jeff quadratic root 2

Percentage Accurate: 72.4% → 90.7%

Time: 4.9s

Alternatives: 11

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-4 \cdot \frac{c}{a}}\\ t_1 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;b \leq -3 \cdot 10^{+157}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{2 \cdot b} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{b}{a}\right) \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq -6.4 \cdot 10^{-288}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{+60}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{t\_1 + b} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot t\_0\right) \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{-2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* -4.0 (/ c a)))) (t_1 (sqrt (fma (* -4.0 a) c (* b b)))))
   (if (<= b -3e+157)
     (if (>= b 0.0) (* (/ c (* 2.0 b)) -2.0) (* (* -2.0 (/ b a)) 0.5))
     (if (<= b -6.4e-288)
       (if (>= b 0.0) (* -1.0 (/ c b)) (* (/ (- t_1 b) a) 0.5))
       (if (<= b 5.4e+60)
         (if (>= b 0.0) (* (/ c (+ t_1 b)) -2.0) (* (* -1.0 t_0) 0.5))
         (if (>= b 0.0) (/ (* 2.0 c) (* -2.0 b)) (* -0.5 t_0)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((-4.0 * (c / a)));
	double t_1 = sqrt(fma((-4.0 * a), c, (b * b)));
	double tmp_1;
	if (b <= -3e+157) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c / (2.0 * b)) * -2.0;
		} else {
			tmp_2 = (-2.0 * (b / a)) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= -6.4e-288) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -1.0 * (c / b);
		} else {
			tmp_3 = ((t_1 - b) / a) * 0.5;
		}
		tmp_1 = tmp_3;
	} else if (b <= 5.4e+60) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c / (t_1 + b)) * -2.0;
		} else {
			tmp_4 = (-1.0 * t_0) * 0.5;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-2.0 * b);
	} else {
		tmp_1 = -0.5 * t_0;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(-4.0 * Float64(c / a)))
	t_1 = sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))
	tmp_1 = 0.0
	if (b <= -3e+157)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(c / Float64(2.0 * b)) * -2.0);
		else
			tmp_2 = Float64(Float64(-2.0 * Float64(b / a)) * 0.5);
		end
		tmp_1 = tmp_2;
	elseif (b <= -6.4e-288)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(-1.0 * Float64(c / b));
		else
			tmp_3 = Float64(Float64(Float64(t_1 - b) / a) * 0.5);
		end
		tmp_1 = tmp_3;
	elseif (b <= 5.4e+60)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(c / Float64(t_1 + b)) * -2.0);
		else
			tmp_4 = Float64(Float64(-1.0 * t_0) * 0.5);
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(-2.0 * b));
	else
		tmp_1 = Float64(-0.5 * t_0);
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -3e+157], If[GreaterEqual[b, 0.0], N[(N[(c / N[(2.0 * b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(-2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, -6.4e-288], If[GreaterEqual[b, 0.0], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, 5.4e+60], If[GreaterEqual[b, 0.0], N[(N[(c / N[(t$95$1 + b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(-1.0 * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(-2.0 * b), $MachinePrecision]), $MachinePrecision], N[(-0.5 * t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -3.0000000000000001e157

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 43.4

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

   7. Applied rewrites 43.4%

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

   8. Taylor expanded in b around inf

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

      1. lower-*.f64 41.0

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

   10. Applied rewrites 41.0%

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

   11. Taylor expanded in b around -inf

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

       1. lower-*.f64 N/A

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

       2. lift-/.f64 67.6

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

   13. Applied rewrites 67.6%

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

   if -3.0000000000000001e157 < b < -6.4000000000000001e-288

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 70.1

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

   7. Applied rewrites 70.1%

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

   if -6.4000000000000001e-288 < b < 5.3999999999999999e60

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 43.4

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

   7. Applied rewrites 43.4%

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

   if 5.3999999999999999e60 < b

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.0

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

   4. Applied rewrites 60.0%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.0

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

   7. Applied rewrites 48.0%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 31.3

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

   10. Applied rewrites 31.3%

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

   11. Taylor expanded in b around inf

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

       1. lower-*.f64 58.1

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

   13. Applied rewrites 58.1%

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

   14. Taylor expanded in a around -inf

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

       1. lower-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-sqrt.f64 41.0

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

   16. Applied rewrites 41.0%

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

Alternative 2: 90.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-2 \cdot \frac{b}{a}\right) \cdot 0.5\\ t_1 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;b \leq -3 \cdot 10^{+157}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{2 \cdot b} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \leq 2.35 \cdot 10^{-242}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(c \cdot \frac{-1}{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{+60}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{t\_1 + b} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{-2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* -2.0 (/ b a)) 0.5)) (t_1 (sqrt (fma (* -4.0 a) c (* b b)))))
   (if (<= b -3e+157)
     (if (>= b 0.0) (* (/ c (* 2.0 b)) -2.0) t_0)
     (if (<= b 2.35e-242)
       (if (>= b 0.0)
         (* (* c (/ -1.0 (* c (sqrt (* -4.0 (/ a c)))))) -2.0)
         (* (/ (- t_1 b) a) 0.5))
       (if (<= b 5.4e+60)
         (if (>= b 0.0) (* (/ c (+ t_1 b)) -2.0) t_0)
         (if (>= b 0.0)
           (/ (* 2.0 c) (* -2.0 b))
           (* -0.5 (sqrt (* -4.0 (/ c a))))))))))

