jeff quadratic root 1

Percentage Accurate: 72.9% → 91.3%

Time: 5.5s

Alternatives: 17

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-b\right) + \left(-b\right)\\ \mathbf{if}\;b \leq -1 \cdot 10^{+148}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-\left(\frac{b}{2 \cdot a} + \frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{2 \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ (- b) (- b))))
   (if (<= b -1e+148)
     (if (>= b 0.0) (/ (+ b b) (* 2.0 (- a))) (/ (+ c c) t_0))
     (if (<= b 1.1e+99)
       (if (>= b 0.0)
         (-
          (+ (/ b (* 2.0 a)) (/ (sqrt (fma (* -4.0 a) c (* b b))) (* 2.0 a))))
         (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))))
       (if (>= b 0.0) (fma -1.0 (/ b a) (/ c b)) (/ (* 2.0 c) t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = -b + -b;
	double tmp_1;
	if (b <= -1e+148) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (b + b) / (2.0 * -a);
		} else {
			tmp_2 = (c + c) / t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.1e+99) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -((b / (2.0 * a)) + (sqrt(fma((-4.0 * a), c, (b * b))) / (2.0 * a)));
		} else {
			tmp_3 = (2.0 * c) / (-b + sqrt(((b * b) - ((4.0 * a) * c))));
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = fma(-1.0, (b / a), (c / b));
	} else {
		tmp_1 = (2.0 * c) / t_0;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-b) + Float64(-b))
	tmp_1 = 0.0
	if (b <= -1e+148)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(b + b) / Float64(2.0 * Float64(-a)));
		else
			tmp_2 = Float64(Float64(c + c) / t_0);
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.1e+99)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(-Float64(Float64(b / Float64(2.0 * a)) + Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) / Float64(2.0 * a))));
		else
			tmp_3 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = fma(-1.0, Float64(b / a), Float64(c / b));
	else
		tmp_1 = Float64(Float64(2.0 * c) / t_0);
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) + (-b)), $MachinePrecision]}, If[LessEqual[b, -1e+148], If[GreaterEqual[b, 0.0], N[(N[(b + b), $MachinePrecision] / N[(2.0 * (-a)), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / t$95$0), $MachinePrecision]], If[LessEqual[b, 1.1e+99], If[GreaterEqual[b, 0.0], (-N[(N[(b / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(-1.0 * N[(b / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1e148

   1. Initial program 37.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 95.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 95.6

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

   10. Applied rewrites 95.6%

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

   if -1e148 < b < 1.09999999999999989e99

   1. Initial program 86.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. div-sub N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-neg.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

   4. Applied rewrites 86.2%

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

   if 1.09999999999999989e99 < b

   1. Initial program 54.6%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 54.6

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

   5. Applied rewrites 54.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 94.4%

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

   9. Taylor expanded in c around 0

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

       1. lower-fma.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 95.2

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

   11. Applied rewrites 95.2%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+148}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\left(-b\right) + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-\left(\frac{b}{2 \cdot a} + \frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{2 \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ t_1 := \left(-b\right) + \left(-b\right)\\ \mathbf{if}\;b \leq -1 \cdot 10^{+148}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_1}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{\frac{c}{a} \cdot -4} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{t\_0 + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}}{b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 a) c (* b b)))) (t_1 (+ (- b) (- b))))
   (if (<= b -1e+148)
     (if (>= b 0.0) (/ (+ b b) (* 2.0 (- a))) (/ (+ c c) t_1))
     (if (<= b -5e-310)
       (if (>= b 0.0) (* (sqrt (* (/ c a) -4.0)) -0.5) (/ (+ c c) (- t_0 b)))
       (if (<= b 1.1e+99)
         (if (>= b 0.0)
           (* (/ (+ t_0 b) a) -0.5)
           (/ (- (- c) (/ (* a (* c c)) (* b b))) b))
         (if (>= b 0.0) (fma -1.0 (/ b a) (/ c b)) (/ (* 2.0 c) t_1)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-4.0 * a), c, (b * b)));
	double t_1 = -b + -b;
	double tmp_1;
	if (b <= -1e+148) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (b + b) / (2.0 * -a);
		} else {
			tmp_2 = (c + c) / t_1;
		}
		tmp_1 = tmp_2;
	} else if (b <= -5e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = sqrt(((c / a) * -4.0)) * -0.5;
		} else {
			tmp_3 = (c + c) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.1e+99) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = ((t_0 + b) / a) * -0.5;
		} else {
			tmp_4 = (-c - ((a * (c * c)) / (b * b))) / b;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = fma(-1.0, (b / a), (c / b));
	} else {
		tmp_1 = (2.0 * c) / t_1;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))
	t_1 = Float64(Float64(-b) + Float64(-b))
	tmp_1 = 0.0
	if (b <= -1e+148)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(b + b) / Float64(2.0 * Float64(-a)));
		else
			tmp_2 = Float64(Float64(c + c) / t_1);
		end
		tmp_1 = tmp_2;
	elseif (b <= -5e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(sqrt(Float64(Float64(c / a) * -4.0)) * -0.5);
		else
			tmp_3 = Float64(Float64(c + c) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	elseif (b <= 1.1e+99)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(Float64(t_0 + b) / a) * -0.5);
		else
			tmp_4 = Float64(Float64(Float64(-c) - Float64(Float64(a * Float64(c * c)) / Float64(b * b))) / b);
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = fma(-1.0, Float64(b / a), Float64(c / b));
	else
		tmp_1 = Float64(Float64(2.0 * c) / t_1);
	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]}, Block[{t$95$1 = N[((-b) + (-b)), $MachinePrecision]}, If[LessEqual[b, -1e+148], If[GreaterEqual[b, 0.0], N[(N[(b + b), $MachinePrecision] / N[(2.0 * (-a)), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / t$95$1), $MachinePrecision]], If[LessEqual[b, -5e-310], If[GreaterEqual[b, 0.0], N[(N[Sqrt[N[(N[(c / a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.1e+99], If[GreaterEqual[b, 0.0], N[(N[(N[(t$95$0 + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[((-c) - N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(-1.0 * N[(b / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1e148

   1. Initial program 37.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 95.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 95.6

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

   10. Applied rewrites 95.6%

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

   if -1e148 < b < -4.999999999999985e-310

   1. Initial program 93.7%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in a around inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 93.7

