jeff quadratic root 2

Percentage Accurate: 72.6% → 89.7%

Time: 4.8s

Alternatives: 14

Speedup: 0.9×

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: 72.6% 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: 89.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;b \leq -1.35 \cdot 10^{+40}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{\left(-b\right) - \sqrt{b \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{+49}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{t\_0 + b} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{c}{a} \cdot -4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 a) c (* b b)))))
   (if (<= b -1.35e+40)
     (if (>= b 0.0)
       (/ (+ c c) (- (- b) (sqrt (* b b))))
       (fma -1.0 (/ b a) (/ c b)))
     (if (<= b 2.9e+49)
       (if (>= b 0.0) (* (/ c (+ t_0 b)) -2.0) (* (/ (- t_0 b) a) 0.5))
       (if (>= b 0.0)
         (/ (* 2.0 c) (- (- b) b))
         (* 0.5 (sqrt (* (/ c a) -4.0))))))))

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 <= -1.35e+40) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c + c) / (-b - sqrt((b * b)));
		} else {
			tmp_2 = fma(-1.0, (b / a), (c / b));
		}
		tmp_1 = tmp_2;
	} else if (b <= 2.9e+49) {
		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 = (2.0 * c) / (-b - b);
	} else {
		tmp_1 = 0.5 * sqrt(((c / a) * -4.0));
	}
	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 <= -1.35e+40)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(c + c) / Float64(Float64(-b) - sqrt(Float64(b * b))));
		else
			tmp_2 = fma(-1.0, Float64(b / a), Float64(c / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 2.9e+49)
		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(2.0 * c) / Float64(Float64(-b) - b));
	else
		tmp_1 = Float64(0.5 * sqrt(Float64(Float64(c / a) * -4.0)));
	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, -1.35e+40], If[GreaterEqual[b, 0.0], N[(N[(c + c), $MachinePrecision] / N[((-b) - N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(b / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 2.9e+49], 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[(2.0 * c), $MachinePrecision] / N[((-b) - b), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(N[(c / a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.35000000000000005e40

   1. Initial program 62.7%

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

   3. Applied rewrites 62.8%

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

   4. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 62.8

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

   6. Applied rewrites 62.8%

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

   7. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 93.3

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

   9. Applied rewrites 93.3%

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

   10. Taylor expanded in a around inf

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

       1. lower-fma.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 93.7

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

   12. Applied rewrites 93.7%

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

   if -1.35000000000000005e40 < b < 2.9e49

   1. Initial program 85.3%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 85.3%

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

   if 2.9e49 < b

   1. Initial program 60.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in b around -inf

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

      1. lower-*.f64 93.6

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

   7. Applied rewrites 93.6%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 93.6

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

   10. Applied rewrites 93.6%

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

Alternative 2: 89.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c + c}{\left(-b\right) - \sqrt{b \cdot b}}\\ t_1 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;b \leq -1.35 \cdot 10^{+40}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq -6.4 \cdot 10^{-301}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 - b}{a + a}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{+49}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{t\_1 + b} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \sqrt{\frac{c}{a} \cdot -1}, -0.5 \cdot \frac{b}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{c}{a} \cdot -4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (+ c c) (- (- b) (sqrt (* b b)))))
        (t_1 (sqrt (fma (* -4.0 a) c (* b b)))))
   (if (<= b -1.35e+40)
     (if (>= b 0.0) t_0 (fma -1.0 (/ b a) (/ c b)))
     (if (<= b -6.4e-301)
       (if (>= b 0.0) t_0 (/ (- t_1 b) (+ a a)))
       (if (<= b 2.9e+49)
         (if (>= b 0.0)
           (* (/ c (+ t_1 b)) -2.0)
           (fma -1.0 (sqrt (* (/ c a) -1.0)) (* -0.5 (/ b a))))
         (if (>= b 0.0)
           (/ (* 2.0 c) (- (- b) b))
           (* 0.5 (sqrt (* (/ c a) -4.0)))))))))

C

double code(double a, double b, double c) {
	double t_0 = (c + c) / (-b - sqrt((b * b)));
	double t_1 = sqrt(fma((-4.0 * a), c, (b * b)));
	double tmp_1;
	if (b <= -1.35e+40) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = fma(-1.0, (b / a), (c / b));
		}
		tmp_1 = tmp_2;
	} else if (b <= -6.4e-301) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_0;
		} else {
			tmp_3 = (t_1 - b) / (a + a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 2.9e+49) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c / (t_1 + b)) * -2.0;
		} else {
			tmp_4 = fma(-1.0, sqrt(((c / a) * -1.0)), (-0.5 * (b / a)));
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-b - b);
	} else {
		tmp_1 = 0.5 * sqrt(((c / a) * -4.0));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c + c) / Float64(Float64(-b) - sqrt(Float64(b * b))))
	t_1 = sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))
	tmp_1 = 0.0
	if (b <= -1.35e+40)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = fma(-1.0, Float64(b / a), Float64(c / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= -6.4e-301)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_0;
		else
			tmp_3 = Float64(Float64(t_1 - b) / Float64(a + a));
		end
		tmp_1 = tmp_3;
	elseif (b <= 2.9e+49)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(c / Float64(t_1 + b)) * -2.0);
		else
			tmp_4 = fma(-1.0, sqrt(Float64(Float64(c / a) * -1.0)), Float64(-0.5 * Float64(b / a)));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - b));
	else
		tmp_1 = Float64(0.5 * sqrt(Float64(Float64(c / a) * -4.0)));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c + c), $MachinePrecision] / N[((-b) - N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.35e+40], If[GreaterEqual[b, 0.0], t$95$0, N[(-1.0 * N[(b / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -6.4e-301], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(t$95$1 - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 2.9e+49], If[GreaterEqual[b, 0.0], N[(N[(c / N[(t$95$1 + b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(-1.0 * N[Sqrt[N[(N[(c / a), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision] + N[(-0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - b), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(N[(c / a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.35000000000000005e40