C

double code(double a, double b, double c) {
	double t_0 = (-2.0 * (b / a)) * 0.5;
	double t_1 = sqrt(fma((-4.0 * a), c, (b * b)));
	double tmp_1;
	if (b <= -3e+157) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c / (2.0 * b)) * -2.0;
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= 2.35e-242) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c * (-1.0 / (c * sqrt((-4.0 * (a / c)))))) * -2.0;
		} else {
			tmp_3 = ((t_1 - b) / a) * 0.5;
		}
		tmp_1 = tmp_3;
	} else if (b <= 5.4e+60) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c / (t_1 + b)) * -2.0;
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-2.0 * b);
	} else {
		tmp_1 = -0.5 * sqrt((-4.0 * (c / a)));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-2.0 * Float64(b / a)) * 0.5)
	t_1 = sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))
	tmp_1 = 0.0
	if (b <= -3e+157)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(c / Float64(2.0 * b)) * -2.0);
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b <= 2.35e-242)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(c * Float64(-1.0 / Float64(c * sqrt(Float64(-4.0 * Float64(a / c)))))) * -2.0);
		else
			tmp_3 = Float64(Float64(Float64(t_1 - b) / a) * 0.5);
		end
		tmp_1 = tmp_3;
	elseif (b <= 5.4e+60)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(c / Float64(t_1 + b)) * -2.0);
		else
			tmp_4 = t_0;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(-2.0 * b));
	else
		tmp_1 = Float64(-0.5 * sqrt(Float64(-4.0 * Float64(c / a))));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -3e+157], If[GreaterEqual[b, 0.0], N[(N[(c / N[(2.0 * b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], t$95$0], If[LessEqual[b, 2.35e-242], If[GreaterEqual[b, 0.0], N[(N[(c * N[(-1.0 / N[(c * N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(t$95$1 - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, 5.4e+60], If[GreaterEqual[b, 0.0], N[(N[(c / N[(t$95$1 + b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(-2.0 * b), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -3.0000000000000001e157

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 43.4

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

   7. Applied rewrites 43.4%

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

   8. Taylor expanded in b around inf

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

      1. lower-*.f64 41.0

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

   10. Applied rewrites 41.0%

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

   11. Taylor expanded in b around -inf

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

       1. lower-*.f64 N/A

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

       2. lift-/.f64 67.6

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

   13. Applied rewrites 67.6%

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

   if -3.0000000000000001e157 < b < 2.35000000000000018e-242

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. mult-flip N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-+.f64 72.4

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

   6. Applied rewrites 72.4%

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

   7. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 44.8

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

   9. Applied rewrites 44.8%

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

   if 2.35000000000000018e-242 < b < 5.3999999999999999e60

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 69.9

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

   7. Applied rewrites 69.9%

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

   if 5.3999999999999999e60 < b

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.0

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

   4. Applied rewrites 60.0%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.0

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

   7. Applied rewrites 48.0%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 31.3

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

   10. Applied rewrites 31.3%

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

   11. Taylor expanded in b around inf

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

       1. lower-*.f64 58.1

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

   13. Applied rewrites 58.1%

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

   14. Taylor expanded in a around -inf

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

       1. lower-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-sqrt.f64 41.0

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

   16. Applied rewrites 41.0%

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

Alternative 3: 90.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;b \leq -1 \cdot 10^{+116}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) \cdot \mathsf{fma}\left(a \cdot \frac{c}{b \cdot b}, -2, 2\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{+60}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{t\_0 + b} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{-2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 a) c (* b b)))))
   (if (<= b -1e+116)
     (if (>= b 0.0)
       (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))
       (/ (* (- b) (fma (* a (/ c (* b b))) -2.0 2.0)) (* 2.0 a)))
     (if (<= b 5.4e+60)
       (if (>= b 0.0) (* (/ c (+ t_0 b)) -2.0) (* (/ (- t_0 b) a) 0.5))
       (if (>= b 0.0)
         (/ (* 2.0 c) (* -2.0 b))
         (* -0.5 (sqrt (* -4.0 (/ c a)))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-4.0 * a), c, (b * b)));
	double tmp_1;
	if (b <= -1e+116) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (-b - sqrt(((b * b) - ((4.0 * a) * c))));
		} else {
			tmp_2 = (-b * fma((a * (c / (b * b))), -2.0, 2.0)) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 5.4e+60) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c / (t_0 + b)) * -2.0;
		} else {
			tmp_3 = ((t_0 - b) / a) * 0.5;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-2.0 * b);
	} else {
		tmp_1 = -0.5 * sqrt((-4.0 * (c / a)));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))
	tmp_1 = 0.0
	if (b <= -1e+116)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))));
		else
			tmp_2 = Float64(Float64(Float64(-b) * fma(Float64(a * Float64(c / Float64(b * b))), -2.0, 2.0)) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 5.4e+60)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(c / Float64(t_0 + b)) * -2.0);
		else
			tmp_3 = Float64(Float64(Float64(t_0 - b) / a) * 0.5);
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(-2.0 * b));
	else
		tmp_1 = Float64(-0.5 * sqrt(Float64(-4.0 * Float64(c / a))));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1e+116], If[GreaterEqual[b, 0.0], 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], N[(N[((-b) * N[(N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0 + 2.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 5.4e+60], If[GreaterEqual[b, 0.0], N[(N[(c / N[(t$95$0 + b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(t$95$0 - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(-2.0 * b), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.00000000000000002e116

