jeff quadratic root 2

Percentage Accurate: 72.1% → 89.9%

Time: 5.6s

Alternatives: 10

Speedup: 1.2×

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: 72.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ t_1 := \frac{-2 \cdot c}{t\_0 + b}\\ \mathbf{if}\;b \leq -1.72 \cdot 10^{+93}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b}{a}, 2 \cdot \frac{c}{b}\right) \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+22}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 a) c (* b b)))) (t_1 (/ (* -2.0 c) (+ t_0 b))))
   (if (<= b -1.72e+93)
     (if (>= b 0.0) t_1 (* (fma -2.0 (/ b a) (* 2.0 (/ c b))) 0.5))
     (if (<= b 3.1e+22)
       (if (>= b 0.0) t_1 (* (/ (- t_0 b) a) 0.5))
       (if (>= b 0.0) (/ (* -2.0 c) (+ b b)) (/ (- b) a))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-4.0 * a), c, (b * b)));
	double t_1 = (-2.0 * c) / (t_0 + b);
	double tmp_1;
	if (b <= -1.72e+93) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = fma(-2.0, (b / a), (2.0 * (c / b))) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= 3.1e+22) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = ((t_0 - b) / a) * 0.5;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * c) / (b + b);
	} else {
		tmp_1 = -b / a;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))
	t_1 = Float64(Float64(-2.0 * c) / Float64(t_0 + b))
	tmp_1 = 0.0
	if (b <= -1.72e+93)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(fma(-2.0, Float64(b / a), Float64(2.0 * Float64(c / b))) * 0.5);
		end
		tmp_1 = tmp_2;
	elseif (b <= 3.1e+22)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_1;
		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(b + b));
	else
		tmp_1 = Float64(Float64(-b) / 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]}, Block[{t$95$1 = N[(N[(-2.0 * c), $MachinePrecision] / N[(t$95$0 + b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.72e+93], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(-2.0 * N[(b / a), $MachinePrecision] + N[(2.0 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, 3.1e+22], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(N[(t$95$0 - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * c), $MachinePrecision] / N[(b + b), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.7199999999999999e93

   1. Initial program 44.5%

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

   3. 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. Step-by-step derivation

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 97.5

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

   8. Applied rewrites 97.5%

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

   9. Taylor expanded in a around inf

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

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 98.3

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

   11. Applied rewrites 98.3%

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

   if -1.7199999999999999e93 < b < 3.1000000000000002e22

   1. Initial program 85.6%

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

   3. 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. Step-by-step derivation

   5. Applied rewrites 85.6%

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

   if 3.1000000000000002e22 < b

   1. Initial program 57.0%

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

   3. 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. Step-by-step derivation