   1. Initial program 62.7%

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

   3. Applied rewrites 62.8%

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

   4. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 62.8

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

   6. Applied rewrites 62.8%

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

   7. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 93.3

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

   9. Applied rewrites 93.3%

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

   10. Taylor expanded in a around inf

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

       1. lower-fma.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 93.7

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

   12. Applied rewrites 93.7%

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

   if -1.35000000000000005e40 < b < -6.3999999999999998e-301

   1. Initial program 85.3%

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

   3. Applied rewrites 85.3%

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

   4. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 85.3

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

   6. Applied rewrites 85.3%

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

   if -6.3999999999999998e-301 < b < 2.9e49

   1. Initial program 85.2%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 85.3%

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

   5. Taylor expanded in a around -inf

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

      1. lower-fma.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 84.5

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

   7. Applied rewrites 84.5%

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

   if 2.9e49 < b

   1. Initial program 60.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in b around -inf

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

      1. lower-*.f64 93.6

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

   7. Applied rewrites 93.6%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 93.6

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

   10. Applied rewrites 93.6%

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

Alternative 3: 83.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c + c}{\left(-b\right) - \sqrt{b \cdot b}}\\ t_1 := \sqrt{\frac{c}{a} \cdot -4}\\ \mathbf{if}\;b \leq -1.35 \cdot 10^{+40}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}{a + a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.86 \cdot 10^{+46}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, 0.5, \frac{b}{a} \cdot -0.5\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (+ c c) (- (- b) (sqrt (* b b)))))
        (t_1 (sqrt (* (/ c a) -4.0))))
   (if (<= b -1.35e+40)
     (if (>= b 0.0) t_0 (fma -1.0 (/ b a) (/ c b)))
     (if (<= b -2e-310)
       (if (>= b 0.0) t_0 (/ (- (sqrt (fma (* -4.0 a) c (* b b))) b) (+ a a)))
       (if (<= b 1.86e+46)
         (if (>= b 0.0)
           (/ (* 2.0 c) (- (- b) (sqrt (* -4.0 (* a c)))))
           (fma t_1 0.5 (* (/ b a) -0.5)))
         (if (>= b 0.0) (/ (* 2.0 c) (- (- b) b)) (* 0.5 t_1)))))))

C

double code(double a, double b, double c) {
	double t_0 = (c + c) / (-b - sqrt((b * b)));
	double t_1 = sqrt(((c / a) * -4.0));
	double tmp_1;
	if (b <= -1.35e+40) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = fma(-1.0, (b / a), (c / b));
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_0;
		} else {
			tmp_3 = (sqrt(fma((-4.0 * a), c, (b * b))) - b) / (a + a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.86e+46) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (2.0 * c) / (-b - sqrt((-4.0 * (a * c))));
		} else {
			tmp_4 = fma(t_1, 0.5, ((b / a) * -0.5));
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-b - b);
	} else {
		tmp_1 = 0.5 * t_1;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c + c) / Float64(Float64(-b) - sqrt(Float64(b * b))))
	t_1 = sqrt(Float64(Float64(c / a) * -4.0))
	tmp_1 = 0.0
	if (b <= -1.35e+40)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = fma(-1.0, Float64(b / a), Float64(c / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= -2e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_0;
		else
			tmp_3 = Float64(Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) - b) / Float64(a + a));
		end
		tmp_1 = tmp_3;
	elseif (b <= 1.86e+46)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - sqrt(Float64(-4.0 * Float64(a * c)))));
		else
			tmp_4 = fma(t_1, 0.5, Float64(Float64(b / a) * -0.5));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - b));
	else
		tmp_1 = Float64(0.5 * t_1);
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c + c), $MachinePrecision] / N[((-b) - N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(c / a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.35e+40], If[GreaterEqual[b, 0.0], t$95$0, N[(-1.0 * N[(b / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -2e-310], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.86e+46], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * 0.5 + N[(N[(b / a), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - b), $MachinePrecision]), $MachinePrecision], N[(0.5 * t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.35000000000000005e40

   1. Initial program 62.7%

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

   3. Applied rewrites 62.8%

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

   4. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 62.8

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

   6. Applied rewrites 62.8%

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

   7. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 93.3

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

   9. Applied rewrites 93.3%

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

   10. Taylor expanded in a around inf

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

       1. lower-fma.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 93.7

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

   12. Applied rewrites 93.7%

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

   if -1.35000000000000005e40 < b < -1.999999999999994e-310

   1. Initial program 85.2%

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

   3. Applied rewrites 85.2%

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

   4. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 85.2

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

   6. Applied rewrites 85.2%

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

   if -1.999999999999994e-310 < b < 1.8600000000000001e46

   1. Initial program 85.3%

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

   2. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 85.3

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

   4. Applied rewrites 85.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-*.f64 56.9

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

   7. Applied rewrites 56.9%

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

   if 1.8600000000000001e46 < b

   1. Initial program 60.6%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 93.5%

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

   5. Taylor expanded in b around -inf

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

      1. lower-*.f64 93.5

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

   7. Applied rewrites 93.5%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 93.5

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

   10. Applied rewrites 93.5%

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

Alternative 4: 79.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-4 \cdot \left(a \cdot c\right)}\\ t_1 := \frac{c + c}{\left(-b\right) - \sqrt{b \cdot b}}\\ t_2 := \sqrt{\frac{c}{a} \cdot -4}\\ \mathbf{if}\;b \leq -2.05 \cdot 10^{-37}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a + a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.86 \cdot 10^{+46}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, 0.5, \frac{b}{a} \cdot -0.5\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* -4.0 (* a c))))
        (t_1 (/ (+ c c) (- (- b) (sqrt (* b b)))))
        (t_2 (sqrt (* (/ c a) -4.0))))
   (if (<= b -2.05e-37)
     (if (>= b 0.0) t_1 (fma -1.0 (/ b a) (/ c b)))
     (if (<= b -2e-310)
       (if (>= b 0.0) t_1 (/ (- t_0 b) (+ a a)))
       (if (<= b 1.86e+46)
         (if (>= b 0.0)
           (/ (* 2.0 c) (- (- b) t_0))
           (fma t_2 0.5 (* (/ b a) -0.5)))
         (if (>= b 0.0) (/ (* 2.0 c) (- (- b) b)) (* 0.5 t_2)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((-4.0 * (a * c)));
	double t_1 = (c + c) / (-b - sqrt((b * b)));
	double t_2 = sqrt(((c / a) * -4.0));
	double tmp_1;
	if (b <= -2.05e-37) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = fma(-1.0, (b / a), (c / b));
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = (t_0 - b) / (a + a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.86e+46) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_4 = fma(t_2, 0.5, ((b / a) * -0.5));
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-b - b);
	} else {
		tmp_1 = 0.5 * t_2;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(-4.0 * Float64(a * c)))
	t_1 = Float64(Float64(c + c) / Float64(Float64(-b) - sqrt(Float64(b * b))))
	t_2 = sqrt(Float64(Float64(c / a) * -4.0))
	tmp_1 = 0.0
	if (b <= -2.05e-37)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = fma(-1.0, Float64(b / a), Float64(c / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= -2e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(a + a));
		end
		tmp_1 = tmp_3;
	elseif (b <= 1.86e+46)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0));
		else
			tmp_4 = fma(t_2, 0.5, Float64(Float64(b / a) * -0.5));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - b));
	else
		tmp_1 = Float64(0.5 * t_2);
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(c + c), $MachinePrecision] / N[((-b) - N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(c / a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2.05e-37], If[GreaterEqual[b, 0.0], t$95$1, N[(-1.0 * N[(b / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -2e-310], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(t$95$0 - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.86e+46], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * 0.5 + N[(N[(b / a), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - b), $MachinePrecision]), $MachinePrecision], N[(0.5 * t$95$2), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.0499999999999999e-37