   1. Initial program 72.4%

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

   2. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.4

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

   4. Applied rewrites 69.4%

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

   if -1.00000000000000002e116 < b < 5.3999999999999999e60

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.5%

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

   if 5.3999999999999999e60 < b

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.0

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

   4. Applied rewrites 60.0%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.0

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

   7. Applied rewrites 48.0%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 31.3

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

   10. Applied rewrites 31.3%

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

   11. Taylor expanded in b around inf

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

       1. lower-*.f64 58.1

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

   13. Applied rewrites 58.1%

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

   14. Taylor expanded in a around -inf

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

       1. lower-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-sqrt.f64 41.0

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

   16. Applied rewrites 41.0%

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

Alternative 4: 89.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;b \leq -3 \cdot 10^{+157}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{2 \cdot b} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{b}{a}\right) \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{+60}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{t\_0 + b} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{-2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 a) c (* b b)))))
   (if (<= b -3e+157)
     (if (>= b 0.0) (* (/ c (* 2.0 b)) -2.0) (* (* -2.0 (/ b a)) 0.5))
     (if (<= b 5.4e+60)
       (if (>= b 0.0) (* (/ c (+ t_0 b)) -2.0) (* (/ (- t_0 b) a) 0.5))
       (if (>= b 0.0)
         (/ (* 2.0 c) (* -2.0 b))
         (* -0.5 (sqrt (* -4.0 (/ c a)))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-4.0 * a), c, (b * b)));
	double tmp_1;
	if (b <= -3e+157) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c / (2.0 * b)) * -2.0;
		} else {
			tmp_2 = (-2.0 * (b / a)) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= 5.4e+60) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c / (t_0 + b)) * -2.0;
		} else {
			tmp_3 = ((t_0 - b) / a) * 0.5;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-2.0 * b);
	} else {
		tmp_1 = -0.5 * sqrt((-4.0 * (c / a)));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))
	tmp_1 = 0.0
	if (b <= -3e+157)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(c / Float64(2.0 * b)) * -2.0);
		else
			tmp_2 = Float64(Float64(-2.0 * Float64(b / a)) * 0.5);
		end
		tmp_1 = tmp_2;
	elseif (b <= 5.4e+60)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(c / Float64(t_0 + b)) * -2.0);
		else
			tmp_3 = Float64(Float64(Float64(t_0 - b) / a) * 0.5);
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(-2.0 * b));
	else
		tmp_1 = Float64(-0.5 * sqrt(Float64(-4.0 * Float64(c / a))));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -3e+157], If[GreaterEqual[b, 0.0], N[(N[(c / N[(2.0 * b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(-2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, 5.4e+60], If[GreaterEqual[b, 0.0], N[(N[(c / N[(t$95$0 + b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(t$95$0 - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(-2.0 * b), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.0000000000000001e157

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 43.4

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

   7. Applied rewrites 43.4%

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

   8. Taylor expanded in b around inf

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

      1. lower-*.f64 41.0

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

   10. Applied rewrites 41.0%

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

   11. Taylor expanded in b around -inf

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

       1. lower-*.f64 N/A

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

       2. lift-/.f64 67.6

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

   13. Applied rewrites 67.6%

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

   if -3.0000000000000001e157 < b < 5.3999999999999999e60

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.5%

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

   if 5.3999999999999999e60 < b

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.0

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

   4. Applied rewrites 60.0%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.0

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

   7. Applied rewrites 48.0%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 31.3

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

   10. Applied rewrites 31.3%

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

   11. Taylor expanded in b around inf

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

       1. lower-*.f64 58.1

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

   13. Applied rewrites 58.1%

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

   14. Taylor expanded in a around -inf

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

       1. lower-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-sqrt.f64 41.0

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

   16. Applied rewrites 41.0%

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

Alternative 5: 85.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{if}\;b \leq -3 \cdot 10^{+157}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{2 \cdot b} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{b}{a}\right) \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq -6.4 \cdot 10^{-288}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-84}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot t\_0\right) \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{-2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* -4.0 (/ c a)))))
   (if (<= b -3e+157)
     (if (>= b 0.0) (* (/ c (* 2.0 b)) -2.0) (* (* -2.0 (/ b a)) 0.5))
     (if (<= b -6.4e-288)
       (if (>= b 0.0)
         (* -1.0 (/ c b))
         (* (/ (- (sqrt (fma (* -4.0 a) c (* b b))) b) a) 0.5))
       (if (<= b 1.25e-84)
         (if (>= b 0.0)
           (* (/ c (sqrt (* -4.0 (* a c)))) -2.0)
           (* (* -1.0 t_0) 0.5))
         (if (>= b 0.0) (/ (* 2.0 c) (* -2.0 b)) (* -0.5 t_0)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((-4.0 * (c / a)));
	double tmp_1;
	if (b <= -3e+157) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c / (2.0 * b)) * -2.0;
		} else {
			tmp_2 = (-2.0 * (b / a)) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= -6.4e-288) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -1.0 * (c / b);
		} else {
			tmp_3 = ((sqrt(fma((-4.0 * a), c, (b * b))) - b) / a) * 0.5;
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.25e-84) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c / sqrt((-4.0 * (a * c)))) * -2.0;
		} else {
			tmp_4 = (-1.0 * t_0) * 0.5;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-2.0 * b);
	} else {
		tmp_1 = -0.5 * t_0;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(-4.0 * Float64(c / a)))
	tmp_1 = 0.0
	if (b <= -3e+157)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(c / Float64(2.0 * b)) * -2.0);
		else
			tmp_2 = Float64(Float64(-2.0 * Float64(b / a)) * 0.5);
		end
		tmp_1 = tmp_2;
	elseif (b <= -6.4e-288)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(-1.0 * Float64(c / b));
		else
			tmp_3 = Float64(Float64(Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) - b) / a) * 0.5);
		end
		tmp_1 = tmp_3;
	elseif (b <= 1.25e-84)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(c / sqrt(Float64(-4.0 * Float64(a * c)))) * -2.0);
		else
			tmp_4 = Float64(Float64(-1.0 * t_0) * 0.5);
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(-2.0 * b));
	else
		tmp_1 = Float64(-0.5 * t_0);
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -3e+157], If[GreaterEqual[b, 0.0], N[(N[(c / N[(2.0 * b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(-2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, -6.4e-288], If[GreaterEqual[b, 0.0], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, 1.25e-84], If[GreaterEqual[b, 0.0], N[(N[(c / N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(-1.0 * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(-2.0 * b), $MachinePrecision]), $MachinePrecision], N[(-0.5 * t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -3.0000000000000001e157