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

   8. Applied rewrites 93.7%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 93.7

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

   10. Applied rewrites 93.7%

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

   if -4.999999999999985e-310 < b < 1.09999999999999989e99

   1. Initial program 78.8%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 78.8%

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

   6. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 78.8

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

   8. Applied rewrites 78.8%

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

   if 1.09999999999999989e99 < b

   1. Initial program 54.6%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 54.6

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

   5. Applied rewrites 54.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 94.4%

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

   9. Taylor expanded in c around 0

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

       1. lower-fma.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 95.2

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

   11. Applied rewrites 95.2%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+148}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\left(-b\right) + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{\frac{c}{a} \cdot -4} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}}{b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-b\right) + \left(-b\right)\\ t_1 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ t_2 := 2 \cdot \left(-a\right)\\ \mathbf{if}\;b \leq -1 \cdot 10^{+148}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + t\_1}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t\_1}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ (- b) (- b)))
        (t_1 (sqrt (- (* b b) (* (* 4.0 a) c))))
        (t_2 (* 2.0 (- a))))
   (if (<= b -1e+148)
     (if (>= b 0.0) (/ (+ b b) t_2) (/ (+ c c) t_0))
     (if (<= b 1.1e+99)
       (if (>= b 0.0) (/ (+ b t_1) t_2) (/ (* 2.0 c) (+ (- b) t_1)))
       (if (>= b 0.0) (fma -1.0 (/ b a) (/ c b)) (/ (* 2.0 c) t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = -b + -b;
	double t_1 = sqrt(((b * b) - ((4.0 * a) * c)));
	double t_2 = 2.0 * -a;
	double tmp_1;
	if (b <= -1e+148) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (b + b) / t_2;
		} else {
			tmp_2 = (c + c) / t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.1e+99) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (b + t_1) / t_2;
		} else {
			tmp_3 = (2.0 * c) / (-b + t_1);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = fma(-1.0, (b / a), (c / b));
	} else {
		tmp_1 = (2.0 * c) / t_0;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-b) + Float64(-b))
	t_1 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	t_2 = Float64(2.0 * Float64(-a))
	tmp_1 = 0.0
	if (b <= -1e+148)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(b + b) / t_2);
		else
			tmp_2 = Float64(Float64(c + c) / t_0);
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.1e+99)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(b + t_1) / t_2);
		else
			tmp_3 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_1));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = fma(-1.0, Float64(b / a), Float64(c / b));
	else
		tmp_1 = Float64(Float64(2.0 * c) / t_0);
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) + (-b)), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * (-a)), $MachinePrecision]}, If[LessEqual[b, -1e+148], If[GreaterEqual[b, 0.0], N[(N[(b + b), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / t$95$0), $MachinePrecision]], If[LessEqual[b, 1.1e+99], If[GreaterEqual[b, 0.0], N[(N[(b + t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$1), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(-1.0 * N[(b / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1e148

   1. Initial program 37.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 95.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 95.6

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

   10. Applied rewrites 95.6%

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

   if -1e148 < b < 1.09999999999999989e99

   1. Initial program 86.2%

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

   if 1.09999999999999989e99 < b

   1. Initial program 54.6%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 54.6

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

   5. Applied rewrites 54.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 94.4%

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

   9. Taylor expanded in c around 0

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

       1. lower-fma.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 95.2

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

   11. Applied rewrites 95.2%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+148}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\left(-b\right) + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-b\right) + \left(-b\right)\\ t_1 := 2 \cdot \left(-a\right)\\ \mathbf{if}\;b \leq -1 \cdot 10^{+148}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0}\\ \end{array}\\ \mathbf{elif}\;b \leq -7.1 \cdot 10^{-202}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{\frac{c}{a} \cdot -4} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-46}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot a\right) \cdot c}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, b, \sqrt{\left(c \cdot a\right) \cdot -1}\right)}{-a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ (- b) (- b))) (t_1 (* 2.0 (- a))))
   (if (<= b -1e+148)
     (if (>= b 0.0) (/ (+ b b) t_1) (/ (+ c c) t_0))
     (if (<= b -7.1e-202)
       (if (>= b 0.0)
         (* (sqrt (* (/ c a) -4.0)) -0.5)
         (/ (+ c c) (- (sqrt (fma (* -4.0 a) c (* b b))) b)))
       (if (<= b 1.35e-46)
         (if (>= b 0.0)
           (/ (+ b (sqrt (* (* -4.0 a) c))) t_1)
           (/ (fma 0.5 b (sqrt (* (* c a) -1.0))) (- a)))
         (if (>= b 0.0) (fma -1.0 (/ b a) (/ c b)) (/ (* 2.0 c) t_0)))))))

C

double code(double a, double b, double c) {
	double t_0 = -b + -b;
	double t_1 = 2.0 * -a;
	double tmp_1;
	if (b <= -1e+148) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (b + b) / t_1;
		} else {
			tmp_2 = (c + c) / t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -7.1e-202) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = sqrt(((c / a) * -4.0)) * -0.5;
		} else {
			tmp_3 = (c + c) / (sqrt(fma((-4.0 * a), c, (b * b))) - b);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.35e-46) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (b + sqrt(((-4.0 * a) * c))) / t_1;
		} else {
			tmp_4 = fma(0.5, b, sqrt(((c * a) * -1.0))) / -a;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = fma(-1.0, (b / a), (c / b));
	} else {
		tmp_1 = (2.0 * c) / t_0;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-b) + Float64(-b))
	t_1 = Float64(2.0 * Float64(-a))
	tmp_1 = 0.0
	if (b <= -1e+148)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(b + b) / t_1);
		else
			tmp_2 = Float64(Float64(c + c) / t_0);
		end
		tmp_1 = tmp_2;
	elseif (b <= -7.1e-202)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(sqrt(Float64(Float64(c / a) * -4.0)) * -0.5);
		else
			tmp_3 = Float64(Float64(c + c) / Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) - b));
		end
		tmp_1 = tmp_3;
	elseif (b <= 1.35e-46)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(b + sqrt(Float64(Float64(-4.0 * a) * c))) / t_1);
		else
			tmp_4 = Float64(fma(0.5, b, sqrt(Float64(Float64(c * a) * -1.0))) / Float64(-a));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = fma(-1.0, Float64(b / a), Float64(c / b));
	else
		tmp_1 = Float64(Float64(2.0 * c) / t_0);
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) + (-b)), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * (-a)), $MachinePrecision]}, If[LessEqual[b, -1e+148], If[GreaterEqual[b, 0.0], N[(N[(b + b), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / t$95$0), $MachinePrecision]], If[LessEqual[b, -7.1e-202], If[GreaterEqual[b, 0.0], N[(N[Sqrt[N[(N[(c / a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.35e-46], If[GreaterEqual[b, 0.0], N[(N[(b + N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(0.5 * b + N[Sqrt[N[(N[(c * a), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-a)), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(-1.0 * N[(b / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1e148

   1. Initial program 37.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 95.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 95.6