   5. Applied rewrites 57.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 91.6%

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

   9. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. distribute-frac-neg N/A

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

       3. lift-/.f64 N/A

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

       4. lift-neg.f64 91.6

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

   11. Applied rewrites 91.6%

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

Alternative 2: 89.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ t_1 := \frac{-2 \cdot c}{t\_0 + b}\\ \mathbf{if}\;b \leq -1.72 \cdot 10^{+93}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b}{a}, 2 \cdot \frac{c}{b}\right) \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{\frac{c}{a} \cdot -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+22}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-b\right) \cdot \mathsf{fma}\left(-2, \frac{c}{b \cdot b}, \frac{2}{a}\right)\right) \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 a) c (* b b)))) (t_1 (/ (* -2.0 c) (+ t_0 b))))
   (if (<= b -1.72e+93)
     (if (>= b 0.0) t_1 (* (fma -2.0 (/ b a) (* 2.0 (/ c b))) 0.5))
     (if (<= b -4e-310)
       (if (>= b 0.0) (sqrt (* (/ c a) -1.0)) (* (/ (- t_0 b) a) 0.5))
       (if (<= b 3.1e+22)
         (if (>= b 0.0)
           t_1
           (* (* (- b) (fma -2.0 (/ c (* b b)) (/ 2.0 a))) 0.5))
         (if (>= b 0.0) (/ (* -2.0 c) (+ b b)) (/ (- b) a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-4.0 * a), c, (b * b)));
	double t_1 = (-2.0 * c) / (t_0 + b);
	double tmp_1;
	if (b <= -1.72e+93) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = fma(-2.0, (b / a), (2.0 * (c / b))) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= -4e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = sqrt(((c / a) * -1.0));
		} else {
			tmp_3 = ((t_0 - b) / a) * 0.5;
		}
		tmp_1 = tmp_3;
	} else if (b <= 3.1e+22) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = t_1;
		} else {
			tmp_4 = (-b * fma(-2.0, (c / (b * b)), (2.0 / a))) * 0.5;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * c) / (b + b);
	} else {
		tmp_1 = -b / a;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))
	t_1 = Float64(Float64(-2.0 * c) / Float64(t_0 + b))
	tmp_1 = 0.0
	if (b <= -1.72e+93)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(fma(-2.0, Float64(b / a), Float64(2.0 * Float64(c / b))) * 0.5);
		end
		tmp_1 = tmp_2;
	elseif (b <= -4e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = sqrt(Float64(Float64(c / a) * -1.0));
		else
			tmp_3 = Float64(Float64(Float64(t_0 - b) / a) * 0.5);
		end
		tmp_1 = tmp_3;
	elseif (b <= 3.1e+22)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = t_1;
		else
			tmp_4 = Float64(Float64(Float64(-b) * fma(-2.0, Float64(c / Float64(b * b)), Float64(2.0 / a))) * 0.5);
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-2.0 * c) / Float64(b + b));
	else
		tmp_1 = Float64(Float64(-b) / 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]}, Block[{t$95$1 = N[(N[(-2.0 * c), $MachinePrecision] / N[(t$95$0 + b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.72e+93], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(-2.0 * N[(b / a), $MachinePrecision] + N[(2.0 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, -4e-310], If[GreaterEqual[b, 0.0], N[Sqrt[N[(N[(c / a), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision], N[(N[(N[(t$95$0 - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, 3.1e+22], If[GreaterEqual[b, 0.0], t$95$1, N[(N[((-b) * N[(-2.0 * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(2.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * c), $MachinePrecision] / N[(b + b), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.7199999999999999e93

   1. Initial program 44.5%

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

   3. 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. Step-by-step derivation

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 97.5

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

   8. Applied rewrites 97.5%

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

   9. Taylor expanded in a around inf

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

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 98.3

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

   11. Applied rewrites 98.3%

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

   if -1.7199999999999999e93 < b < -3.999999999999988e-310

   1. Initial program 86.2%

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

   3. 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. Step-by-step derivation

   5. Applied rewrites 86.2%

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

   6. Taylor expanded in a around -inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 86.2

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

   8. Applied rewrites 86.2%

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

   if -3.999999999999988e-310 < b < 3.1000000000000002e22

   1. Initial program 85.0%

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

   3. 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. Step-by-step derivation

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 85.0

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

   8. Applied rewrites 85.0%

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

   if 3.1000000000000002e22 < b

   1. Initial program 57.0%

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

   3. 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. Step-by-step derivation