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 43.4

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

   7. Applied rewrites 43.4%

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

   8. Taylor expanded in b around inf

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

      1. lower-*.f64 41.0

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

   10. Applied rewrites 41.0%

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

   11. Taylor expanded in b around -inf

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

       1. lower-*.f64 N/A

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

       2. lift-/.f64 67.6

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

   13. Applied rewrites 67.6%

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

   if -3.0000000000000001e157 < b < -6.4000000000000001e-288

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 70.1

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

   7. Applied rewrites 70.1%

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

   if -6.4000000000000001e-288 < b < 1.25e-84

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 43.4

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

   7. Applied rewrites 43.4%

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

   8. Taylor expanded in a around inf

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 14.3

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

   10. Applied rewrites 14.3%

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

   11. Taylor expanded in a around 0

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

       1. lower-sqrt.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 21.0

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

   13. Applied rewrites 21.0%

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

   if 1.25e-84 < b

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.0

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

   4. Applied rewrites 60.0%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.0

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

   7. Applied rewrites 48.0%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 31.3

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

   10. Applied rewrites 31.3%

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

   11. Taylor expanded in b around inf

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

       1. lower-*.f64 58.1

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

   13. Applied rewrites 58.1%

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

   14. Taylor expanded in a around -inf

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

       1. lower-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-sqrt.f64 41.0

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

   16. Applied rewrites 41.0%

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

Alternative 6: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{+157}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{2 \cdot b} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{b}{a}\right) \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-84}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-1 \cdot \frac{1}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{-2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e+157)
   (if (>= b 0.0) (* (/ c (* 2.0 b)) -2.0) (* (* -2.0 (/ b a)) 0.5))
   (if (<= b 1.25e-84)
     (if (>= b 0.0)
       (* (* -1.0 (/ 1.0 (* a (sqrt (/ -4.0 (* a c)))))) -2.0)
       (* (/ (- (sqrt (fma (* -4.0 a) c (* b b))) b) a) 0.5))
     (if (>= b 0.0)
       (/ (* 2.0 c) (* -2.0 b))
       (* -0.5 (sqrt (* -4.0 (/ c a))))))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -3e+157) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c / (2.0 * b)) * -2.0;
		} else {
			tmp_2 = (-2.0 * (b / a)) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.25e-84) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-1.0 * (1.0 / (a * sqrt((-4.0 / (a * c)))))) * -2.0;
		} else {
			tmp_3 = ((sqrt(fma((-4.0 * a), c, (b * b))) - b) / a) * 0.5;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-2.0 * b);
	} else {
		tmp_1 = -0.5 * sqrt((-4.0 * (c / a)));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -3e+157)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(c / Float64(2.0 * b)) * -2.0);
		else
			tmp_2 = Float64(Float64(-2.0 * Float64(b / a)) * 0.5);
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.25e-84)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-1.0 * Float64(1.0 / Float64(a * sqrt(Float64(-4.0 / Float64(a * c)))))) * -2.0);
		else
			tmp_3 = Float64(Float64(Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) - b) / a) * 0.5);
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(-2.0 * b));
	else
		tmp_1 = Float64(-0.5 * sqrt(Float64(-4.0 * Float64(c / a))));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3e+157], If[GreaterEqual[b, 0.0], N[(N[(c / N[(2.0 * b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(-2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, 1.25e-84], If[GreaterEqual[b, 0.0], N[(N[(-1.0 * N[(1.0 / N[(a * N[Sqrt[N[(-4.0 / N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(-2.0 * b), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.0000000000000001e157