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

   10. Applied rewrites 95.6%

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

   if -1e148 < b < -7.1e-202

   1. Initial program 93.1%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 93.0%

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

   6. Taylor expanded in a around inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 93.0

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

   8. Applied rewrites 93.0%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 93.0

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

   10. Applied rewrites 93.0%

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

   if -7.1e-202 < b < 1.35e-46

   1. Initial program 76.9%

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

   3. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 76.9

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 32.2

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

   8. Applied rewrites 32.2%

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

   9. Taylor expanded in a around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 68.1

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

   11. Applied rewrites 68.1%

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

   if 1.35e-46 < b

   1. Initial program 66.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 66.3

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 85.0%

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

   9. Taylor expanded in c around 0

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

       1. lower-fma.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 85.7

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

   11. Applied rewrites 85.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+148}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\left(-b\right) + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq -7.1 \cdot 10^{-202}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{\frac{c}{a} \cdot -4} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-46}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot a\right) \cdot c}}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, b, \sqrt{\left(c \cdot a\right) \cdot -1}\right)}{-a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-b\right) + \left(-b\right)\\ t_1 := 2 \cdot \left(-a\right)\\ t_2 := \frac{2 \cdot c}{t\_0}\\ \mathbf{if}\;b \leq -2.8 \cdot 10^{-91}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{\frac{c}{a} \cdot -4} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-46}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{-4 \cdot \left(a \cdot c\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ (- b) (- b))) (t_1 (* 2.0 (- a))) (t_2 (/ (* 2.0 c) t_0)))
   (if (<= b -2.8e-91)
     (if (>= b 0.0) (/ (+ b b) t_1) (/ (+ c c) t_0))
     (if (<= b -5e-310)
       (if (>= b 0.0)
         (* (sqrt (* (/ c a) -4.0)) -0.5)
         (/ (* 2.0 c) (- (sqrt (* (* -4.0 a) c)) b)))
       (if (<= b 1.35e-46)
         (if (>= b 0.0) (/ (+ b (sqrt (* -4.0 (* a c)))) t_1) t_2)
         (if (>= b 0.0) (fma -1.0 (/ b a) (/ c b)) t_2))))))

C

double code(double a, double b, double c) {
	double t_0 = -b + -b;
	double t_1 = 2.0 * -a;
	double t_2 = (2.0 * c) / t_0;
	double tmp_1;
	if (b <= -2.8e-91) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (b + b) / t_1;
		} else {
			tmp_2 = (c + c) / t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -5e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = sqrt(((c / a) * -4.0)) * -0.5;
		} else {
			tmp_3 = (2.0 * c) / (sqrt(((-4.0 * a) * c)) - b);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.35e-46) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (b + sqrt((-4.0 * (a * c)))) / t_1;
		} else {
			tmp_4 = t_2;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = fma(-1.0, (b / a), (c / b));
	} else {
		tmp_1 = t_2;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-b) + Float64(-b))
	t_1 = Float64(2.0 * Float64(-a))
	t_2 = Float64(Float64(2.0 * c) / t_0)
	tmp_1 = 0.0
	if (b <= -2.8e-91)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(b + b) / t_1);
		else
			tmp_2 = Float64(Float64(c + c) / t_0);
		end
		tmp_1 = tmp_2;
	elseif (b <= -5e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(sqrt(Float64(Float64(c / a) * -4.0)) * -0.5);
		else
			tmp_3 = Float64(Float64(2.0 * c) / Float64(sqrt(Float64(Float64(-4.0 * a) * c)) - b));
		end
		tmp_1 = tmp_3;
	elseif (b <= 1.35e-46)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(b + sqrt(Float64(-4.0 * Float64(a * c)))) / t_1);
		else
			tmp_4 = t_2;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = fma(-1.0, Float64(b / a), Float64(c / b));
	else
		tmp_1 = t_2;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) + (-b)), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * (-a)), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * c), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[b, -2.8e-91], If[GreaterEqual[b, 0.0], N[(N[(b + b), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / t$95$0), $MachinePrecision]], If[LessEqual[b, -5e-310], If[GreaterEqual[b, 0.0], N[(N[Sqrt[N[(N[(c / a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.35e-46], If[GreaterEqual[b, 0.0], N[(N[(b + N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$2], If[GreaterEqual[b, 0.0], N[(-1.0 * N[(b / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.8e-91

   1. Initial program 68.8%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 90.2

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 90.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 90.2

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

   10. Applied rewrites 90.2%

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

   if -2.8e-91 < b < -4.999999999999985e-310

   1. Initial program 87.6%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 87.6%

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

   6. Taylor expanded in a around inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 87.6

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

   8. Applied rewrites 87.6%

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

   9. Taylor expanded in a around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 86.2

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

   11. Applied rewrites 86.2%

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

   if -4.999999999999985e-310 < b < 1.35e-46

   1. Initial program 72.6%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 72.6

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

   5. Applied rewrites 72.6%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 62.0

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

   8. Applied rewrites 62.0%

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

   if 1.35e-46 < b

   1. Initial program 66.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 66.3

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 85.0%

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

   9. Taylor expanded in c around 0

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

       1. lower-fma.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 85.7

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

   11. Applied rewrites 85.7%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{-91}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\left(-b\right) + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{\frac{c}{a} \cdot -4} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-46}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-b\right) + \left(-b\right)\\ \mathbf{if}\;b \leq -1 \cdot 10^{+148}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ (- b) (- b))))
   (if (<= b -1e+148)
     (if (>= b 0.0) (/ (+ b b) (* 2.0 (- a))) (/ (+ c c) t_0))
     (if (<= b 1.1e+99)
       (if (>= b 0.0)
         (* (/ (+ (sqrt (fma (* -4.0 a) c (* b b))) b) a) -0.5)
         (/ (* 2.0 c) (- (sqrt (fma -4.0 (* a c) (* b b))) b)))
       (if (>= b 0.0) (fma -1.0 (/ b a) (/ c b)) (/ (* 2.0 c) t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = -b + -b;
	double tmp_1;
	if (b <= -1e+148) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (b + b) / (2.0 * -a);
		} else {
			tmp_2 = (c + c) / t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.1e+99) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = ((sqrt(fma((-4.0 * a), c, (b * b))) + b) / a) * -0.5;
		} else {
			tmp_3 = (2.0 * c) / (sqrt(fma(-4.0, (a * c), (b * b))) - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = fma(-1.0, (b / a), (c / b));
	} else {
		tmp_1 = (2.0 * c) / t_0;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-b) + Float64(-b))
	tmp_1 = 0.0
	if (b <= -1e+148)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(b + b) / Float64(2.0 * Float64(-a)));
		else
			tmp_2 = Float64(Float64(c + c) / t_0);
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.1e+99)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) + b) / a) * -0.5);
		else
			tmp_3 = Float64(Float64(2.0 * c) / Float64(sqrt(fma(-4.0, Float64(a * c), Float64(b * b))) - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = fma(-1.0, Float64(b / a), Float64(c / b));
	else
		tmp_1 = Float64(Float64(2.0 * c) / t_0);
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) + (-b)), $MachinePrecision]}, If[LessEqual[b, -1e+148], If[GreaterEqual[b, 0.0], N[(N[(b + b), $MachinePrecision] / N[(2.0 * (-a)), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / t$95$0), $MachinePrecision]], If[LessEqual[b, 1.1e+99], If[GreaterEqual[b, 0.0], N[(N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[(N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(-1.0 * N[(b / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1e148

   1. Initial program 37.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 95.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 95.6