   5. Applied rewrites 57.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 91.6%

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

   9. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. distribute-frac-neg N/A

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

       3. lift-/.f64 N/A

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

       4. lift-neg.f64 91.6

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

   11. Applied rewrites 91.6%

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

Alternative 3: 89.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ t_1 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{t\_0 + b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b}{a}, 2 \cdot \frac{c}{b}\right) \cdot 0.5\\ \end{array}\\ \mathbf{if}\;b \leq -1.72 \cdot 10^{+93}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{\frac{c}{a} \cdot -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 a) c (* b b))))
        (t_1
         (if (>= b 0.0)
           (/ (* -2.0 c) (+ t_0 b))
           (* (fma -2.0 (/ b a) (* 2.0 (/ c b))) 0.5))))
   (if (<= b -1.72e+93)
     t_1
     (if (<= b -4e-310)
       (if (>= b 0.0) (sqrt (* (/ c a) -1.0)) (* (/ (- t_0 b) a) 0.5))
       (if (<= b 3.1e+22)
         t_1
         (if (>= b 0.0) (/ (* -2.0 c) (+ b b)) (/ (- b) a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-4.0 * a), c, (b * b)));
	double tmp;
	if (b >= 0.0) {
		tmp = (-2.0 * c) / (t_0 + b);
	} else {
		tmp = fma(-2.0, (b / a), (2.0 * (c / b))) * 0.5;
	}
	double t_1 = tmp;
	double tmp_1;
	if (b <= -1.72e+93) {
		tmp_1 = t_1;
	} else if (b <= -4e-310) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = sqrt(((c / a) * -1.0));
		} else {
			tmp_2 = ((t_0 - b) / a) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= 3.1e+22) {
		tmp_1 = t_1;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * c) / (b + b);
	} else {
		tmp_1 = -b / a;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(-2.0 * c) / Float64(t_0 + b));
	else
		tmp = Float64(fma(-2.0, Float64(b / a), Float64(2.0 * Float64(c / b))) * 0.5);
	end
	t_1 = tmp
	tmp_1 = 0.0
	if (b <= -1.72e+93)
		tmp_1 = t_1;
	elseif (b <= -4e-310)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = sqrt(Float64(Float64(c / a) * -1.0));
		else
			tmp_2 = Float64(Float64(Float64(t_0 - b) / a) * 0.5);
		end
		tmp_1 = tmp_2;
	elseif (b <= 3.1e+22)
		tmp_1 = t_1;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-2.0 * c) / Float64(b + b));
	else
		tmp_1 = Float64(Float64(-b) / 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]}, Block[{t$95$1 = If[GreaterEqual[b, 0.0], N[(N[(-2.0 * c), $MachinePrecision] / N[(t$95$0 + b), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(b / a), $MachinePrecision] + N[(2.0 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]}, If[LessEqual[b, -1.72e+93], t$95$1, If[LessEqual[b, -4e-310], If[GreaterEqual[b, 0.0], N[Sqrt[N[(N[(c / a), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision], N[(N[(N[(t$95$0 - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, 3.1e+22], t$95$1, If[GreaterEqual[b, 0.0], N[(N[(-2.0 * c), $MachinePrecision] / N[(b + b), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.7199999999999999e93 or -3.999999999999988e-310 < b < 3.1000000000000002e22

   1. Initial program 68.9%

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

   3. 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. Step-by-step derivation

   5. Applied rewrites 69.0%

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

   6. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 90.0

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

   8. Applied rewrites 90.0%

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

   9. Taylor expanded in a around inf

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

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 90.3

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

   11. Applied rewrites 90.3%

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

   if -1.7199999999999999e93 < b < -3.999999999999988e-310

   1. Initial program 86.2%

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

   3. 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. Step-by-step derivation

   5. Applied rewrites 86.2%

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

   6. Taylor expanded in a around -inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 86.2

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

   8. Applied rewrites 86.2%

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

   if 3.1000000000000002e22 < b

   1. Initial program 57.0%

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

   3. 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. Step-by-step derivation