   1. Initial program 67.8%

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

   3. Applied rewrites 67.9%

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

   4. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 67.9

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

   6. Applied rewrites 67.9%

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

   7. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 88.9

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

   9. Applied rewrites 88.9%

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

   10. Taylor expanded in a around inf

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

       1. lower-fma.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 89.3

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

   12. Applied rewrites 89.3%

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

   if -2.0499999999999999e-37 < b < -1.999999999999994e-310

   1. Initial program 82.9%

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

   3. Applied rewrites 82.9%

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

   4. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 82.9

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

   6. Applied rewrites 82.9%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-*.f64 66.6

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

   9. Applied rewrites 66.6%

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

   if -1.999999999999994e-310 < b < 1.8600000000000001e46

   1. Initial program 85.3%

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

   2. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 85.3

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

   4. Applied rewrites 85.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-*.f64 56.9

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

   7. Applied rewrites 56.9%

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

   if 1.8600000000000001e46 < b

   1. Initial program 60.6%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 93.5%

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

   5. Taylor expanded in b around -inf

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

      1. lower-*.f64 93.5

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

   7. Applied rewrites 93.5%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 93.5

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

   10. Applied rewrites 93.5%

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

Alternative 5: 78.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b}\\ t_1 := \frac{c + c}{\left(-b\right) - t\_0}\\ \mathbf{if}\;b \leq -2.05 \cdot 10^{-37}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}{a + a}\\ \end{array}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{+24}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -1} - 0.5 \cdot b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + t\_0}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* b b))) (t_1 (/ (+ c c) (- (- b) t_0))))
   (if (<= b -2.05e-37)
     (if (>= b 0.0) t_1 (fma -1.0 (/ b a) (/ c b)))
     (if (<= b -2e-310)
       (if (>= b 0.0) t_1 (/ (- (sqrt (* -4.0 (* a c))) b) (+ a a)))
       (if (<= b 3e+24)
         (if (>= b 0.0)
           (/ (- (sqrt (* (* a c) -1.0)) (* 0.5 b)) a)
           (/ (+ (- b) t_0) (* 2.0 a)))
         (if (>= b 0.0) (* -1.0 (/ c b)) (* -1.0 (/ b a))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((b * b));
	double t_1 = (c + c) / (-b - t_0);
	double tmp_1;
	if (b <= -2.05e-37) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = fma(-1.0, (b / a), (c / b));
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = (sqrt((-4.0 * (a * c))) - b) / (a + a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 3e+24) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (sqrt(((a * c) * -1.0)) - (0.5 * b)) / a;
		} else {
			tmp_4 = (-b + t_0) / (2.0 * a);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = -1.0 * (c / b);
	} else {
		tmp_1 = -1.0 * (b / a);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(b * b))
	t_1 = Float64(Float64(c + c) / Float64(Float64(-b) - t_0))
	tmp_1 = 0.0
	if (b <= -2.05e-37)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = fma(-1.0, Float64(b / a), Float64(c / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= -2e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = Float64(Float64(sqrt(Float64(-4.0 * Float64(a * c))) - b) / Float64(a + a));
		end
		tmp_1 = tmp_3;
	elseif (b <= 3e+24)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(sqrt(Float64(Float64(a * c) * -1.0)) - Float64(0.5 * b)) / a);
		else
			tmp_4 = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(-1.0 * Float64(c / b));
	else
		tmp_1 = Float64(-1.0 * Float64(b / a));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(c + c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.05e-37], If[GreaterEqual[b, 0.0], t$95$1, N[(-1.0 * N[(b / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -2e-310], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 3e+24], If[GreaterEqual[b, 0.0], N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision] - N[(0.5 * b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.0499999999999999e-37

   1. Initial program 67.8%

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

   3. Applied rewrites 67.9%

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

   4. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 67.9

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

   6. Applied rewrites 67.9%

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

   7. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 88.9

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

   9. Applied rewrites 88.9%

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

   10. Taylor expanded in a around inf

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

       1. lower-fma.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 89.3

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

   12. Applied rewrites 89.3%

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

   if -2.0499999999999999e-37 < b < -1.999999999999994e-310

   1. Initial program 82.9%

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

   3. Applied rewrites 82.9%

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

   4. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 82.9

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

   6. Applied rewrites 82.9%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-*.f64 66.6

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

   9. Applied rewrites 66.6%

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

   if -1.999999999999994e-310 < b < 2.99999999999999995e24

   1. Initial program 84.6%

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

   2. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. sqrt-unprod N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 55.1