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

   10. Applied rewrites 95.6%

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

   if -1e148 < b < 1.09999999999999989e99

   1. Initial program 86.2%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 86.2%

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

   6. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 86.2

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

   8. Applied rewrites 86.2%

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

   if 1.09999999999999989e99 < b

   1. Initial program 54.6%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 54.6

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

   5. Applied rewrites 54.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 94.4%

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

   9. Taylor expanded in c around 0

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

       1. lower-fma.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 95.2

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

   11. Applied rewrites 95.2%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+148}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\left(-b\right) + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-b\right) + \left(-b\right)\\ t_1 := \frac{2 \cdot c}{t\_0}\\ \mathbf{if}\;b \leq -2.8 \cdot 10^{-91}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{\frac{c}{a} \cdot -4} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{-48}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-\sqrt{\left(a \cdot c\right) \cdot -4}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ (- b) (- b))) (t_1 (/ (* 2.0 c) t_0)))
   (if (<= b -2.8e-91)
     (if (>= b 0.0) (/ (+ b b) (* 2.0 (- a))) (/ (+ c c) t_0))
     (if (<= b -5e-310)
       (if (>= b 0.0)
         (* (sqrt (* (/ c a) -4.0)) -0.5)
         (/ (* 2.0 c) (- (sqrt (* (* -4.0 a) c)) b)))
       (if (<= b 4.9e-48)
         (if (>= b 0.0) (/ (- (sqrt (* (* a c) -4.0))) (* 2.0 a)) t_1)
         (if (>= b 0.0) (fma -1.0 (/ b a) (/ c b)) t_1))))))

C

double code(double a, double b, double c) {
	double t_0 = -b + -b;
	double t_1 = (2.0 * c) / t_0;
	double tmp_1;
	if (b <= -2.8e-91) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (b + b) / (2.0 * -a);
		} else {
			tmp_2 = (c + c) / t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -5e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = sqrt(((c / a) * -4.0)) * -0.5;
		} else {
			tmp_3 = (2.0 * c) / (sqrt(((-4.0 * a) * c)) - b);
		}
		tmp_1 = tmp_3;
	} else if (b <= 4.9e-48) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = -sqrt(((a * c) * -4.0)) / (2.0 * a);
		} else {
			tmp_4 = t_1;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = fma(-1.0, (b / a), (c / b));
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-b) + Float64(-b))
	t_1 = Float64(Float64(2.0 * c) / t_0)
	tmp_1 = 0.0
	if (b <= -2.8e-91)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(b + b) / Float64(2.0 * Float64(-a)));
		else
			tmp_2 = Float64(Float64(c + c) / t_0);
		end
		tmp_1 = tmp_2;
	elseif (b <= -5e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(sqrt(Float64(Float64(c / a) * -4.0)) * -0.5);
		else
			tmp_3 = Float64(Float64(2.0 * c) / Float64(sqrt(Float64(Float64(-4.0 * a) * c)) - b));
		end
		tmp_1 = tmp_3;
	elseif (b <= 4.9e-48)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(-sqrt(Float64(Float64(a * c) * -4.0))) / Float64(2.0 * a));
		else
			tmp_4 = t_1;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = fma(-1.0, Float64(b / a), Float64(c / b));
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) + (-b)), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * c), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[b, -2.8e-91], If[GreaterEqual[b, 0.0], N[(N[(b + b), $MachinePrecision] / N[(2.0 * (-a)), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / t$95$0), $MachinePrecision]], If[LessEqual[b, -5e-310], If[GreaterEqual[b, 0.0], N[(N[Sqrt[N[(N[(c / a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 4.9e-48], If[GreaterEqual[b, 0.0], N[((-N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]) / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], t$95$1], If[GreaterEqual[b, 0.0], N[(-1.0 * N[(b / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.8e-91

   1. Initial program 68.8%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 90.2

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 90.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 90.2

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

   10. Applied rewrites 90.2%

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

   if -2.8e-91 < b < -4.999999999999985e-310

   1. Initial program 87.6%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 87.6%

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

   6. Taylor expanded in a around inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 87.6

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

   8. Applied rewrites 87.6%

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

   9. Taylor expanded in a around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 86.2

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

   11. Applied rewrites 86.2%

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

   if -4.999999999999985e-310 < b < 4.9000000000000002e-48

   1. Initial program 72.6%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 72.6

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

   5. Applied rewrites 72.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 27.0%

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

   9. Taylor expanded in a around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. sqrt-unprod N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 57.9

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

   11. Applied rewrites 57.9%

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

   if 4.9000000000000002e-48 < b

   1. Initial program 66.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 66.3

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 85.0%

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

   9. Taylor expanded in c around 0

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

       1. lower-fma.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 85.7

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

   11. Applied rewrites 85.7%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{-91}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\left(-b\right) + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{\frac{c}{a} \cdot -4} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{-48}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-\sqrt{\left(a \cdot c\right) \cdot -4}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-b\right) + \left(-b\right)\\ t_1 := 2 \cdot \left(-a\right)\\ t_2 := \frac{2 \cdot c}{t\_0}\\ \mathbf{if}\;b \leq -1.85 \cdot 10^{-91}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, b, \sqrt{\left(-a\right) \cdot c}\right)}{-a}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{-48}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-\sqrt{\left(a \cdot c\right) \cdot -4}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ (- b) (- b))) (t_1 (* 2.0 (- a))) (t_2 (/ (* 2.0 c) t_0)))
   (if (<= b -1.85e-91)
     (if (>= b 0.0) (/ (+ b b) t_1) (/ (+ c c) t_0))
     (if (<= b -5e-310)
       (if (>= b 0.0)
         (/ (+ b (sqrt (* b b))) t_1)
         (/ (fma 0.5 b (sqrt (* (- a) c))) (- a)))
       (if (<= b 4.9e-48)
         (if (>= b 0.0) (/ (- (sqrt (* (* a c) -4.0))) (* 2.0 a)) t_2)
         (if (>= b 0.0) (fma -1.0 (/ b a) (/ c b)) t_2))))))