   5. Applied rewrites 57.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 91.6%

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

   9. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. distribute-frac-neg N/A

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

       3. lift-/.f64 N/A

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

       4. lift-neg.f64 91.6

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

   11. Applied rewrites 91.6%

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

Alternative 4: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{c}{a} \cdot -1}\\ t_1 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\\ \mathbf{if}\;b \leq -5 \cdot 10^{+153}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -1.12 \cdot 10^{-302}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{-66}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-\sqrt{\left(-a\right) \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;-t\_0\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* (/ c a) -1.0)))
        (t_1 (if (>= b 0.0) (/ (* -2.0 c) (+ b b)) (/ (- b) a))))
   (if (<= b -5e+153)
     t_1
     (if (<= b -1.12e-302)
       (if (>= b 0.0)
         t_0
         (* (/ (- (sqrt (fma (* -4.0 a) c (* b b))) b) a) 0.5))
       (if (<= b 4.6e-66)
         (if (>= b 0.0) (/ (- (sqrt (* (- a) c))) (- a)) (- t_0))
         t_1)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((c / a) * -1.0));
	double tmp;
	if (b >= 0.0) {
		tmp = (-2.0 * c) / (b + b);
	} else {
		tmp = -b / a;
	}
	double t_1 = tmp;
	double tmp_1;
	if (b <= -5e+153) {
		tmp_1 = t_1;
	} else if (b <= -1.12e-302) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = ((sqrt(fma((-4.0 * a), c, (b * b))) - b) / a) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= 4.6e-66) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -sqrt((-a * c)) / -a;
		} else {
			tmp_3 = -t_0;
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(c / a) * -1.0))
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(-2.0 * c) / Float64(b + b));
	else
		tmp = Float64(Float64(-b) / a);
	end
	t_1 = tmp
	tmp_1 = 0.0
	if (b <= -5e+153)
		tmp_1 = t_1;
	elseif (b <= -1.12e-302)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(Float64(Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) - b) / a) * 0.5);
		end
		tmp_1 = tmp_2;
	elseif (b <= 4.6e-66)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-sqrt(Float64(Float64(-a) * c))) / Float64(-a));
		else
			tmp_3 = Float64(-t_0);
		end
		tmp_1 = tmp_3;
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(c / a), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = If[GreaterEqual[b, 0.0], N[(N[(-2.0 * c), $MachinePrecision] / N[(b + b), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]}, If[LessEqual[b, -5e+153], t$95$1, If[LessEqual[b, -1.12e-302], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, 4.6e-66], If[GreaterEqual[b, 0.0], N[((-N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision]) / (-a)), $MachinePrecision], (-t$95$0)], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.00000000000000018e153 or 4.59999999999999984e-66 < b

   1. Initial program 56.0%

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

   3. 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. Step-by-step derivation

   5. Applied rewrites 56.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 70.4%

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

   9. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. distribute-frac-neg N/A

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

       3. lift-/.f64 N/A

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

       4. lift-neg.f64 87.0

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

   11. Applied rewrites 87.0%

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

   if -5.00000000000000018e153 < b < -1.12e-302

   1. Initial program 87.2%

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

   3. 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. Step-by-step derivation

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in a around -inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 87.2

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

   8. Applied rewrites 87.2%

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

   if -1.12e-302 < b < 4.59999999999999984e-66

   1. Initial program 84.3%

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

   3. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. sqrt-unprod N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 73.2