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 43.4

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

   7. Applied rewrites 43.4%

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

   8. Taylor expanded in b around inf

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

      1. lower-*.f64 41.0

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

   10. Applied rewrites 41.0%

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

   11. Taylor expanded in b around -inf

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

       1. lower-*.f64 N/A

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

       2. lift-/.f64 67.6

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

   13. Applied rewrites 67.6%

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

   if -3.0000000000000001e157 < b < 1.25e-84

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 43.9

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

   7. Applied rewrites 43.9%

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

   8. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 51.2

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

   10. Applied rewrites 51.2%

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

   if 1.25e-84 < b

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.0

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

   4. Applied rewrites 60.0%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.0

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

   7. Applied rewrites 48.0%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 31.3

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

   10. Applied rewrites 31.3%

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

   11. Taylor expanded in b around inf

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

       1. lower-*.f64 58.1

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

   13. Applied rewrites 58.1%

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

   14. Taylor expanded in a around -inf

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

       1. lower-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-sqrt.f64 41.0

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

   16. Applied rewrites 41.0%

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

Alternative 7: 81.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b \leq -1.8 \cdot 10^{-73}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{2 \cdot b} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{b}{a}\right) \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-84}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{t\_0 + b} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{-2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* -4.0 (* a c)))))
   (if (<= b -1.8e-73)
     (if (>= b 0.0) (* (/ c (* 2.0 b)) -2.0) (* (* -2.0 (/ b a)) 0.5))
     (if (<= b 2.3e-84)
       (if (>= b 0.0) (* (/ c (+ t_0 b)) -2.0) (* (/ (- t_0 b) a) 0.5))
       (if (>= b 0.0)
         (/ (* 2.0 c) (* -2.0 b))
         (* -0.5 (sqrt (* -4.0 (/ c a)))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((-4.0 * (a * c)));
	double tmp_1;
	if (b <= -1.8e-73) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c / (2.0 * b)) * -2.0;
		} else {
			tmp_2 = (-2.0 * (b / a)) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= 2.3e-84) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c / (t_0 + b)) * -2.0;
		} else {
			tmp_3 = ((t_0 - b) / a) * 0.5;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-2.0 * b);
	} else {
		tmp_1 = -0.5 * sqrt((-4.0 * (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
    real(8) :: tmp_3
    t_0 = sqrt(((-4.0d0) * (a * c)))
    if (b <= (-1.8d-73)) then
        if (b >= 0.0d0) then
            tmp_2 = (c / (2.0d0 * b)) * (-2.0d0)
        else
            tmp_2 = ((-2.0d0) * (b / a)) * 0.5d0
        end if
        tmp_1 = tmp_2
    else if (b <= 2.3d-84) then
        if (b >= 0.0d0) then
            tmp_3 = (c / (t_0 + b)) * (-2.0d0)
        else
            tmp_3 = ((t_0 - b) / a) * 0.5d0
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = (2.0d0 * c) / ((-2.0d0) * b)
    else
        tmp_1 = (-0.5d0) * sqrt(((-4.0d0) * (c / a)))
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt((-4.0 * (a * c)));
	double tmp_1;
	if (b <= -1.8e-73) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c / (2.0 * b)) * -2.0;
		} else {
			tmp_2 = (-2.0 * (b / a)) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= 2.3e-84) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c / (t_0 + b)) * -2.0;
		} else {
			tmp_3 = ((t_0 - b) / a) * 0.5;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-2.0 * b);
	} else {
		tmp_1 = -0.5 * Math.sqrt((-4.0 * (c / a)));
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt((-4.0 * (a * c)))
	tmp_1 = 0
	if b <= -1.8e-73:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (c / (2.0 * b)) * -2.0
		else:
			tmp_2 = (-2.0 * (b / a)) * 0.5
		tmp_1 = tmp_2
	elif b <= 2.3e-84:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (c / (t_0 + b)) * -2.0
		else:
			tmp_3 = ((t_0 - b) / a) * 0.5
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = (2.0 * c) / (-2.0 * b)
	else:
		tmp_1 = -0.5 * math.sqrt((-4.0 * (c / a)))
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(-4.0 * Float64(a * c)))
	tmp_1 = 0.0
	if (b <= -1.8e-73)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(c / Float64(2.0 * b)) * -2.0);
		else
			tmp_2 = Float64(Float64(-2.0 * Float64(b / a)) * 0.5);
		end
		tmp_1 = tmp_2;
	elseif (b <= 2.3e-84)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(c / Float64(t_0 + b)) * -2.0);
		else
			tmp_3 = Float64(Float64(Float64(t_0 - b) / a) * 0.5);
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(-2.0 * b));
	else
		tmp_1 = Float64(-0.5 * sqrt(Float64(-4.0 * Float64(c / a))));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt((-4.0 * (a * c)));
	tmp_2 = 0.0;
	if (b <= -1.8e-73)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (c / (2.0 * b)) * -2.0;
		else
			tmp_3 = (-2.0 * (b / a)) * 0.5;
		end
		tmp_2 = tmp_3;
	elseif (b <= 2.3e-84)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (c / (t_0 + b)) * -2.0;
		else
			tmp_4 = ((t_0 - b) / a) * 0.5;
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = (2.0 * c) / (-2.0 * b);
	else
		tmp_2 = -0.5 * sqrt((-4.0 * (c / a)));
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.8e-73], If[GreaterEqual[b, 0.0], N[(N[(c / N[(2.0 * b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(-2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, 2.3e-84], If[GreaterEqual[b, 0.0], N[(N[(c / N[(t$95$0 + b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(t$95$0 - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(-2.0 * b), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.8e-73