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

   4. Applied rewrites 55.1%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 55.1

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

   7. Applied rewrites 55.1%

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

   9. Applied rewrites 55.1%

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

   if 2.99999999999999995e24 < b

   1. Initial program 62.7%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 62.8%

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

   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. lift-/.f64 62.8

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

   7. Applied rewrites 62.8%

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

   8. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 92.3

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

   10. Applied rewrites 92.3%

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

   11. Taylor expanded in b around -inf

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

       1. lower-*.f64 N/A

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

       2. lift-/.f64 92.3

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

   13. Applied rewrites 92.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 78.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b}\\ t_1 := \frac{c + c}{\left(-b\right) - t\_0}\\ \mathbf{if}\;b \leq -2.05 \cdot 10^{-37}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}{a + a}\\ \end{array}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{+24}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-\frac{-1 \cdot \sqrt{\left(a \cdot c\right) \cdot -1}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + t\_0}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* b b))) (t_1 (/ (+ c c) (- (- b) t_0))))
   (if (<= b -2.05e-37)
     (if (>= b 0.0) t_1 (fma -1.0 (/ b a) (/ c b)))
     (if (<= b -2e-310)
       (if (>= b 0.0) t_1 (/ (- (sqrt (* -4.0 (* a c))) b) (+ a a)))
       (if (<= b 3e+24)
         (if (>= b 0.0)
           (- (/ (* -1.0 (sqrt (* (* a c) -1.0))) a))
           (/ (+ (- b) t_0) (* 2.0 a)))
         (if (>= b 0.0) (* -1.0 (/ c b)) (* -1.0 (/ b a))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((b * b));
	double t_1 = (c + c) / (-b - t_0);
	double tmp_1;
	if (b <= -2.05e-37) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = fma(-1.0, (b / a), (c / b));
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = (sqrt((-4.0 * (a * c))) - b) / (a + a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 3e+24) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = -((-1.0 * sqrt(((a * c) * -1.0))) / a);
		} else {
			tmp_4 = (-b + t_0) / (2.0 * a);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = -1.0 * (c / b);
	} else {
		tmp_1 = -1.0 * (b / a);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(b * b))
	t_1 = Float64(Float64(c + c) / Float64(Float64(-b) - t_0))
	tmp_1 = 0.0
	if (b <= -2.05e-37)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = fma(-1.0, Float64(b / a), Float64(c / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= -2e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = Float64(Float64(sqrt(Float64(-4.0 * Float64(a * c))) - b) / Float64(a + a));
		end
		tmp_1 = tmp_3;
	elseif (b <= 3e+24)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(-Float64(Float64(-1.0 * sqrt(Float64(Float64(a * c) * -1.0))) / a));
		else
			tmp_4 = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(-1.0 * Float64(c / b));
	else
		tmp_1 = Float64(-1.0 * Float64(b / a));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(c + c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.05e-37], If[GreaterEqual[b, 0.0], t$95$1, N[(-1.0 * N[(b / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -2e-310], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 3e+24], If[GreaterEqual[b, 0.0], (-N[(N[(-1.0 * N[Sqrt[N[(N[(a * c), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.0499999999999999e-37

   1. Initial program 67.8%

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

   3. Applied rewrites 67.9%

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

   4. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 67.9

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

   6. Applied rewrites 67.9%

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

   7. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 88.9

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

   9. Applied rewrites 88.9%

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

   10. Taylor expanded in a around inf

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

       1. lower-fma.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 89.3

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

   12. Applied rewrites 89.3%

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

   if -2.0499999999999999e-37 < b < -1.999999999999994e-310

   1. Initial program 82.9%

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

   3. Applied rewrites 82.9%

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

   4. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 82.9

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

   6. Applied rewrites 82.9%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-*.f64 66.6

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

   9. Applied rewrites 66.6%

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

   if -1.999999999999994e-310 < b < 2.99999999999999995e24

   1. Initial program 84.6%

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

   2. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. sqrt-unprod N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 55.1

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

   4. Applied rewrites 55.1%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 55.1

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

   7. Applied rewrites 55.1%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-prod N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sqrt.f64 55.6

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

   10. Applied rewrites 55.6%

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

   if 2.99999999999999995e24 < b

   1. Initial program 62.7%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 62.8%

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

   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. lift-/.f64 62.8

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

   7. Applied rewrites 62.8%

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

   8. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 92.3

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

   10. Applied rewrites 92.3%

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

   11. Taylor expanded in b around -inf

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

       1. lower-*.f64 N/A

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

       2. lift-/.f64 92.3

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

   13. Applied rewrites 92.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 78.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b}\\ \mathbf{if}\;b \leq -2.05 \cdot 10^{-37}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{+24}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-\frac{-1 \cdot \sqrt{\left(a \cdot c\right) \cdot -1}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + t\_0}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* b b))))
   (if (<= b -2.05e-37)
     (if (>= b 0.0) (/ (+ c c) (- (- b) t_0)) (fma -1.0 (/ b a) (/ c b)))
     (if (<= b -2e-310)
       (if (>= b 0.0)
         (/ (* 2.0 c) (- (- b) b))
         (/ (sqrt (* (* a c) -4.0)) (* 2.0 a)))
       (if (<= b 3e+24)
         (if (>= b 0.0)
           (- (/ (* -1.0 (sqrt (* (* a c) -1.0))) a))
           (/ (+ (- b) t_0) (* 2.0 a)))
         (if (>= b 0.0) (* -1.0 (/ c b)) (* -1.0 (/ b a))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((b * b));
	double tmp_1;
	if (b <= -2.05e-37) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c + c) / (-b - t_0);
		} else {
			tmp_2 = fma(-1.0, (b / a), (c / b));
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * c) / (-b - b);
		} else {
			tmp_3 = sqrt(((a * c) * -4.0)) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 3e+24) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = -((-1.0 * sqrt(((a * c) * -1.0))) / a);
		} else {
			tmp_4 = (-b + t_0) / (2.0 * a);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = -1.0 * (c / b);
	} else {
		tmp_1 = -1.0 * (b / a);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(b * b))
	tmp_1 = 0.0
	if (b <= -2.05e-37)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(c + c) / Float64(Float64(-b) - t_0));
		else
			tmp_2 = fma(-1.0, Float64(b / a), Float64(c / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= -2e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - b));
		else
			tmp_3 = Float64(sqrt(Float64(Float64(a * c) * -4.0)) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (b <= 3e+24)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(-Float64(Float64(-1.0 * sqrt(Float64(Float64(a * c) * -1.0))) / a));
		else
			tmp_4 = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(-1.0 * Float64(c / b));
	else
		tmp_1 = Float64(-1.0 * Float64(b / a));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2.05e-37], If[GreaterEqual[b, 0.0], N[(N[(c + c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(b / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -2e-310], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - b), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 3e+24], If[GreaterEqual[b, 0.0], (-N[(N[(-1.0 * N[Sqrt[N[(N[(a * c), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.0499999999999999e-37

   1. Initial program 67.8%

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

   3. Applied rewrites 67.9%

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

   4. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 67.9

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

   6. Applied rewrites 67.9%

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

   7. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 88.9

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

   9. Applied rewrites 88.9%

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

   10. Taylor expanded in a around inf

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

       1. lower-fma.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 89.3

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

   12. Applied rewrites 89.3%

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

   if -2.0499999999999999e-37 < b < -1.999999999999994e-310

   1. Initial program 82.9%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 82.9%

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

   5. Taylor expanded in a around inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-*.f64 63.6

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

   7. Applied rewrites 63.6%

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

   if -1.999999999999994e-310 < b < 2.99999999999999995e24

   1. Initial program 84.6%

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

   2. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. sqrt-unprod N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 55.1