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

   5. Applied rewrites 73.2%

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

   6. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 73.4

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

   8. Applied rewrites 73.4%

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

   9. Taylor expanded in a around inf

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

       1. mul-1-neg N/A

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

       2. sqrt-prod N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-neg.f64 73.4

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

   11. Applied rewrites 73.4%

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

   12. Taylor expanded in a around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lift-*.f64 73.4

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

   14. Applied rewrites 73.4%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq -1.12 \cdot 10^{-302}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{\frac{c}{a} \cdot -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{-66}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-\sqrt{\left(-a\right) \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{c}{a} \cdot -1}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1300000 \lor \neg \left(b \leq 4.6 \cdot 10^{-66}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, b, -\sqrt{\left(a \cdot c\right) \cdot -1}\right)}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left(-4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (or (<= b -1300000.0) (not (<= b 4.6e-66)))
   (if (>= b 0.0) (/ (* -2.0 c) (+ b b)) (/ (- b) a))
   (if (>= b 0.0)
     (/ (fma 0.5 b (- (sqrt (* (* a c) -1.0)))) (- a))
     (/ (+ (- b) (sqrt (* (* -4.0 a) c))) (* 2.0 a)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if ((b <= -1300000.0) || !(b <= 4.6e-66)) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-2.0 * c) / (b + b);
		} else {
			tmp_2 = -b / a;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = fma(0.5, b, -sqrt(((a * c) * -1.0))) / -a;
	} else {
		tmp_1 = (-b + sqrt(((-4.0 * a) * c))) / (2.0 * a);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if ((b <= -1300000.0) || !(b <= 4.6e-66))
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-2.0 * c) / Float64(b + b));
		else
			tmp_2 = Float64(Float64(-b) / a);
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(fma(0.5, b, Float64(-sqrt(Float64(Float64(a * c) * -1.0)))) / Float64(-a));
	else
		tmp_1 = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(-4.0 * a) * c))) / Float64(2.0 * a));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[Or[LessEqual[b, -1300000.0], N[Not[LessEqual[b, 4.6e-66]], $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * c), $MachinePrecision] / N[(b + b), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(0.5 * b + (-N[Sqrt[N[(N[(a * c), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / (-a)), $MachinePrecision], N[(N[((-b) + N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.3e6 or 4.59999999999999984e-66 < b

   1. Initial program 61.2%

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

   3. 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. Step-by-step derivation

   5. Applied rewrites 61.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 73.7%

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

   9. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. distribute-frac-neg N/A

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

       3. lift-/.f64 N/A

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

       4. lift-neg.f64 86.7

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

   11. Applied rewrites 86.7%

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

   if -1.3e6 < b < 4.59999999999999984e-66

   1. Initial program 84.2%

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

   3. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. sqrt-unprod N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 79.2

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lift-*.f64 65.8

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

   8. Applied rewrites 65.8%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1300000 \lor \neg \left(b \leq 4.6 \cdot 10^{-66}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, b, -\sqrt{\left(a \cdot c\right) \cdot -1}\right)}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left(-4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-2 \cdot c}{b + b}\\ t_1 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\\ \mathbf{if}\;b \leq -1.1 \cdot 10^{-97}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -1.12 \cdot 10^{-302}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4}}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{-66}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-\sqrt{\left(-a\right) \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{c}{a} \cdot -1}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* -2.0 c) (+ b b))) (t_1 (if (>= b 0.0) t_0 (/ (- b) a))))
   (if (<= b -1.1e-97)
     t_1
     (if (<= b -1.12e-302)
       (if (>= b 0.0) t_0 (* (/ (sqrt (* (* a c) -4.0)) a) 0.5))
       (if (<= b 4.6e-66)
         (if (>= b 0.0)
           (/ (- (sqrt (* (- a) c))) (- a))
           (- (sqrt (* (/ c a) -1.0))))
         t_1)))))