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 43.4

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

   7. Applied rewrites 43.4%

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

   8. Taylor expanded in b around inf

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

      1. lower-*.f64 41.0

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

   10. Applied rewrites 41.0%

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

   11. Taylor expanded in b around -inf

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

       1. lower-*.f64 N/A

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

       2. lift-/.f64 67.6

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

   13. Applied rewrites 67.6%

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

   if -1.8e-73 < b < 2.29999999999999981e-84

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.9

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

   7. Applied rewrites 56.9%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 40.7

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

   10. Applied rewrites 40.7%

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

   if 2.29999999999999981e-84 < b

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.0

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

   4. Applied rewrites 60.0%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.0

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

   7. Applied rewrites 48.0%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 31.3

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

   10. Applied rewrites 31.3%

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

   11. Taylor expanded in b around inf

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

       1. lower-*.f64 58.1

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

   13. Applied rewrites 58.1%

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

   14. Taylor expanded in a around -inf

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

       1. lower-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-sqrt.f64 41.0

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

   16. Applied rewrites 41.0%

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

Alternative 8: 80.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-4 \cdot \frac{c}{a}}\\ t_1 := \frac{c}{2 \cdot b} \cdot -2\\ t_2 := \sqrt{-4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b \leq -5.6 \cdot 10^{-83}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{b}{a}\right) \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq -6.4 \cdot 10^{-288}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{t\_2}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-84}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{t\_2} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot t\_0\right) \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{-2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* -4.0 (/ c a))))
        (t_1 (* (/ c (* 2.0 b)) -2.0))
        (t_2 (sqrt (* -4.0 (* a c)))))
   (if (<= b -5.6e-83)
     (if (>= b 0.0) t_1 (* (* -2.0 (/ b a)) 0.5))
     (if (<= b -6.4e-288)
       (if (>= b 0.0) t_1 (* 0.5 (/ t_2 a)))
       (if (<= b 1.25e-84)
         (if (>= b 0.0) (* (/ c t_2) -2.0) (* (* -1.0 t_0) 0.5))
         (if (>= b 0.0) (/ (* 2.0 c) (* -2.0 b)) (* -0.5 t_0)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((-4.0 * (c / a)));
	double t_1 = (c / (2.0 * b)) * -2.0;
	double t_2 = sqrt((-4.0 * (a * c)));
	double tmp_1;
	if (b <= -5.6e-83) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = (-2.0 * (b / a)) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= -6.4e-288) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = 0.5 * (t_2 / a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.25e-84) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c / t_2) * -2.0;
		} else {
			tmp_4 = (-1.0 * t_0) * 0.5;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-2.0 * b);
	} else {
		tmp_1 = -0.5 * t_0;
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    real(8) :: tmp_4
    t_0 = sqrt(((-4.0d0) * (c / a)))
    t_1 = (c / (2.0d0 * b)) * (-2.0d0)
    t_2 = sqrt(((-4.0d0) * (a * c)))
    if (b <= (-5.6d-83)) then
        if (b >= 0.0d0) then
            tmp_2 = t_1
        else
            tmp_2 = ((-2.0d0) * (b / a)) * 0.5d0
        end if
        tmp_1 = tmp_2
    else if (b <= (-6.4d-288)) then
        if (b >= 0.0d0) then
            tmp_3 = t_1
        else
            tmp_3 = 0.5d0 * (t_2 / a)
        end if
        tmp_1 = tmp_3
    else if (b <= 1.25d-84) then
        if (b >= 0.0d0) then
            tmp_4 = (c / t_2) * (-2.0d0)
        else
            tmp_4 = ((-1.0d0) * t_0) * 0.5d0
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = (2.0d0 * c) / ((-2.0d0) * b)
    else
        tmp_1 = (-0.5d0) * t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt((-4.0 * (c / a)));
	double t_1 = (c / (2.0 * b)) * -2.0;
	double t_2 = Math.sqrt((-4.0 * (a * c)));
	double tmp_1;
	if (b <= -5.6e-83) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = (-2.0 * (b / a)) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= -6.4e-288) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = 0.5 * (t_2 / a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.25e-84) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (c / t_2) * -2.0;
		} else {
			tmp_4 = (-1.0 * t_0) * 0.5;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-2.0 * b);
	} else {
		tmp_1 = -0.5 * t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt((-4.0 * (c / a)))
	t_1 = (c / (2.0 * b)) * -2.0
	t_2 = math.sqrt((-4.0 * (a * c)))
	tmp_1 = 0
	if b <= -5.6e-83:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_1
		else:
			tmp_2 = (-2.0 * (b / a)) * 0.5
		tmp_1 = tmp_2
	elif b <= -6.4e-288:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = t_1
		else:
			tmp_3 = 0.5 * (t_2 / a)
		tmp_1 = tmp_3
	elif b <= 1.25e-84:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = (c / t_2) * -2.0
		else:
			tmp_4 = (-1.0 * t_0) * 0.5
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = (2.0 * c) / (-2.0 * b)
	else:
		tmp_1 = -0.5 * t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(-4.0 * Float64(c / a)))
	t_1 = Float64(Float64(c / Float64(2.0 * b)) * -2.0)
	t_2 = sqrt(Float64(-4.0 * Float64(a * c)))
	tmp_1 = 0.0
	if (b <= -5.6e-83)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(Float64(-2.0 * Float64(b / a)) * 0.5);
		end
		tmp_1 = tmp_2;
	elseif (b <= -6.4e-288)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = Float64(0.5 * Float64(t_2 / a));
		end
		tmp_1 = tmp_3;
	elseif (b <= 1.25e-84)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(c / t_2) * -2.0);
		else
			tmp_4 = Float64(Float64(-1.0 * t_0) * 0.5);
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(-2.0 * b));
	else
		tmp_1 = Float64(-0.5 * t_0);
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = sqrt((-4.0 * (c / a)));
	t_1 = (c / (2.0 * b)) * -2.0;
	t_2 = sqrt((-4.0 * (a * c)));
	tmp_2 = 0.0;
	if (b <= -5.6e-83)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = (-2.0 * (b / a)) * 0.5;
		end
		tmp_2 = tmp_3;
	elseif (b <= -6.4e-288)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = t_1;
		else
			tmp_4 = 0.5 * (t_2 / a);
		end
		tmp_2 = tmp_4;
	elseif (b <= 1.25e-84)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = (c / t_2) * -2.0;
		else
			tmp_5 = (-1.0 * t_0) * 0.5;
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = (2.0 * c) / (-2.0 * b);
	else
		tmp_2 = -0.5 * t_0;
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(c / N[(2.0 * b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -5.6e-83], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(-2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, -6.4e-288], If[GreaterEqual[b, 0.0], t$95$1, N[(0.5 * N[(t$95$2 / a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.25e-84], If[GreaterEqual[b, 0.0], N[(N[(c / t$95$2), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(-1.0 * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(-2.0 * b), $MachinePrecision]), $MachinePrecision], N[(-0.5 * t$95$0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -5.6000000000000002e-83

