5)) (t_1 (sqrt (* -4.0 (* a c)))))
   (if (<= b -2.05e-29)
     (if (>= b 0.0) (* (* 0.5 (/ b a)) -2.0) t_0)
     (if (<= b -5e-310)
       (if (>= b 0.0)
         (* (/ -1.0 (sqrt (* -4.0 (/ a c)))) -2.0)
         (* (/ t_1 a) 0.5))
       (if (<= b 1.06e-75)
         (if (>= b 0.0) (* (/ c t_1) -2.0) t_0)
         (if (>= b 0.0) (* (/ c (+ b (fabs b))) -2.0) t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = ((sqrt((b * b)) - b) / a) * 0.5;
	double t_1 = sqrt((-4.0 * (a * c)));
	double tmp_1;
	if (b <= -2.05e-29) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (0.5 * (b / a)) * -2.0;
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -5e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-1.0 / sqrt((-4.0 * (a / c)))) * -2.0;
		} else {
			tmp_3 = (t_1 / a) * 0.5;
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.06e-75) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c / t_1) * -2.0;
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / (b + fabs(b))) * -2.0;
	} 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) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    real(8) :: tmp_4
    t_0 = ((sqrt((b * b)) - b) / a) * 0.5d0
    t_1 = sqrt(((-4.0d0) * (a * c)))
    if (b <= (-2.05d-29)) then
        if (b >= 0.0d0) then
            tmp_2 = (0.5d0 * (b / a)) * (-2.0d0)
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b <= (-5d-310)) then
        if (b >= 0.0d0) then
            tmp_3 = ((-1.0d0) / sqrt(((-4.0d0) * (a / c)))) * (-2.0d0)
        else
            tmp_3 = (t_1 / a) * 0.5d0
        end if
        tmp_1 = tmp_3
    else if (b <= 1.06d-75) then
        if (b >= 0.0d0) then
            tmp_4 = (c / t_1) * (-2.0d0)
        else
            tmp_4 = t_0
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = (c / (b + abs(b))) * (-2.0d0)
    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 = ((Math.sqrt((b * b)) - b) / a) * 0.5;
	double t_1 = Math.sqrt((-4.0 * (a * c)));
	double tmp_1;
	if (b <= -2.05e-29) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (0.5 * (b / a)) * -2.0;
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -5e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-1.0 / Math.sqrt((-4.0 * (a / c)))) * -2.0;
		} else {
			tmp_3 = (t_1 / a) * 0.5;
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.06e-75) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c / t_1) * -2.0;
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / (b + Math.abs(b))) * -2.0;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = ((math.sqrt((b * b)) - b) / a) * 0.5
	t_1 = math.sqrt((-4.0 * (a * c)))
	tmp_1 = 0
	if b <= -2.05e-29:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (0.5 * (b / a)) * -2.0
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b <= -5e-310:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (-1.0 / math.sqrt((-4.0 * (a / c)))) * -2.0
		else:
			tmp_3 = (t_1 / a) * 0.5
		tmp_1 = tmp_3
	elif b <= 1.06e-75:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = (c / t_1) * -2.0
		else:
			tmp_4 = t_0
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = (c / (b + math.fabs(b))) * -2.0
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(sqrt(Float64(b * b)) - b) / a) * 0.5)
	t_1 = sqrt(Float64(-4.0 * Float64(a * c)))
	tmp_1 = 0.0
	if (b <= -2.05e-29)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(0.5 * Float64(b / a)) * -2.0);
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b <= -5e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-1.0 / sqrt(Float64(-4.0 * Float64(a / c)))) * -2.0);
		else
			tmp_3 = Float64(Float64(t_1 / a) * 0.5);
		end
		tmp_1 = tmp_3;
	elseif (b <= 1.06e-75)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(c / t_1) * -2.0);
		else
			tmp_4 = t_0;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / Float64(b + abs(b))) * -2.0);
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = ((sqrt((b * b)) - b) / a) * 0.5;
	t_1 = sqrt((-4.0 * (a * c)));
	tmp_2 = 0.0;
	if (b <= -2.05e-29)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (0.5 * (b / a)) * -2.0;
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b <= -5e-310)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (-1.0 / sqrt((-4.0 * (a / c)))) * -2.0;
		else
			tmp_4 = (t_1 / a) * 0.5;
		end
		tmp_2 = tmp_4;
	elseif (b <= 1.06e-75)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = (c / t_1) * -2.0;
		else
			tmp_5 = t_0;
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = (c / (b + abs(b))) * -2.0;
	else
		tmp_2 = t_0;
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2.05e-29], If[GreaterEqual[b, 0.0], N[(N[(0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], t$95$0], If[LessEqual[b, -5e-310], If[GreaterEqual[b, 0.0], N[(N[(-1.0 / N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(t$95$1 / a), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, 1.06e-75], If[GreaterEqual[b, 0.0], N[(N[(c / t$95$1), $MachinePrecision] * -2.0), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(c / N[(b + N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.0499999999999999e-29