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

   4. Applied rewrites 55.1%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 55.1

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

   7. Applied rewrites 55.1%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-prod N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sqrt.f64 55.6

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

   10. Applied rewrites 55.6%

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

   if 2.99999999999999995e24 < b

   1. Initial program 62.7%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 62.8%

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

   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. lift-/.f64 62.8

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

   7. Applied rewrites 62.8%

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

   8. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 92.3

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

   10. Applied rewrites 92.3%

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

   11. Taylor expanded in b around -inf

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

       1. lower-*.f64 N/A

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

       2. lift-/.f64 92.3

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

   13. Applied rewrites 92.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 78.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left(a \cdot c\right) \cdot -4}\\ \mathbf{if}\;b \leq -2.05 \cdot 10^{-37}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{\left(-b\right) - \sqrt{b \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{+24}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{t\_0} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{b}{a}\right) \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* (* a c) -4.0))))
   (if (<= b -2.05e-37)
     (if (>= b 0.0)
       (/ (+ c c) (- (- b) (sqrt (* b b))))
       (fma -1.0 (/ b a) (/ c b)))
     (if (<= b -2e-310)
       (if (>= b 0.0) (/ (* 2.0 c) (- (- b) b)) (/ t_0 (* 2.0 a)))
       (if (<= b 3e+24)
         (if (>= b 0.0) (* (/ c t_0) -2.0) (* (* -2.0 (/ b a)) 0.5))
         (if (>= b 0.0) (* -1.0 (/ c b)) (* -1.0 (/ b a))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((a * c) * -4.0));
	double tmp_1;
	if (b <= -2.05e-37) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c + c) / (-b - sqrt((b * b)));
		} else {
			tmp_2 = fma(-1.0, (b / a), (c / b));
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * c) / (-b - b);
		} else {
			tmp_3 = t_0 / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 3e+24) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c / t_0) * -2.0;
		} else {
			tmp_4 = (-2.0 * (b / a)) * 0.5;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = -1.0 * (c / b);
	} else {
		tmp_1 = -1.0 * (b / a);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(a * c) * -4.0))
	tmp_1 = 0.0
	if (b <= -2.05e-37)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(c + c) / Float64(Float64(-b) - sqrt(Float64(b * b))));
		else
			tmp_2 = fma(-1.0, Float64(b / a), Float64(c / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= -2e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - b));
		else
			tmp_3 = Float64(t_0 / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (b <= 3e+24)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(c / t_0) * -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(-1.0 * Float64(c / b));
	else
		tmp_1 = Float64(-1.0 * Float64(b / a));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2.05e-37], If[GreaterEqual[b, 0.0], N[(N[(c + c), $MachinePrecision] / N[((-b) - N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(b / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -2e-310], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - b), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 3e+24], If[GreaterEqual[b, 0.0], N[(N[(c / t$95$0), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(-2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.0499999999999999e-37

   1. Initial program 67.8%

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

   3. Applied rewrites 67.9%

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

   4. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 67.9

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

   6. Applied rewrites 67.9%

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

   7. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 88.9

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

   9. Applied rewrites 88.9%

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

   10. Taylor expanded in a around inf

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

       1. lower-fma.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 89.3

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

   12. Applied rewrites 89.3%

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

   if -2.0499999999999999e-37 < b < -1.999999999999994e-310

   1. Initial program 82.9%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 82.9%

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

   5. Taylor expanded in a around inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-*.f64 63.6

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

   7. Applied rewrites 63.6%

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

   if -1.999999999999994e-310 < b < 2.99999999999999995e24

   1. Initial program 84.6%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 84.6%

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

   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. lift-/.f64 84.6

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

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in a around inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-*.f64 55.1