C

double code(double a, double b, double c) {
	double t_0 = -b + -b;
	double t_1 = 2.0 * -a;
	double t_2 = (2.0 * c) / t_0;
	double tmp_1;
	if (b <= -1.85e-91) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (b + b) / t_1;
		} else {
			tmp_2 = (c + c) / t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -5e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (b + sqrt((b * b))) / t_1;
		} else {
			tmp_3 = fma(0.5, b, sqrt((-a * c))) / -a;
		}
		tmp_1 = tmp_3;
	} else if (b <= 4.9e-48) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = -sqrt(((a * c) * -4.0)) / (2.0 * a);
		} else {
			tmp_4 = t_2;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = fma(-1.0, (b / a), (c / b));
	} else {
		tmp_1 = t_2;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-b) + Float64(-b))
	t_1 = Float64(2.0 * Float64(-a))
	t_2 = Float64(Float64(2.0 * c) / t_0)
	tmp_1 = 0.0
	if (b <= -1.85e-91)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(b + b) / t_1);
		else
			tmp_2 = Float64(Float64(c + c) / t_0);
		end
		tmp_1 = tmp_2;
	elseif (b <= -5e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(b + sqrt(Float64(b * b))) / t_1);
		else
			tmp_3 = Float64(fma(0.5, b, sqrt(Float64(Float64(-a) * c))) / Float64(-a));
		end
		tmp_1 = tmp_3;
	elseif (b <= 4.9e-48)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(-sqrt(Float64(Float64(a * c) * -4.0))) / Float64(2.0 * a));
		else
			tmp_4 = t_2;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = fma(-1.0, Float64(b / a), Float64(c / b));
	else
		tmp_1 = t_2;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) + (-b)), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * (-a)), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * c), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[b, -1.85e-91], If[GreaterEqual[b, 0.0], N[(N[(b + b), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / t$95$0), $MachinePrecision]], If[LessEqual[b, -5e-310], If[GreaterEqual[b, 0.0], N[(N[(b + N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(0.5 * b + N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-a)), $MachinePrecision]], If[LessEqual[b, 4.9e-48], If[GreaterEqual[b, 0.0], N[((-N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]) / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], t$95$2], If[GreaterEqual[b, 0.0], N[(-1.0 * N[(b / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.8500000000000001e-91

   1. Initial program 68.8%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 90.2

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 90.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 90.2

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

   10. Applied rewrites 90.2%

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

   if -1.8500000000000001e-91 < b < -4.999999999999985e-310

   1. Initial program 87.6%

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

   3. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 85.6

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

   8. Applied rewrites 85.6%

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

   9. Taylor expanded in a around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lower-*.f64 85.6

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

   11. Applied rewrites 85.6%

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

   if -4.999999999999985e-310 < b < 4.9000000000000002e-48

   1. Initial program 72.6%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 72.6

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

   5. Applied rewrites 72.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 27.0%

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

   9. Taylor expanded in a around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. sqrt-unprod N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 57.9

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

   11. Applied rewrites 57.9%

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

   if 4.9000000000000002e-48 < b

   1. Initial program 66.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 66.3

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 85.0%

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

   9. Taylor expanded in c around 0

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

       1. lower-fma.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 85.7

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

   11. Applied rewrites 85.7%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{-91}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\left(-b\right) + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b}}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, b, \sqrt{\left(-a\right) \cdot c}\right)}{-a}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{-48}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-\sqrt{\left(a \cdot c\right) \cdot -4}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-b\right) + \left(-b\right)\\ t_1 := 2 \cdot \left(-a\right)\\ \mathbf{if}\;b \leq -1.85 \cdot 10^{-91}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-46}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot a\right) \cdot c}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, b, \sqrt{\left(c \cdot a\right) \cdot -1}\right)}{-a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ (- b) (- b))) (t_1 (* 2.0 (- a))))
   (if (<= b -1.85e-91)
     (if (>= b 0.0) (/ (+ b b) t_1) (/ (+ c c) t_0))
     (if (<= b 1.35e-46)
       (if (>= b 0.0)
         (/ (+ b (sqrt (* (* -4.0 a) c))) t_1)
         (/ (fma 0.5 b (sqrt (* (* c a) -1.0))) (- a)))
       (if (>= b 0.0) (fma -1.0 (/ b a) (/ c b)) (/ (* 2.0 c) t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = -b + -b;
	double t_1 = 2.0 * -a;
	double tmp_1;
	if (b <= -1.85e-91) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (b + b) / t_1;
		} else {
			tmp_2 = (c + c) / t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.35e-46) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (b + sqrt(((-4.0 * a) * c))) / t_1;
		} else {
			tmp_3 = fma(0.5, b, sqrt(((c * a) * -1.0))) / -a;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = fma(-1.0, (b / a), (c / b));
	} else {
		tmp_1 = (2.0 * c) / t_0;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-b) + Float64(-b))
	t_1 = Float64(2.0 * Float64(-a))
	tmp_1 = 0.0
	if (b <= -1.85e-91)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(b + b) / t_1);
		else
			tmp_2 = Float64(Float64(c + c) / t_0);
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.35e-46)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(b + sqrt(Float64(Float64(-4.0 * a) * c))) / t_1);
		else
			tmp_3 = Float64(fma(0.5, b, sqrt(Float64(Float64(c * a) * -1.0))) / Float64(-a));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = fma(-1.0, Float64(b / a), Float64(c / b));
	else
		tmp_1 = Float64(Float64(2.0 * c) / t_0);
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) + (-b)), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * (-a)), $MachinePrecision]}, If[LessEqual[b, -1.85e-91], If[GreaterEqual[b, 0.0], N[(N[(b + b), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / t$95$0), $MachinePrecision]], If[LessEqual[b, 1.35e-46], If[GreaterEqual[b, 0.0], N[(N[(b + N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(0.5 * b + N[Sqrt[N[(N[(c * a), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-a)), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(-1.0 * N[(b / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.8500000000000001e-91

   1. Initial program 68.8%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 90.2

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 90.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 90.2

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

   10. Applied rewrites 90.2%

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

   if -1.8500000000000001e-91 < b < 1.35e-46

   1. Initial program 78.7%

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

   3. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 77.9

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

   5. Applied rewrites 77.9%

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

   6. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 45.9

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

   8. Applied rewrites 45.9%

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

   9. Taylor expanded in a around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 71.5

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

   11. Applied rewrites 71.5%

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

   if 1.35e-46 < b

   1. Initial program 66.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 66.3

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 85.0%

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

   9. Taylor expanded in c around 0

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

       1. lower-fma.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 85.7

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

   11. Applied rewrites 85.7%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{-91}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\left(-b\right) + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-46}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot a\right) \cdot c}}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, b, \sqrt{\left(c \cdot a\right) \cdot -1}\right)}{-a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-b\right) + \left(-b\right)\\ t_1 := 2 \cdot \left(-a\right)\\ t_2 := \frac{2 \cdot c}{t\_0}\\ \mathbf{if}\;b \leq -1.85 \cdot 10^{-91}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-303}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, b, \sqrt{\left(-a\right) \cdot c}\right)}{-a}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{-220}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \sqrt{\frac{c}{a} \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ (- b) (- b))) (t_1 (* 2.0 (- a))) (t_2 (/ (* 2.0 c) t_0)))
   (if (<= b -1.85e-91)
     (if (>= b 0.0) (/ (+ b b) t_1) (/ (+ c c) t_0))
     (if (<= b 2.1e-303)
       (if (>= b 0.0)
         (/ (+ b (sqrt (* b b))) t_1)
         (/ (fma 0.5 b (sqrt (* (- a) c))) (- a)))
       (if (<= b 1e-220)
         (if (>= b 0.0) (* -0.5 (sqrt (* (/ c a) -4.0))) t_2)
         (if (>= b 0.0) (fma -1.0 (/ b a) (/ c b)) t_2))))))