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-sqrt.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 32.7

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

   10. Applied rewrites 32.7%

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

   11. Taylor expanded in b around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 26.0

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

   13. Applied rewrites 26.0%

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

   if -2.0499999999999999e-29 < b < -4.999999999999985e-310

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-sqrt.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 32.7

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

   10. Applied rewrites 32.7%

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

   11. Taylor expanded in b around 0

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

       1. lower-/.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 20.9

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

   13. Applied rewrites 20.9%

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

   if -4.999999999999985e-310 < b < 1.06e-75

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-sqrt.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 32.7

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

   10. Applied rewrites 32.7%

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

   11. Taylor expanded in b around 0

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

       1. lower-/.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 37.6

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

   13. Applied rewrites 37.6%

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

   if 1.06e-75 < b

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-sqrt.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 32.7

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

   10. Applied rewrites 32.7%

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

   11. Taylor expanded in a around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. pow2 N/A

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

       4. rem-sqrt-square N/A

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

       5. lower-fabs.f64 58.7

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

   13. Applied rewrites 58.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b + \left|b\right|} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b} - b}{a} \cdot 0.5\\ \end{array} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 11: 66.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b} - b}{a} \cdot 0.5\\ \mathbf{if}\;b \leq -3.3 \cdot 10^{-209}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(0.5 \cdot \frac{b}{a}\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{\sqrt{-4 \cdot \frac{a}{c}}} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot \sqrt{-4 \cdot \frac{c}{a}}\right) \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 1.06 \cdot 10^{-75}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b + \left|b\right|} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (/ (- (sqrt (* b b)) b) a) 0.5)))
   (if (<= b -3.3e-209)
     (if (>= b 0.0) (* (* 0.5 (/ b a)) -2.0) t_0)
     (if (<= b -5e-310)
       (if (>= b 0.0)
         (* (/ -1.0 (sqrt (* -4.0 (/ a c)))) -2.0)
         (* (* -1.0 (sqrt (* -4.0 (/ c a)))) 0.5))
       (if (<= b 1.06e-75)
         (if (>= b 0.0) (* (/ c (sqrt (* -4.0 (* a c)))) -2.0) t_0)
         (if (>= b 0.0) (* (/ c (+ b (fabs b))) -2.0) t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = ((sqrt((b * b)) - b) / a) * 0.5;
	double tmp_1;
	if (b <= -3.3e-209) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (0.5 * (b / a)) * -2.0;
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -5e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-1.0 / sqrt((-4.0 * (a / c)))) * -2.0;
		} else {
			tmp_3 = (-1.0 * sqrt((-4.0 * (c / a)))) * 0.5;
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.06e-75) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c / sqrt((-4.0 * (a * c)))) * -2.0;
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / (b + fabs(b))) * -2.0;
	} 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
    real(8) :: tmp_3
    real(8) :: tmp_4
    t_0 = ((sqrt((b * b)) - b) / a) * 0.5d0
    if (b <= (-3.3d-209)) then
        if (b >= 0.0d0) then
            tmp_2 = (0.5d0 * (b / a)) * (-2.0d0)
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b <= (-5d-310)) then
        if (b >= 0.0d0) then
            tmp_3 = ((-1.0d0) / sqrt(((-4.0d0) * (a / c)))) * (-2.0d0)
        else
            tmp_3 = ((-1.0d0) * sqrt(((-4.0d0) * (c / a)))) * 0.5d0
        end if
        tmp_1 = tmp_3
    else if (b <= 1.06d-75) then
        if (b >= 0.0d0) then
            tmp_4 = (c / sqrt(((-4.0d0) * (a * c)))) * (-2.0d0)
        else
            tmp_4 = t_0
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = (c / (b + abs(b))) * (-2.0d0)
    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 = ((Math.sqrt((b * b)) - b) / a) * 0.5;
	double tmp_1;
	if (b <= -3.3e-209) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (0.5 * (b / a)) * -2.0;
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -5e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-1.0 / Math.sqrt((-4.0 * (a / c)))) * -2.0;
		} else {
			tmp_3 = (-1.0 * Math.sqrt((-4.0 * (c / a)))) * 0.5;
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.06e-75) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c / Math.sqrt((-4.0 * (a * c)))) * -2.0;