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

   10. Applied rewrites 55.1%

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

   if 2.99999999999999995e24 < b

   1. Initial program 62.7%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 62.8%

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

   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. lift-/.f64 62.8

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

   7. Applied rewrites 62.8%

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

   8. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 92.3

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

   10. Applied rewrites 92.3%

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

   11. Taylor expanded in b around -inf

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

       1. lower-*.f64 N/A

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

       2. lift-/.f64 92.3

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

   13. Applied rewrites 92.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 78.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\\ t_1 := \sqrt{\left(a \cdot c\right) \cdot -4}\\ \mathbf{if}\;b \leq -2.05 \cdot 10^{-37}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{+24}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{t\_1} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{b}{a}\right) \cdot 0.5\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (if (>= b 0.0) (* -1.0 (/ c b)) (* -1.0 (/ b a))))
        (t_1 (sqrt (* (* a c) -4.0))))
   (if (<= b -2.05e-37)
     t_0
     (if (<= b -2e-310)
       (if (>= b 0.0) (/ (* 2.0 c) (- (- b) b)) (/ t_1 (* 2.0 a)))
       (if (<= b 3e+24)
         (if (>= b 0.0) (* (/ c t_1) -2.0) (* (* -2.0 (/ b a)) 0.5))
         t_0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -1.0 * (c / b);
	} else {
		tmp = -1.0 * (b / a);
	}
	double t_0 = tmp;
	double t_1 = sqrt(((a * c) * -4.0));
	double tmp_1;
	if (b <= -2.05e-37) {
		tmp_1 = t_0;
	} else if (b <= -2e-310) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (-b - b);
		} else {
			tmp_2 = t_1 / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 3e+24) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c / t_1) * -2.0;
		} else {
			tmp_3 = (-2.0 * (b / a)) * 0.5;
		}
		tmp_1 = tmp_3;
	} 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
    if (b >= 0.0d0) then
        tmp = (-1.0d0) * (c / b)
    else
        tmp = (-1.0d0) * (b / a)
    end if
    t_0 = tmp
    t_1 = sqrt(((a * c) * (-4.0d0)))
    if (b <= (-2.05d-37)) then
        tmp_1 = t_0
    else if (b <= (-2d-310)) then
        if (b >= 0.0d0) then
            tmp_2 = (2.0d0 * c) / (-b - b)
        else
            tmp_2 = t_1 / (2.0d0 * a)
        end if
        tmp_1 = tmp_2
    else if (b <= 3d+24) then
        if (b >= 0.0d0) then
            tmp_3 = (c / t_1) * (-2.0d0)
        else
            tmp_3 = ((-2.0d0) * (b / a)) * 0.5d0
        end if
        tmp_1 = tmp_3
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -1.0 * (c / b);
	} else {
		tmp = -1.0 * (b / a);
	}
	double t_0 = tmp;
	double t_1 = Math.sqrt(((a * c) * -4.0));
	double tmp_1;
	if (b <= -2.05e-37) {
		tmp_1 = t_0;
	} else if (b <= -2e-310) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (-b - b);
		} else {
			tmp_2 = t_1 / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 3e+24) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c / t_1) * -2.0;
		} else {
			tmp_3 = (-2.0 * (b / a)) * 0.5;
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = -1.0 * (c / b)
	else:
		tmp = -1.0 * (b / a)
	t_0 = tmp
	t_1 = math.sqrt(((a * c) * -4.0))
	tmp_1 = 0
	if b <= -2.05e-37:
		tmp_1 = t_0
	elif b <= -2e-310:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (2.0 * c) / (-b - b)
		else:
			tmp_2 = t_1 / (2.0 * a)
		tmp_1 = tmp_2
	elif b <= 3e+24:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (c / t_1) * -2.0
		else:
			tmp_3 = (-2.0 * (b / a)) * 0.5
		tmp_1 = tmp_3
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(-1.0 * Float64(c / b));
	else
		tmp = Float64(-1.0 * Float64(b / a));
	end
	t_0 = tmp
	t_1 = sqrt(Float64(Float64(a * c) * -4.0))
	tmp_1 = 0.0
	if (b <= -2.05e-37)
		tmp_1 = t_0;
	elseif (b <= -2e-310)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - b));
		else
			tmp_2 = Float64(t_1 / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 3e+24)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(c / t_1) * -2.0);
		else
			tmp_3 = Float64(Float64(-2.0 * Float64(b / a)) * 0.5);
		end
		tmp_1 = tmp_3;
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = -1.0 * (c / b);
	else
		tmp = -1.0 * (b / a);
	end
	t_0 = tmp;
	t_1 = sqrt(((a * c) * -4.0));
	tmp_2 = 0.0;
	if (b <= -2.05e-37)
		tmp_2 = t_0;
	elseif (b <= -2e-310)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (2.0 * c) / (-b - b);
		else
			tmp_3 = t_1 / (2.0 * a);
		end
		tmp_2 = tmp_3;
	elseif (b <= 3e+24)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (c / t_1) * -2.0;
		else
			tmp_4 = (-2.0 * (b / a)) * 0.5;
		end
		tmp_2 = tmp_4;
	else
		tmp_2 = t_0;
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = If[GreaterEqual[b, 0.0], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]]}, Block[{t$95$1 = N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2.05e-37], t$95$0, If[LessEqual[b, -2e-310], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - b), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 3e+24], If[GreaterEqual[b, 0.0], N[(N[(c / t$95$1), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(-2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.0499999999999999e-37 or 2.99999999999999995e24 < b

   1. Initial program 65.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 65.6%

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

   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. lift-/.f64 77.0

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

   7. Applied rewrites 77.0%

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

   8. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 90.4

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

   10. Applied rewrites 90.4%

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

   11. Taylor expanded in b around -inf

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

       1. lower-*.f64 N/A

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

       2. lift-/.f64 90.4

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

   13. Applied rewrites 90.4%

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

   if -2.0499999999999999e-37 < b < -1.999999999999994e-310

   1. Initial program 82.9%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 82.9%

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

   5. Taylor expanded in a around inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-*.f64 63.6

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

   7. Applied rewrites 63.6%

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

   if -1.999999999999994e-310 < b < 2.99999999999999995e24

   1. Initial program 84.6%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 84.6%

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

   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. lift-/.f64 84.6

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

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in a around inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-*.f64 55.1

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

   10. Applied rewrites 55.1%

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

Alternative 10: 74.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\\ \mathbf{if}\;b \leq -3.3 \cdot 10^{-156}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \sqrt{\frac{c}{a} \cdot -1}\\ \end{array}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{+24}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{\sqrt{\left(a \cdot c\right) \cdot -4}} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{b}{a}\right) \cdot 0.5\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (if (>= b 0.0) (* -1.0 (/ c b)) (* -1.0 (/ b a)))))
   (if (<= b -3.3e-156)
     t_0
     (if (<= b -2e-310)
       (if (>= b 0.0)
         (/ (* 2.0 c) (- (- b) b))
         (* -1.0 (sqrt (* (/ c a) -1.0))))
       (if (<= b 3e+24)
         (if (>= b 0.0)
           (* (/ c (sqrt (* (* a c) -4.0))) -2.0)
           (* (* -2.0 (/ b a)) 0.5))
         t_0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -1.0 * (c / b);
	} else {
		tmp = -1.0 * (b / a);
	}
	double t_0 = tmp;
	double tmp_1;
	if (b <= -3.3e-156) {
		tmp_1 = t_0;
	} else if (b <= -2e-310) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (-b - b);
		} else {
			tmp_2 = -1.0 * sqrt(((c / a) * -1.0));
		}
		tmp_1 = tmp_2;
	} else if (b <= 3e+24) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c / sqrt(((a * c) * -4.0))) * -2.0;
		} else {
			tmp_3 = (-2.0 * (b / a)) * 0.5;
		}
		tmp_1 = tmp_3;
	} 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
    if (b >= 0.0d0) then
        tmp = (-1.0d0) * (c / b)
    else
        tmp = (-1.0d0) * (b / a)
    end if
    t_0 = tmp
    if (b <= (-3.3d-156)) then
        tmp_1 = t_0
    else if (b <= (-2d-310)) then
        if (b >= 0.0d0) then
            tmp_2 = (2.0d0 * c) / (-b - b)
        else
            tmp_2 = (-1.0d0) * sqrt(((c / a) * (-1.0d0)))
        end if
        tmp_1 = tmp_2
    else if (b <= 3d+24) then
        if (b >= 0.0d0) then
            tmp_3 = (c / sqrt(((a * c) * (-4.0d0)))) * (-2.0d0)
        else
            tmp_3 = ((-2.0d0) * (b / a)) * 0.5d0
        end if
        tmp_1 = tmp_3
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -1.0 * (c / b);
	} else {
		tmp = -1.0 * (b / a);
	}
	double t_0 = tmp;
	double tmp_1;
	if (b <= -3.3e-156) {
		tmp_1 = t_0;
	} else if (b <= -2e-310) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (-b - b);
		} else {
			tmp_2 = -1.0 * Math.sqrt(((c / a) * -1.0));
		}
		tmp_1 = tmp_2;
	} else if (b <= 3e+24) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c / Math.sqrt(((a * c) * -4.0))) * -2.0;
		} else {
			tmp_3 = (-2.0 * (b / a)) * 0.5;
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = -1.0 * (c / b)
	else:
		tmp = -1.0 * (b / a)
	t_0 = tmp
	tmp_1 = 0
	if b <= -3.3e-156:
		tmp_1 = t_0
	elif b <= -2e-310:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (2.0 * c) / (-b - b)
		else:
			tmp_2 = -1.0 * math.sqrt(((c / a) * -1.0))
		tmp_1 = tmp_2
	elif b <= 3e+24:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (c / math.sqrt(((a * c) * -4.0))) * -2.0
		else:
			tmp_3 = (-2.0 * (b / a)) * 0.5
		tmp_1 = tmp_3
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(-1.0 * Float64(c / b));
	else
		tmp = Float64(-1.0 * Float64(b / a));
	end
	t_0 = tmp
	tmp_1 = 0.0
	if (b <= -3.3e-156)
		tmp_1 = t_0;
	elseif (b <= -2e-310)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - b));
		else
			tmp_2 = Float64(-1.0 * sqrt(Float64(Float64(c / a) * -1.0)));
		end
		tmp_1 = tmp_2;
	elseif (b <= 3e+24)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(c / sqrt(Float64(Float64(a * c) * -4.0))) * -2.0);
		else
			tmp_3 = Float64(Float64(-2.0 * Float64(b / a)) * 0.5);
		end
		tmp_1 = tmp_3;
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = -1.0 * (c / b);
	else
		tmp = -1.0 * (b / a);
	end
	t_0 = tmp;
	tmp_2 = 0.0;
	if (b <= -3.3e-156)
		tmp_2 = t_0;
	elseif (b <= -2e-310)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (2.0 * c) / (-b - b);
		else
			tmp_3 = -1.0 * sqrt(((c / a) * -1.0));
		end
		tmp_2 = tmp_3;
	elseif (b <= 3e+24)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (c / sqrt(((a * c) * -4.0))) * -2.0;
		else
			tmp_4 = (-2.0 * (b / a)) * 0.5;
		end
		tmp_2 = tmp_4;
	else
		tmp_2 = t_0;
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = If[GreaterEqual[b, 0.0], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]]}, If[LessEqual[b, -3.3e-156], t$95$0, If[LessEqual[b, -2e-310], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - b), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[Sqrt[N[(N[(c / a), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 3e+24], If[GreaterEqual[b, 0.0], N[(N[(c / N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(-2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.2999999999999999e-156 or 2.99999999999999995e24 < b