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 43.4

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

   7. Applied rewrites 43.4%

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

   8. Taylor expanded in b around inf

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

      1. lower-*.f64 41.0

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

   10. Applied rewrites 41.0%

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

   11. Taylor expanded in b around -inf

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

       1. lower-*.f64 N/A

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

       2. lift-/.f64 67.6

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

   13. Applied rewrites 67.6%

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

   if -5.6000000000000002e-83 < b < -6.4000000000000001e-288

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 43.4

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

   7. Applied rewrites 43.4%

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

   8. Taylor expanded in b around inf

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

      1. lower-*.f64 41.0

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

   10. Applied rewrites 41.0%

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

   11. Taylor expanded in b around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 46.8

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

   13. Applied rewrites 46.8%

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

   if -6.4000000000000001e-288 < b < 1.25e-84

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 43.4

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

   7. Applied rewrites 43.4%

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

   8. Taylor expanded in a around inf

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 14.3

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

   10. Applied rewrites 14.3%

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

   11. Taylor expanded in a around 0

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

       1. lower-sqrt.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 21.0

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

   13. Applied rewrites 21.0%

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

   if 1.25e-84 < b

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.0

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

   4. Applied rewrites 60.0%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.0

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

   7. Applied rewrites 48.0%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 31.3

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

   10. Applied rewrites 31.3%

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

   11. Taylor expanded in b around inf

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

       1. lower-*.f64 58.1

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

   13. Applied rewrites 58.1%

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

   14. Taylor expanded in a around -inf

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

       1. lower-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-sqrt.f64 41.0

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

   16. Applied rewrites 41.0%

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

Alternative 9: 76.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{if}\;b \leq -2.8 \cdot 10^{-110}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{2 \cdot b} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{b}{a}\right) \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-84}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot t\_0\right) \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{-2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* -4.0 (/ c a)))))
   (if (<= b -2.8e-110)
     (if (>= b 0.0) (* (/ c (* 2.0 b)) -2.0) (* (* -2.0 (/ b a)) 0.5))
     (if (<= b 1.25e-84)
       (if (>= b 0.0)
         (* (/ c (sqrt (* -4.0 (* a c)))) -2.0)
         (* (* -1.0 t_0) 0.5))
       (if (>= b 0.0) (/ (* 2.0 c) (* -2.0 b)) (* -0.5 t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((-4.0 * (c / a)));
	double tmp_1;
	if (b <= -2.8e-110) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c / (2.0 * b)) * -2.0;
		} else {
			tmp_2 = (-2.0 * (b / a)) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.25e-84) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c / sqrt((-4.0 * (a * c)))) * -2.0;
		} else {
			tmp_3 = (-1.0 * t_0) * 0.5;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-2.0 * b);
	} else {
		tmp_1 = -0.5 * t_0;
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = sqrt(((-4.0d0) * (c / a)))
    if (b <= (-2.8d-110)) then
        if (b >= 0.0d0) then
            tmp_2 = (c / (2.0d0 * b)) * (-2.0d0)
        else
            tmp_2 = ((-2.0d0) * (b / a)) * 0.5d0
        end if
        tmp_1 = tmp_2
    else if (b <= 1.25d-84) then
        if (b >= 0.0d0) then
            tmp_3 = (c / sqrt(((-4.0d0) * (a * c)))) * (-2.0d0)
        else
            tmp_3 = ((-1.0d0) * t_0) * 0.5d0
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = (2.0d0 * c) / ((-2.0d0) * b)
    else
        tmp_1 = (-0.5d0) * t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt((-4.0 * (c / a)));
	double tmp_1;
	if (b <= -2.8e-110) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c / (2.0 * b)) * -2.0;
		} else {
			tmp_2 = (-2.0 * (b / a)) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.25e-84) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c / Math.sqrt((-4.0 * (a * c)))) * -2.0;
		} else {
			tmp_3 = (-1.0 * t_0) * 0.5;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-2.0 * b);
	} else {
		tmp_1 = -0.5 * t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt((-4.0 * (c / a)))
	tmp_1 = 0
	if b <= -2.8e-110:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (c / (2.0 * b)) * -2.0
		else:
			tmp_2 = (-2.0 * (b / a)) * 0.5
		tmp_1 = tmp_2
	elif b <= 1.25e-84:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (c / math.sqrt((-4.0 * (a * c)))) * -2.0
		else:
			tmp_3 = (-1.0 * t_0) * 0.5
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = (2.0 * c) / (-2.0 * b)
	else:
		tmp_1 = -0.5 * t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(-4.0 * Float64(c / a)))
	tmp_1 = 0.0
	if (b <= -2.8e-110)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(c / Float64(2.0 * b)) * -2.0);
		else
			tmp_2 = Float64(Float64(-2.0 * Float64(b / a)) * 0.5);
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.25e-84)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(c / sqrt(Float64(-4.0 * Float64(a * c)))) * -2.0);
		else
			tmp_3 = Float64(Float64(-1.0 * t_0) * 0.5);
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(-2.0 * b));
	else
		tmp_1 = Float64(-0.5 * t_0);
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt((-4.0 * (c / a)));
	tmp_2 = 0.0;
	if (b <= -2.8e-110)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (c / (2.0 * b)) * -2.0;
		else
			tmp_3 = (-2.0 * (b / a)) * 0.5;
		end
		tmp_2 = tmp_3;
	elseif (b <= 1.25e-84)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (c / sqrt((-4.0 * (a * c)))) * -2.0;
		else
			tmp_4 = (-1.0 * t_0) * 0.5;
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = (2.0 * c) / (-2.0 * b);
	else
		tmp_2 = -0.5 * t_0;
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2.8e-110], If[GreaterEqual[b, 0.0], N[(N[(c / N[(2.0 * b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(-2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, 1.25e-84], If[GreaterEqual[b, 0.0], N[(N[(c / N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(-1.0 * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(-2.0 * b), $MachinePrecision]), $MachinePrecision], N[(-0.5 * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.8e-110