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c / (b + Math.abs(b))) * -2.0;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = ((math.sqrt((b * b)) - b) / a) * 0.5
	tmp_1 = 0
	if b <= -3.3e-209:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (0.5 * (b / a)) * -2.0
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b <= -5e-310:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (-1.0 / math.sqrt((-4.0 * (a / c)))) * -2.0
		else:
			tmp_3 = (-1.0 * math.sqrt((-4.0 * (c / a)))) * 0.5
		tmp_1 = tmp_3
	elif b <= 1.06e-75:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = (c / math.sqrt((-4.0 * (a * c)))) * -2.0
		else:
			tmp_4 = t_0
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = (c / (b + math.fabs(b))) * -2.0
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(sqrt(Float64(b * b)) - b) / a) * 0.5)
	tmp_1 = 0.0
	if (b <= -3.3e-209)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(0.5 * Float64(b / a)) * -2.0);
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b <= -5e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-1.0 / sqrt(Float64(-4.0 * Float64(a / c)))) * -2.0);
		else
			tmp_3 = Float64(Float64(-1.0 * sqrt(Float64(-4.0 * Float64(c / a)))) * 0.5);
		end
		tmp_1 = tmp_3;
	elseif (b <= 1.06e-75)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(c / sqrt(Float64(-4.0 * Float64(a * c)))) * -2.0);
		else
			tmp_4 = t_0;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / Float64(b + abs(b))) * -2.0);
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = ((sqrt((b * b)) - b) / a) * 0.5;
	tmp_2 = 0.0;
	if (b <= -3.3e-209)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (0.5 * (b / a)) * -2.0;
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b <= -5e-310)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (-1.0 / sqrt((-4.0 * (a / c)))) * -2.0;
		else
			tmp_4 = (-1.0 * sqrt((-4.0 * (c / a)))) * 0.5;
		end
		tmp_2 = tmp_4;
	elseif (b <= 1.06e-75)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = (c / sqrt((-4.0 * (a * c)))) * -2.0;
		else
			tmp_5 = t_0;
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = (c / (b + abs(b))) * -2.0;
	else
		tmp_2 = t_0;
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[b, -3.3e-209], If[GreaterEqual[b, 0.0], N[(N[(0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], t$95$0], If[LessEqual[b, -5e-310], If[GreaterEqual[b, 0.0], N[(N[(-1.0 / N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(-1.0 * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, 1.06e-75], If[GreaterEqual[b, 0.0], N[(N[(c / N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(c / N[(b + N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -3.29999999999999974e-209

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-sqrt.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 32.7

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

   10. Applied rewrites 32.7%

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

   11. Taylor expanded in b around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 26.0

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

   13. Applied rewrites 26.0%

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

   if -3.29999999999999974e-209 < b < -4.999999999999985e-310

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-sqrt.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 32.7

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

   10. Applied rewrites 32.7%

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

   11. Taylor expanded in a around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 15.7

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

   13. Applied rewrites 15.7%

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

   if -4.999999999999985e-310 < b < 1.06e-75

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-sqrt.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 32.7

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

   10. Applied rewrites 32.7%

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

   11. Taylor expanded in b around 0

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

       1. lower-/.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 37.6

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

   13. Applied rewrites 37.6%

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

   if 1.06e-75 < b

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-sqrt.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 32.7

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

   10. Applied rewrites 32.7%

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

   11. Taylor expanded in a around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. pow2 N/A

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

       4. rem-sqrt-square N/A

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

       5. lower-fabs.f64 58.7

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

   13. Applied rewrites 58.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b + \left|b\right|} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b} - b}{a} \cdot 0.5\\ \end{array} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 64.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b} - b}{a} \cdot 0.5\\ \mathbf{if}\;b \leq 1.06 \cdot 10^{-75}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b + \left|b\right|} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (/ (- (sqrt (* b b)) b) a) 0.5)))
   (if (<= b 1.06e-75)
     (if (>= b 0.0) (* (/ c (sqrt (* -4.0 (* a c)))) -2.0) t_0)
     (if (>= b 0.0) (* (/ c (+ b (fabs b))) -2.0) t_0))))