   1. Initial program 68.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 68.2%

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

   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. lift-/.f64 72.7

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

   7. Applied rewrites 72.7%

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

   8. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 84.6

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

   10. Applied rewrites 84.6%

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

   11. Taylor expanded in b around -inf

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

       1. lower-*.f64 N/A

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

       2. lift-/.f64 84.6

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

   13. Applied rewrites 84.6%

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

   if -3.2999999999999999e-156 < b < -1.999999999999994e-310

   1. Initial program 77.7%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 77.7%

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

   5. Taylor expanded in b around -inf

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

      1. lower-*.f64 11.2

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

   7. Applied rewrites 11.2%

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

   8. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 35.6

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

   10. Applied rewrites 35.6%

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

   if -1.999999999999994e-310 < b < 2.99999999999999995e24

   1. Initial program 84.6%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 84.6%

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

   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. lift-/.f64 84.6

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

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in a around inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-*.f64 55.1

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

   10. Applied rewrites 55.1%

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

Alternative 11: 69.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{-156}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \sqrt{\frac{c}{a} \cdot -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.3e-156)
   (if (>= b 0.0) (* -1.0 (/ c b)) (* -1.0 (/ b a)))
   (if (>= b 0.0) (/ (* 2.0 c) (- (- b) b)) (* -1.0 (sqrt (* (/ c a) -1.0))))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -3.3e-156) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -1.0 * (c / b);
		} else {
			tmp_2 = -1.0 * (b / a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-b - b);
	} else {
		tmp_1 = -1.0 * sqrt(((c / a) * -1.0));
	}
	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) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    if (b <= (-3.3d-156)) then
        if (b >= 0.0d0) then
            tmp_2 = (-1.0d0) * (c / b)
        else
            tmp_2 = (-1.0d0) * (b / a)
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (2.0d0 * c) / (-b - b)
    else
        tmp_1 = (-1.0d0) * sqrt(((c / a) * (-1.0d0)))
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -3.3e-156) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -1.0 * (c / b);
		} else {
			tmp_2 = -1.0 * (b / a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-b - b);
	} else {
		tmp_1 = -1.0 * Math.sqrt(((c / a) * -1.0));
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if b <= -3.3e-156:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -1.0 * (c / b)
		else:
			tmp_2 = -1.0 * (b / a)
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = (2.0 * c) / (-b - b)
	else:
		tmp_1 = -1.0 * math.sqrt(((c / a) * -1.0))
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -3.3e-156)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-1.0 * Float64(c / b));
		else
			tmp_2 = Float64(-1.0 * Float64(b / a));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - b));
	else
		tmp_1 = Float64(-1.0 * sqrt(Float64(Float64(c / a) * -1.0)));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp_2 = 0.0;
	if (b <= -3.3e-156)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -1.0 * (c / b);
		else
			tmp_3 = -1.0 * (b / a);
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = (2.0 * c) / (-b - b);
	else
		tmp_2 = -1.0 * sqrt(((c / a) * -1.0));
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.3e-156], If[GreaterEqual[b, 0.0], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - b), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[Sqrt[N[(N[(c / a), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.2999999999999999e-156