C

double code(double a, double b, double c) {
	double t_0 = -b + -b;
	double t_1 = 2.0 * -a;
	double t_2 = (2.0 * c) / t_0;
	double tmp_1;
	if (b <= -1.85e-91) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (b + b) / t_1;
		} else {
			tmp_2 = (c + c) / t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= 2.1e-303) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (b + sqrt((b * b))) / t_1;
		} else {
			tmp_3 = fma(0.5, b, sqrt((-a * c))) / -a;
		}
		tmp_1 = tmp_3;
	} else if (b <= 1e-220) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = -0.5 * sqrt(((c / a) * -4.0));
		} else {
			tmp_4 = t_2;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = fma(-1.0, (b / a), (c / b));
	} else {
		tmp_1 = t_2;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-b) + Float64(-b))
	t_1 = Float64(2.0 * Float64(-a))
	t_2 = Float64(Float64(2.0 * c) / t_0)
	tmp_1 = 0.0
	if (b <= -1.85e-91)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(b + b) / t_1);
		else
			tmp_2 = Float64(Float64(c + c) / t_0);
		end
		tmp_1 = tmp_2;
	elseif (b <= 2.1e-303)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(b + sqrt(Float64(b * b))) / t_1);
		else
			tmp_3 = Float64(fma(0.5, b, sqrt(Float64(Float64(-a) * c))) / Float64(-a));
		end
		tmp_1 = tmp_3;
	elseif (b <= 1e-220)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(-0.5 * sqrt(Float64(Float64(c / a) * -4.0)));
		else
			tmp_4 = t_2;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = fma(-1.0, Float64(b / a), Float64(c / b));
	else
		tmp_1 = t_2;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) + (-b)), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * (-a)), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * c), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[b, -1.85e-91], If[GreaterEqual[b, 0.0], N[(N[(b + b), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / t$95$0), $MachinePrecision]], If[LessEqual[b, 2.1e-303], If[GreaterEqual[b, 0.0], N[(N[(b + N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(0.5 * b + N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-a)), $MachinePrecision]], If[LessEqual[b, 1e-220], If[GreaterEqual[b, 0.0], N[(-0.5 * N[Sqrt[N[(N[(c / a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2], If[GreaterEqual[b, 0.0], N[(-1.0 * N[(b / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.8500000000000001e-91

   1. Initial program 68.8%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 90.2

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 90.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 90.2

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

   10. Applied rewrites 90.2%

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

   if -1.8500000000000001e-91 < b < 2.1e-303

   1. Initial program 88.0%

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

   3. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 86.1

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

   5. Applied rewrites 86.1%

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

   6. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 83.1

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

   8. Applied rewrites 83.1%

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

   9. Taylor expanded in a around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lower-*.f64 83.1

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

   11. Applied rewrites 83.1%

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

   if 2.1e-303 < b < 9.99999999999999992e-221

   1. Initial program 72.5%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 72.5

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

   5. Applied rewrites 72.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 6.8%

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

   9. Taylor expanded in a around inf

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

       1. lower-*.f64 N/A

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

       2. sqrt-prod N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-sqrt.f64 54.4

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

   11. Applied rewrites 54.4%

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

   if 9.99999999999999992e-221 < b

   1. Initial program 67.6%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 67.6

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

   5. Applied rewrites 67.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 74.4%

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

   9. Taylor expanded in c around 0

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

       1. lower-fma.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 74.9

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

   11. Applied rewrites 74.9%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{-91}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\left(-b\right) + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-303}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b}}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, b, \sqrt{\left(-a\right) \cdot c}\right)}{-a}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{-220}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \sqrt{\frac{c}{a} \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-b\right) + \left(-b\right)\\ t_1 := \frac{2 \cdot c}{t\_0}\\ t_2 := \frac{b + b}{2 \cdot \left(-a\right)}\\ \mathbf{if}\;b \leq -1.42 \cdot 10^{-95}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0}\\ \end{array}\\ \mathbf{elif}\;b \leq -4.4 \cdot 10^{-298}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{-c}{a}}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{-220}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \sqrt{\frac{c}{a} \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ (- b) (- b)))
        (t_1 (/ (* 2.0 c) t_0))
        (t_2 (/ (+ b b) (* 2.0 (- a)))))
   (if (<= b -1.42e-95)
     (if (>= b 0.0) t_2 (/ (+ c c) t_0))
     (if (<= b -4.4e-298)
       (if (>= b 0.0) t_2 (- (sqrt (/ (- c) a))))
       (if (<= b 1e-220)
         (if (>= b 0.0) (* -0.5 (sqrt (* (/ c a) -4.0))) t_1)
         (if (>= b 0.0) (fma -1.0 (/ b a) (/ c b)) t_1))))))

C

double code(double a, double b, double c) {
	double t_0 = -b + -b;
	double t_1 = (2.0 * c) / t_0;
	double t_2 = (b + b) / (2.0 * -a);
	double tmp_1;
	if (b <= -1.42e-95) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_2;
		} else {
			tmp_2 = (c + c) / t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -4.4e-298) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_2;
		} else {
			tmp_3 = -sqrt((-c / a));
		}
		tmp_1 = tmp_3;
	} else if (b <= 1e-220) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = -0.5 * sqrt(((c / a) * -4.0));
		} else {
			tmp_4 = t_1;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = fma(-1.0, (b / a), (c / b));
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-b) + Float64(-b))
	t_1 = Float64(Float64(2.0 * c) / t_0)
	t_2 = Float64(Float64(b + b) / Float64(2.0 * Float64(-a)))
	tmp_1 = 0.0
	if (b <= -1.42e-95)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_2;
		else
			tmp_2 = Float64(Float64(c + c) / t_0);
		end
		tmp_1 = tmp_2;
	elseif (b <= -4.4e-298)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_2;
		else
			tmp_3 = Float64(-sqrt(Float64(Float64(-c) / a)));
		end
		tmp_1 = tmp_3;
	elseif (b <= 1e-220)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(-0.5 * sqrt(Float64(Float64(c / a) * -4.0)));
		else
			tmp_4 = t_1;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = fma(-1.0, Float64(b / a), Float64(c / b));
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) + (-b)), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * c), $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b + b), $MachinePrecision] / N[(2.0 * (-a)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.42e-95], If[GreaterEqual[b, 0.0], t$95$2, N[(N[(c + c), $MachinePrecision] / t$95$0), $MachinePrecision]], If[LessEqual[b, -4.4e-298], If[GreaterEqual[b, 0.0], t$95$2, (-N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision])], If[LessEqual[b, 1e-220], If[GreaterEqual[b, 0.0], N[(-0.5 * N[Sqrt[N[(N[(c / a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1], If[GreaterEqual[b, 0.0], N[(-1.0 * N[(b / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.42000000000000007e-95

   1. Initial program 68.8%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 90.2

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 90.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 90.2