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 43.4

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

   7. Applied rewrites 43.4%

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

   8. Taylor expanded in b around inf

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

      1. lower-*.f64 41.0

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

   10. Applied rewrites 41.0%

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

   11. Taylor expanded in b around -inf

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

       1. lower-*.f64 N/A

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

       2. lift-/.f64 67.6

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

   13. Applied rewrites 67.6%

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

   if -2.8e-110 < b < 1.25e-84

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 43.4

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

   7. Applied rewrites 43.4%

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

   8. Taylor expanded in a around inf

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 14.3

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

   10. Applied rewrites 14.3%

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

   11. Taylor expanded in a around 0

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

       1. lower-sqrt.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 21.0

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

   13. Applied rewrites 21.0%

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

   if 1.25e-84 < b

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.0

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

   4. Applied rewrites 60.0%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.0

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

   7. Applied rewrites 48.0%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 31.3

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

   10. Applied rewrites 31.3%

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

   11. Taylor expanded in b around inf

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

       1. lower-*.f64 58.1

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

   13. Applied rewrites 58.1%

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

   14. Taylor expanded in a around -inf

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

       1. lower-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-sqrt.f64 41.0

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

   16. Applied rewrites 41.0%

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

Alternative 10: 69.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{-110}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{2 \cdot b} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{b}{a}\right) \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{-2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.8e-110)
   (if (>= b 0.0) (* (/ c (* 2.0 b)) -2.0) (* (* -2.0 (/ b a)) 0.5))
   (if (>= b 0.0) (/ (* 2.0 c) (* -2.0 b)) (* -0.5 (sqrt (* -4.0 (/ c a)))))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -2.8e-110) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c / (2.0 * b)) * -2.0;
		} else {
			tmp_2 = (-2.0 * (b / a)) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-2.0 * b);
	} else {
		tmp_1 = -0.5 * sqrt((-4.0 * (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) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    if (b <= (-2.8d-110)) then
        if (b >= 0.0d0) then
            tmp_2 = (c / (2.0d0 * b)) * (-2.0d0)
        else
            tmp_2 = ((-2.0d0) * (b / a)) * 0.5d0
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (2.0d0 * c) / ((-2.0d0) * b)
    else
        tmp_1 = (-0.5d0) * sqrt(((-4.0d0) * (c / a)))
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -2.8e-110) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c / (2.0 * b)) * -2.0;
		} else {
			tmp_2 = (-2.0 * (b / a)) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-2.0 * b);
	} else {
		tmp_1 = -0.5 * Math.sqrt((-4.0 * (c / a)));
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if b <= -2.8e-110:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (c / (2.0 * b)) * -2.0
		else:
			tmp_2 = (-2.0 * (b / a)) * 0.5
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = (2.0 * c) / (-2.0 * b)
	else:
		tmp_1 = -0.5 * math.sqrt((-4.0 * (c / a)))
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -2.8e-110)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(c / Float64(2.0 * b)) * -2.0);
		else
			tmp_2 = Float64(Float64(-2.0 * Float64(b / a)) * 0.5);
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(-2.0 * b));
	else
		tmp_1 = Float64(-0.5 * sqrt(Float64(-4.0 * Float64(c / a))));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp_2 = 0.0;
	if (b <= -2.8e-110)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (c / (2.0 * b)) * -2.0;
		else
			tmp_3 = (-2.0 * (b / a)) * 0.5;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = (2.0 * c) / (-2.0 * b);
	else
		tmp_2 = -0.5 * sqrt((-4.0 * (c / a)));
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.8e-110], If[GreaterEqual[b, 0.0], N[(N[(c / N[(2.0 * b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(-2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(-2.0 * b), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.8e-110

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 43.4

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

   7. Applied rewrites 43.4%

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

   8. Taylor expanded in b around inf

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

      1. lower-*.f64 41.0

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

   10. Applied rewrites 41.0%

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

   11. Taylor expanded in b around -inf

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

       1. lower-*.f64 N/A

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

       2. lift-/.f64 67.6

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

   13. Applied rewrites 67.6%

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

   if -2.8e-110 < b

   1. Initial program 72.4%

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

   2. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.0

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

   4. Applied rewrites 60.0%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.0

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

   7. Applied rewrites 48.0%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 31.3

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

   10. Applied rewrites 31.3%

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

   11. Taylor expanded in b around inf

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

       1. lower-*.f64 58.1

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

   13. Applied rewrites 58.1%

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

   14. Taylor expanded in a around -inf

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

       1. lower-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-sqrt.f64 41.0

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

   16. Applied rewrites 41.0%

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

Alternative 11: 67.6% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0) (* (/ c (* 2.0 b)) -2.0) (* (* -2.0 (/ b a)) 0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (c / (2.0 * b)) * -2.0;
	} else {
		tmp = (-2.0 * (b / a)) * 0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b >= 0.0d0) then
        tmp = (c / (2.0d0 * b)) * (-2.0d0)
    else
        tmp = ((-2.0d0) * (b / a)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (c / (2.0 * b)) * -2.0;
	} else {
		tmp = (-2.0 * (b / a)) * 0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = (c / (2.0 * b)) * -2.0
	else:
		tmp = (-2.0 * (b / a)) * 0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(c / Float64(2.0 * b)) * -2.0);
	else
		tmp = Float64(Float64(-2.0 * Float64(b / a)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (c / (2.0 * b)) * -2.0;
	else
		tmp = (-2.0 * (b / a)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Initial program 72.4%

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

2. Taylor expanded in a around 0

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

4. Applied rewrites 72.5%

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

5. Taylor expanded in a around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lift-/.f64 43.4

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

7. Applied rewrites 43.4%

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

8. Taylor expanded in b around inf

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

   1. lower-*.f64 41.0

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

10. Applied rewrites 41.0%

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

11. Taylor expanded in b around -inf

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

    1. lower-*.f64 N/A

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

    2. lift-/.f64 67.6

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

13. Applied rewrites 67.6%

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

Reproduce

herbie shell --seed 2025136 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))