   1. Initial program 71.8%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 71.9%

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

   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. lift-/.f64 79.4

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

   7. Applied rewrites 79.4%

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

   8. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 79.4

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

   10. Applied rewrites 79.4%

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

   11. Taylor expanded in b around -inf

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

       1. lower-*.f64 N/A

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

       2. lift-/.f64 79.4

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

   13. Applied rewrites 79.4%

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

   if -3.2999999999999999e-156 < b

   1. Initial program 73.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 68.5%

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

   5. Taylor expanded in b around -inf

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

      1. lower-*.f64 58.8

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

   7. Applied rewrites 58.8%

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

   8. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 62.4

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

   10. Applied rewrites 62.4%

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

Alternative 12: 69.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-136}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \sqrt{\frac{c}{a} \cdot -1}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{b}{a}\right) \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5e-136)
   (if (>= b 0.0) (* -1.0 (sqrt (* (/ c a) -1.0))) (* (* -2.0 (/ b a)) 0.5))
   (if (>= b 0.0) (* -1.0 (/ c b)) (* -1.0 (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= 5e-136) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -1.0 * sqrt(((c / a) * -1.0));
		} else {
			tmp_2 = (-2.0 * (b / a)) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = -1.0 * (c / b);
	} else {
		tmp_1 = -1.0 * (b / a);
	}
	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) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    if (b <= 5d-136) then
        if (b >= 0.0d0) then
            tmp_2 = (-1.0d0) * sqrt(((c / a) * (-1.0d0)))
        else
            tmp_2 = ((-2.0d0) * (b / a)) * 0.5d0
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (-1.0d0) * (c / b)
    else
        tmp_1 = (-1.0d0) * (b / a)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if (b <= 5e-136) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -1.0 * Math.sqrt(((c / a) * -1.0));
		} else {
			tmp_2 = (-2.0 * (b / a)) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = -1.0 * (c / b);
	} else {
		tmp_1 = -1.0 * (b / a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if b <= 5e-136:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -1.0 * math.sqrt(((c / a) * -1.0))
		else:
			tmp_2 = (-2.0 * (b / a)) * 0.5
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = -1.0 * (c / b)
	else:
		tmp_1 = -1.0 * (b / a)
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= 5e-136)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-1.0 * sqrt(Float64(Float64(c / a) * -1.0)));
		else
			tmp_2 = Float64(Float64(-2.0 * Float64(b / a)) * 0.5);
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(-1.0 * Float64(c / b));
	else
		tmp_1 = Float64(-1.0 * Float64(b / a));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp_2 = 0.0;
	if (b <= 5e-136)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -1.0 * sqrt(((c / a) * -1.0));
		else
			tmp_3 = (-2.0 * (b / a)) * 0.5;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = -1.0 * (c / b);
	else
		tmp_2 = -1.0 * (b / a);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5e-136], If[GreaterEqual[b, 0.0], N[(-1.0 * N[Sqrt[N[(N[(c / a), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 5.0000000000000002e-136

   1. Initial program 73.6%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 73.6%

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

   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. lift-/.f64 69.4

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

   7. Applied rewrites 69.4%

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

   8. Taylor expanded in a around -inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 62.4

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

   10. Applied rewrites 62.4%

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

   11. Taylor expanded in c around -inf

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

       1. lower-*.f64 N/A

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

       2. sqrt-prod N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-sqrt.f64 62.5

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

   13. Applied rewrites 62.5%

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

   if 5.0000000000000002e-136 < b

   1. Initial program 71.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 71.2%

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

   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. lift-/.f64 71.2

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

   7. Applied rewrites 71.2%

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

   8. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 80.4

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

   10. Applied rewrites 80.4%

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

   11. Taylor expanded in b around -inf

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

       1. lower-*.f64 N/A

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

       2. lift-/.f64 80.4

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

   13. Applied rewrites 80.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 67.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.4 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{\frac{c}{a} \cdot -1}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{b}{a}\right) \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= c -4.4e+154)
   (if (>= b 0.0) (sqrt (* (/ c a) -1.0)) (* (* -2.0 (/ b a)) 0.5))
   (if (>= b 0.0) (* -1.0 (/ c b)) (* -1.0 (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (c <= -4.4e+154) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = sqrt(((c / a) * -1.0));
		} else {
			tmp_2 = (-2.0 * (b / a)) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = -1.0 * (c / b);
	} else {
		tmp_1 = -1.0 * (b / a);
	}
	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) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    if (c <= (-4.4d+154)) then
        if (b >= 0.0d0) then
            tmp_2 = sqrt(((c / a) * (-1.0d0)))
        else
            tmp_2 = ((-2.0d0) * (b / a)) * 0.5d0
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (-1.0d0) * (c / b)
    else
        tmp_1 = (-1.0d0) * (b / a)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if (c <= -4.4e+154) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = Math.sqrt(((c / a) * -1.0));
		} else {
			tmp_2 = (-2.0 * (b / a)) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = -1.0 * (c / b);
	} else {
		tmp_1 = -1.0 * (b / a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if c <= -4.4e+154:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = math.sqrt(((c / a) * -1.0))
		else:
			tmp_2 = (-2.0 * (b / a)) * 0.5
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = -1.0 * (c / b)
	else:
		tmp_1 = -1.0 * (b / a)
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (c <= -4.4e+154)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = sqrt(Float64(Float64(c / a) * -1.0));
		else
			tmp_2 = Float64(Float64(-2.0 * Float64(b / a)) * 0.5);
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(-1.0 * Float64(c / b));
	else
		tmp_1 = Float64(-1.0 * Float64(b / a));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp_2 = 0.0;
	if (c <= -4.4e+154)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = sqrt(((c / a) * -1.0));
		else
			tmp_3 = (-2.0 * (b / a)) * 0.5;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = -1.0 * (c / b);
	else
		tmp_2 = -1.0 * (b / a);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[c, -4.4e+154], If[GreaterEqual[b, 0.0], N[Sqrt[N[(N[(c / a), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision], N[(N[(-2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -4.4000000000000002e154

   1. Initial program 51.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 51.5%

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

   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. lift-/.f64 46.6

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

   7. Applied rewrites 46.6%

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

   8. Taylor expanded in a around -inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 44.1

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

   10. Applied rewrites 44.1%

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

   if -4.4000000000000002e154 < c

   1. Initial program 75.3%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 75.3%

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

   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. lift-/.f64 73.1

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

   7. Applied rewrites 73.1%

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

   8. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 69.8

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

   10. Applied rewrites 69.8%

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

   11. Taylor expanded in b around -inf

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

       1. lower-*.f64 N/A

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

       2. lift-/.f64 69.8

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

   13. Applied rewrites 69.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 66.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -1.0 * (c / b);
	} else {
		tmp = -1.0 * (b / 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) :: tmp
    if (b >= 0.0d0) then
        tmp = (-1.0d0) * (c / b)
    else
        tmp = (-1.0d0) * (b / a)
    end if
    code = tmp
end function

Java

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

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = -1.0 * (c / b)
	else:
		tmp = -1.0 * (b / a)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = -1.0 * (c / b);
	else
		tmp = -1.0 * (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Initial program 72.6%

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

2. Taylor expanded in a around 0

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

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. lift-/.f64 70.1

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

7. Applied rewrites 70.1%

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

8. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 67.4

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

10. Applied rewrites 67.4%

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

11. Taylor expanded in b around -inf

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

    1. lower-*.f64 N/A

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

    2. lift-/.f64 67.4

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

13. Applied rewrites 67.4%

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

Reproduce

herbie shell --seed 2025115 
(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))))