C

double code(double a, double b, double c) {
	double t_0 = (-2.0 * c) / (b + b);
	double tmp;
	if (b >= 0.0) {
		tmp = t_0;
	} else {
		tmp = -b / a;
	}
	double t_1 = tmp;
	double tmp_1;
	if (b <= -1.1e-97) {
		tmp_1 = t_1;
	} else if (b <= -1.12e-302) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = (sqrt(((a * c) * -4.0)) / a) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= 4.6e-66) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -sqrt((-a * c)) / -a;
		} else {
			tmp_3 = -sqrt(((c / a) * -1.0));
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = ((-2.0d0) * c) / (b + b)
    if (b >= 0.0d0) then
        tmp = t_0
    else
        tmp = -b / a
    end if
    t_1 = tmp
    if (b <= (-1.1d-97)) then
        tmp_1 = t_1
    else if (b <= (-1.12d-302)) then
        if (b >= 0.0d0) then
            tmp_2 = t_0
        else
            tmp_2 = (sqrt(((a * c) * (-4.0d0))) / a) * 0.5d0
        end if
        tmp_1 = tmp_2
    else if (b <= 4.6d-66) then
        if (b >= 0.0d0) then
            tmp_3 = -sqrt((-a * c)) / -a
        else
            tmp_3 = -sqrt(((c / a) * (-1.0d0)))
        end if
        tmp_1 = tmp_3
    else
        tmp_1 = t_1
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (-2.0 * c) / (b + b);
	double tmp;
	if (b >= 0.0) {
		tmp = t_0;
	} else {
		tmp = -b / a;
	}
	double t_1 = tmp;
	double tmp_1;
	if (b <= -1.1e-97) {
		tmp_1 = t_1;
	} else if (b <= -1.12e-302) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = (Math.sqrt(((a * c) * -4.0)) / a) * 0.5;
		}
		tmp_1 = tmp_2;
	} else if (b <= 4.6e-66) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -Math.sqrt((-a * c)) / -a;
		} else {
			tmp_3 = -Math.sqrt(((c / a) * -1.0));
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (-2.0 * c) / (b + b)
	tmp = 0
	if b >= 0.0:
		tmp = t_0
	else:
		tmp = -b / a
	t_1 = tmp
	tmp_1 = 0
	if b <= -1.1e-97:
		tmp_1 = t_1
	elif b <= -1.12e-302:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_0
		else:
			tmp_2 = (math.sqrt(((a * c) * -4.0)) / a) * 0.5
		tmp_1 = tmp_2
	elif b <= 4.6e-66:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = -math.sqrt((-a * c)) / -a
		else:
			tmp_3 = -math.sqrt(((c / a) * -1.0))
		tmp_1 = tmp_3
	else:
		tmp_1 = t_1
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-2.0 * c) / Float64(b + b))
	tmp = 0.0
	if (b >= 0.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(-b) / a);
	end
	t_1 = tmp
	tmp_1 = 0.0
	if (b <= -1.1e-97)
		tmp_1 = t_1;
	elseif (b <= -1.12e-302)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(Float64(sqrt(Float64(Float64(a * c) * -4.0)) / a) * 0.5);
		end
		tmp_1 = tmp_2;
	elseif (b <= 4.6e-66)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-sqrt(Float64(Float64(-a) * c))) / Float64(-a));
		else
			tmp_3 = Float64(-sqrt(Float64(Float64(c / a) * -1.0)));
		end
		tmp_1 = tmp_3;
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = (-2.0 * c) / (b + b);
	tmp = 0.0;
	if (b >= 0.0)
		tmp = t_0;
	else
		tmp = -b / a;
	end
	t_1 = tmp;
	tmp_2 = 0.0;
	if (b <= -1.1e-97)
		tmp_2 = t_1;
	elseif (b <= -1.12e-302)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_0;
		else
			tmp_3 = (sqrt(((a * c) * -4.0)) / a) * 0.5;
		end
		tmp_2 = tmp_3;
	elseif (b <= 4.6e-66)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = -sqrt((-a * c)) / -a;
		else
			tmp_4 = -sqrt(((c / a) * -1.0));
		end
		tmp_2 = tmp_4;
	else
		tmp_2 = t_1;
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-2.0 * c), $MachinePrecision] / N[(b + b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = If[GreaterEqual[b, 0.0], t$95$0, N[((-b) / a), $MachinePrecision]]}, If[LessEqual[b, -1.1e-97], t$95$1, If[LessEqual[b, -1.12e-302], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]], If[LessEqual[b, 4.6e-66], If[GreaterEqual[b, 0.0], N[((-N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision]) / (-a)), $MachinePrecision], (-N[Sqrt[N[(N[(c / a), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision])], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.0999999999999999e-97 or 4.59999999999999984e-66 < b

   1. Initial program 66.3%

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

   3. 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. Step-by-step derivation