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

   10. Applied rewrites 90.2%

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

   if -1.42000000000000007e-95 < b < -4.4e-298

   1. Initial program 87.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 8.9

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

   5. Applied rewrites 8.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 8.9%

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

   9. Taylor expanded in a around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. sqrt-prod N/A

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

       4. lift-/.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-sqrt.f64 53.8

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

   11. Applied rewrites 53.8%

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

   12. Taylor expanded in a around 0

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

       1. associate-*r/ N/A

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

       2. mul-1-neg N/A

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

       3. lower-/.f64 N/A

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

       4. lower-neg.f64 53.8

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

   14. Applied rewrites 53.8%

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

   if -4.4e-298 < b < 9.99999999999999992e-221

   1. Initial program 75.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 70.3

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

   5. Applied rewrites 70.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 6.5%

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

   9. Taylor expanded in a around inf

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

       1. lower-*.f64 N/A

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

       2. sqrt-prod N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-sqrt.f64 49.0

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

   11. Applied rewrites 49.0%

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

   if 9.99999999999999992e-221 < b

   1. Initial program 67.6%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 67.6

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

   5. Applied rewrites 67.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 74.4%

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

   9. Taylor expanded in c around 0

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

       1. lower-fma.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 74.9

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

   11. Applied rewrites 74.9%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.42 \cdot 10^{-95}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\left(-b\right) + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq -4.4 \cdot 10^{-298}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{-c}{a}}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{-220}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \sqrt{\frac{c}{a} \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-b\right) + \left(-b\right)\\ t_1 := \frac{2 \cdot c}{t\_0}\\ t_2 := \frac{b + b}{2 \cdot \left(-a\right)}\\ \mathbf{if}\;b \leq -1.42 \cdot 10^{-95}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0}\\ \end{array}\\ \mathbf{elif}\;b \leq -4.4 \cdot 10^{-298}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{-c}{a}}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-213}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{\frac{c}{a} \cdot -1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ (- b) (- b)))
        (t_1 (/ (* 2.0 c) t_0))
        (t_2 (/ (+ b b) (* 2.0 (- a)))))
   (if (<= b -1.42e-95)
     (if (>= b 0.0) t_2 (/ (+ c c) t_0))
     (if (<= b -4.4e-298)
       (if (>= b 0.0) t_2 (- (sqrt (/ (- c) a))))
       (if (<= b 2.7e-213)
         (if (>= b 0.0) (sqrt (* (/ c a) -1.0)) t_1)
         (if (>= b 0.0) (fma -1.0 (/ b a) (/ c b)) t_1))))))

C

double code(double a, double b, double c) {
	double t_0 = -b + -b;
	double t_1 = (2.0 * c) / t_0;
	double t_2 = (b + b) / (2.0 * -a);
	double tmp_1;
	if (b <= -1.42e-95) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_2;
		} else {
			tmp_2 = (c + c) / t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -4.4e-298) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_2;
		} else {
			tmp_3 = -sqrt((-c / a));
		}
		tmp_1 = tmp_3;
	} else if (b <= 2.7e-213) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = sqrt(((c / a) * -1.0));
		} else {
			tmp_4 = t_1;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = fma(-1.0, (b / a), (c / b));
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-b) + Float64(-b))
	t_1 = Float64(Float64(2.0 * c) / t_0)
	t_2 = Float64(Float64(b + b) / Float64(2.0 * Float64(-a)))
	tmp_1 = 0.0
	if (b <= -1.42e-95)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_2;
		else
			tmp_2 = Float64(Float64(c + c) / t_0);
		end
		tmp_1 = tmp_2;
	elseif (b <= -4.4e-298)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_2;
		else
			tmp_3 = Float64(-sqrt(Float64(Float64(-c) / a)));
		end
		tmp_1 = tmp_3;
	elseif (b <= 2.7e-213)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = sqrt(Float64(Float64(c / a) * -1.0));
		else
			tmp_4 = t_1;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = fma(-1.0, Float64(b / a), Float64(c / b));
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) + (-b)), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * c), $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b + b), $MachinePrecision] / N[(2.0 * (-a)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.42e-95], If[GreaterEqual[b, 0.0], t$95$2, N[(N[(c + c), $MachinePrecision] / t$95$0), $MachinePrecision]], If[LessEqual[b, -4.4e-298], If[GreaterEqual[b, 0.0], t$95$2, (-N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision])], If[LessEqual[b, 2.7e-213], If[GreaterEqual[b, 0.0], N[Sqrt[N[(N[(c / a), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision], t$95$1], If[GreaterEqual[b, 0.0], N[(-1.0 * N[(b / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.42000000000000007e-95

   1. Initial program 68.8%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 90.2

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 90.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 90.2

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

   10. Applied rewrites 90.2%

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

   if -1.42000000000000007e-95 < b < -4.4e-298

   1. Initial program 87.3%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 8.9

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

   5. Applied rewrites 8.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 8.9%

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

   9. Taylor expanded in a around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. sqrt-prod N/A

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

       4. lift-/.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-sqrt.f64 53.8

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

   11. Applied rewrites 53.8%

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

   12. Taylor expanded in a around 0

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

       1. associate-*r/ N/A

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

       2. mul-1-neg N/A

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

       3. lower-/.f64 N/A

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

       4. lower-neg.f64 53.8

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

   14. Applied rewrites 53.8%

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

   if -4.4e-298 < b < 2.7000000000000001e-213

   1. Initial program 73.4%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 68.9

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

   5. Applied rewrites 68.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 6.6%

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

   9. Taylor expanded in a around -inf

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

       1. sqrt-prod N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-sqrt.f64 43.6

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

   11. Applied rewrites 43.6%

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

   if 2.7000000000000001e-213 < b

   1. Initial program 67.8%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 67.8

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 75.5%

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

   9. Taylor expanded in c around 0

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

       1. lower-fma.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-/.f64 76.1

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

   11. Applied rewrites 76.1%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.42 \cdot 10^{-95}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\left(-b\right) + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq -4.4 \cdot 10^{-298}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{-c}{a}}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-213}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{\frac{c}{a} \cdot -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b}{a}, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(-b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{-91}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\left(-b\right) + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\left(a \cdot c\right) \cdot -4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.85e-91)
   (if (>= b 0.0) (/ (+ b b) (* 2.0 (- a))) (/ (+ c c) (+ (- b) (- b))))
   (if (>= b 0.0)
     (/ (fma a (/ c b) (- b)) a)
     (/ (* 2.0 c) (sqrt (* (* a c) -4.0))))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1.85e-91) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (b + b) / (2.0 * -a);
		} else {
			tmp_2 = (c + c) / (-b + -b);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = fma(a, (c / b), -b) / a;
	} else {
		tmp_1 = (2.0 * c) / sqrt(((a * c) * -4.0));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -1.85e-91)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(b + b) / Float64(2.0 * Float64(-a)));
		else
			tmp_2 = Float64(Float64(c + c) / Float64(Float64(-b) + Float64(-b)));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(fma(a, Float64(c / b), Float64(-b)) / a);
	else
		tmp_1 = Float64(Float64(2.0 * c) / sqrt(Float64(Float64(a * c) * -4.0)));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.85e-91], If[GreaterEqual[b, 0.0], N[(N[(b + b), $MachinePrecision] / N[(2.0 * (-a)), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[((-b) + (-b)), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision] / a), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.8500000000000001e-91