C

double code(double a, double b, double c) {
	double t_0 = ((sqrt((b * b)) - b) / a) * 0.5;
	double tmp_1;
	if (b <= 1.06e-75) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c / sqrt((-4.0 * (a * c)))) * -2.0;
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (c / (b + fabs(b))) * -2.0;
	} 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 = ((sqrt((b * b)) - b) / a) * 0.5d0
    if (b <= 1.06d-75) then
        if (b >= 0.0d0) then
            tmp_2 = (c / sqrt(((-4.0d0) * (a * c)))) * (-2.0d0)
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (c / (b + abs(b))) * (-2.0d0)
    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 = ((Math.sqrt((b * b)) - b) / a) * 0.5;
	double tmp_1;
	if (b <= 1.06e-75) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c / Math.sqrt((-4.0 * (a * c)))) * -2.0;
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (c / (b + Math.abs(b))) * -2.0;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = ((math.sqrt((b * b)) - b) / a) * 0.5
	tmp_1 = 0
	if b <= 1.06e-75:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (c / math.sqrt((-4.0 * (a * c)))) * -2.0
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = (c / (b + math.fabs(b))) * -2.0
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(sqrt(Float64(b * b)) - b) / a) * 0.5)
	tmp_1 = 0.0
	if (b <= 1.06e-75)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(c / sqrt(Float64(-4.0 * Float64(a * c)))) * -2.0);
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / Float64(b + abs(b))) * -2.0);
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = ((sqrt((b * b)) - b) / a) * 0.5;
	tmp_2 = 0.0;
	if (b <= 1.06e-75)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (c / sqrt((-4.0 * (a * c)))) * -2.0;
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = (c / (b + abs(b))) * -2.0;
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[b, 1.06e-75], If[GreaterEqual[b, 0.0], N[(N[(c / N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(c / N[(b + N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.06e-75

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-sqrt.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 32.7

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

   10. Applied rewrites 32.7%

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

   11. Taylor expanded in b around 0

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

       1. lower-/.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 37.6

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

   13. Applied rewrites 37.6%

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

   if 1.06e-75 < b

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-sqrt.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 32.7

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

   10. Applied rewrites 32.7%

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

   11. Taylor expanded in a around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. pow2 N/A

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

       4. rem-sqrt-square N/A

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

       5. lower-fabs.f64 58.7

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

   13. Applied rewrites 58.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b + \left|b\right|} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b} - b}{a} \cdot 0.5\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 58.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (c / (b + Math.abs(b))) * -2.0;
	} else {
		tmp = ((Math.sqrt((b * b)) - b) / a) * 0.5;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

2. Taylor expanded in a around 0

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

4. Applied rewrites 72.6%

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

5. Taylor expanded in c around -inf

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lift-/.f64 44.5

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

7. Applied rewrites 44.5%

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

8. Taylor expanded in a around 0

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

   1. lower-sqrt.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 32.7

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

10. Applied rewrites 32.7%

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

11. Taylor expanded in a around 0

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

    1. lower-/.f64 N/A

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

    2. lower-+.f64 N/A

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

    3. pow2 N/A

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

    4. rem-sqrt-square N/A

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

    5. lower-fabs.f64 58.7

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

13. Applied rewrites 58.7%

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

Alternative 14: 26.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (0.5 * (b / a)) * -2.0;
	} else {
		tmp = ((Math.sqrt((b * b)) - b) / a) * 0.5;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

2. Taylor expanded in a around 0

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

4. Applied rewrites 72.6%

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

5. Taylor expanded in c around -inf

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lift-/.f64 44.5

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

7. Applied rewrites 44.5%

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

8. Taylor expanded in a around 0

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

   1. lower-sqrt.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 32.7

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

10. Applied rewrites 32.7%

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

11. Taylor expanded in b around -inf

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

    1. lower-*.f64 N/A

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

    2. lower-/.f64 26.0

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

13. Applied rewrites 26.0%

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

Reproduce

herbie shell --seed 2025142 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))