   1. Initial program 68.8%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 90.2

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 90.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 90.2

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

   10. Applied rewrites 90.2%

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

   if -1.8500000000000001e-91 < b

   1. Initial program 72.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-neg.f64 70.1

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

   5. Applied rewrites 70.1%

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

   6. Taylor expanded in a around inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 69.4

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

   8. Applied rewrites 69.4%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{-91}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\left(-b\right) + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\left(a \cdot c\right) \cdot -4}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 69.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.42 \cdot 10^{-95}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\left(-b\right) + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{c}{a} \cdot -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.42e-95)
   (if (>= b 0.0) (/ (+ b b) (* 2.0 (- a))) (/ (+ c c) (+ (- b) (- b))))
   (if (>= b 0.0) (/ (fma a (/ c b) (- b)) a) (- (sqrt (* (/ c a) -1.0))))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1.42e-95) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (b + b) / (2.0 * -a);
		} else {
			tmp_2 = (c + c) / (-b + -b);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = fma(a, (c / b), -b) / a;
	} else {
		tmp_1 = -sqrt(((c / a) * -1.0));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -1.42e-95)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(b + b) / Float64(2.0 * Float64(-a)));
		else
			tmp_2 = Float64(Float64(c + c) / Float64(Float64(-b) + Float64(-b)));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(fma(a, Float64(c / b), Float64(-b)) / a);
	else
		tmp_1 = Float64(-sqrt(Float64(Float64(c / a) * -1.0)));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.42e-95], If[GreaterEqual[b, 0.0], N[(N[(b + b), $MachinePrecision] / N[(2.0 * (-a)), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[((-b) + (-b)), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision] / a), $MachinePrecision], (-N[Sqrt[N[(N[(c / a), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if b < -1.42000000000000007e-95

   1. Initial program 68.8%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 90.2

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 90.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 90.2

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

   10. Applied rewrites 90.2%

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

   if -1.42000000000000007e-95 < b

   1. Initial program 72.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-neg.f64 70.1

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

   5. Applied rewrites 70.1%

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

   6. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 63.5

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

   8. Applied rewrites 63.5%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.42 \cdot 10^{-95}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\left(-b\right) + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{c}{a} \cdot -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 69.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b + b}{2 \cdot \left(-a\right)}\\ \mathbf{if}\;b \leq -1.42 \cdot 10^{-95}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\left(-b\right) + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{-c}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (+ b b) (* 2.0 (- a)))))
   (if (<= b -1.42e-95)
     (if (>= b 0.0) t_0 (/ (+ c c) (+ (- b) (- b))))
     (if (>= b 0.0) t_0 (- (sqrt (/ (- c) a)))))))

C

double code(double a, double b, double c) {
	double t_0 = (b + b) / (2.0 * -a);
	double tmp_1;
	if (b <= -1.42e-95) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = (c + c) / (-b + -b);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = -sqrt((-c / 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) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = (b + b) / (2.0d0 * -a)
    if (b <= (-1.42d-95)) then
        if (b >= 0.0d0) then
            tmp_2 = t_0
        else
            tmp_2 = (c + c) / (-b + -b)
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = t_0
    else
        tmp_1 = -sqrt((-c / a))
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (b + b) / (2.0 * -a);
	double tmp_1;
	if (b <= -1.42e-95) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = (c + c) / (-b + -b);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = -Math.sqrt((-c / a));
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (b + b) / (2.0 * -a)
	tmp_1 = 0
	if b <= -1.42e-95:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_0
		else:
			tmp_2 = (c + c) / (-b + -b)
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = t_0
	else:
		tmp_1 = -math.sqrt((-c / a))
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b + b) / Float64(2.0 * Float64(-a)))
	tmp_1 = 0.0
	if (b <= -1.42e-95)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(Float64(c + c) / Float64(Float64(-b) + Float64(-b)));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = t_0;
	else
		tmp_1 = Float64(-sqrt(Float64(Float64(-c) / a)));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = (b + b) / (2.0 * -a);
	tmp_2 = 0.0;
	if (b <= -1.42e-95)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_0;
		else
			tmp_3 = (c + c) / (-b + -b);
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = t_0;
	else
		tmp_2 = -sqrt((-c / a));
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b + b), $MachinePrecision] / N[(2.0 * (-a)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.42e-95], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(c + c), $MachinePrecision] / N[((-b) + (-b)), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$0, (-N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision])]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.42000000000000007e-95

   1. Initial program 68.8%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 90.2

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 90.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 90.2

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

   10. Applied rewrites 90.2%

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

   if -1.42000000000000007e-95 < b

   1. Initial program 72.0%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 57.4

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

   5. Applied rewrites 57.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 55.0%

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

   9. Taylor expanded in a around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. sqrt-prod N/A

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

       4. lift-/.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-sqrt.f64 63.0

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

   11. Applied rewrites 63.0%

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

   12. Taylor expanded in a around 0

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

       1. associate-*r/ N/A

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

       2. mul-1-neg N/A

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

       3. lower-/.f64 N/A

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

       4. lower-neg.f64 63.0

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

   14. Applied rewrites 63.0%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.42 \cdot 10^{-95}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\left(-b\right) + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{-c}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 68.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (b + b) / (2.0 * -a);
	} else {
		tmp = (c + c) / (-b + -b);
	}
	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 = (b + b) / (2.0d0 * -a)
    else
        tmp = (c + c) / (-b + -b)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.9%

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

3. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. lift-neg.f64 68.7

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

5. Applied rewrites 68.7%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 67.1%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 67.1

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

10. Applied rewrites 67.1%

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

11. Final simplification 67.1%

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

Alternative 17: 35.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-b}{a}\\ \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double a, double b, double c) {
	double t_0 = -b / a;
	double tmp;
	if (b >= 0.0) {
		tmp = t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = -b / a
	tmp = 0
	if b >= 0.0:
		tmp = t_0
	else:
		tmp = t_0
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.9%

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

3. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. lift-neg.f64 68.7

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

5. Applied rewrites 68.7%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 67.1%

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

9. Taylor expanded in a around 0

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

    1. associate-*r/ N/A

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

    2. mul-1-neg N/A

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

    3. lift-neg.f64 N/A

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

    4. lower-/.f64 36.2

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

11. Applied rewrites 36.2%

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

12. Taylor expanded in a around 0

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

    1. mul-1-neg N/A

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

    2. distribute-frac-neg N/A

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

    3. lift-/.f64 N/A

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

    4. lift-neg.f64 36.2

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

14. Applied rewrites 36.2%

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

Reproduce

herbie shell --seed 2025085 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  :precision binary64
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))