   5. Applied rewrites 66.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 76.8%

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

   9. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. distribute-frac-neg N/A

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

       3. lift-/.f64 N/A

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

       4. lift-neg.f64 80.7

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

   11. Applied rewrites 80.7%

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

   if -1.0999999999999999e-97 < b < -1.12e-302

   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. Add Preprocessing

   3. 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. Step-by-step derivation

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 72.4%

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

   9. Taylor expanded in a around inf

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

       1. sqrt-unprod N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lift-*.f64 67.6

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

   11. Applied rewrites 67.6%

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

   if -1.12e-302 < b < 4.59999999999999984e-66

   1. Initial program 84.3%

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

   3. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. sqrt-unprod N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 73.2

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

   5. Applied rewrites 73.2%

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

   6. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 73.4

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

   8. Applied rewrites 73.4%

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

   9. Taylor expanded in a around inf

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

       1. mul-1-neg N/A

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

       2. sqrt-prod N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-neg.f64 73.4

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

   11. Applied rewrites 73.4%

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

   12. Taylor expanded in a around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lift-*.f64 73.4

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

   14. Applied rewrites 73.4%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-97}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq -1.12 \cdot 10^{-302}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4}}{a} \cdot 0.5\\ \end{array}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{-66}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-\sqrt{\left(-a\right) \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{c}{a} \cdot -1}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.75 \cdot 10^{-180} \lor \neg \left(b \leq 4.6 \cdot 10^{-66}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-\sqrt{\left(-a\right) \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{c}{a} \cdot -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (or (<= b -3.75e-180) (not (<= b 4.6e-66)))
   (if (>= b 0.0) (/ (* -2.0 c) (+ b b)) (/ (- b) a))
   (if (>= b 0.0)
     (/ (- (sqrt (* (- a) c))) (- a))
     (- (sqrt (* (/ c a) -1.0))))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if ((b <= -3.75e-180) || !(b <= 4.6e-66)) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-2.0 * c) / (b + b);
		} else {
			tmp_2 = -b / a;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = -sqrt((-a * c)) / -a;
	} else {
		tmp_1 = -sqrt(((c / a) * -1.0));
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    if ((b <= (-3.75d-180)) .or. (.not. (b <= 4.6d-66))) then
        if (b >= 0.0d0) then
            tmp_2 = ((-2.0d0) * c) / (b + b)
        else
            tmp_2 = -b / a
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = -sqrt((-a * c)) / -a
    else
        tmp_1 = -sqrt(((c / a) * (-1.0d0)))
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if ((b <= -3.75e-180) || !(b <= 4.6e-66)) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-2.0 * c) / (b + b);
		} else {
			tmp_2 = -b / a;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = -Math.sqrt((-a * c)) / -a;
	} else {
		tmp_1 = -Math.sqrt(((c / a) * -1.0));
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if (b <= -3.75e-180) or not (b <= 4.6e-66):
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (-2.0 * c) / (b + b)
		else:
			tmp_2 = -b / a
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = -math.sqrt((-a * c)) / -a
	else:
		tmp_1 = -math.sqrt(((c / a) * -1.0))
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if ((b <= -3.75e-180) || !(b <= 4.6e-66))
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-2.0 * c) / Float64(b + b));
		else
			tmp_2 = Float64(Float64(-b) / a);
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-sqrt(Float64(Float64(-a) * c))) / Float64(-a));
	else
		tmp_1 = Float64(-sqrt(Float64(Float64(c / a) * -1.0)));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp_2 = 0.0;
	if ((b <= -3.75e-180) || ~((b <= 4.6e-66)))
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (-2.0 * c) / (b + b);
		else
			tmp_3 = -b / a;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = -sqrt((-a * c)) / -a;
	else
		tmp_2 = -sqrt(((c / a) * -1.0));
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[Or[LessEqual[b, -3.75e-180], N[Not[LessEqual[b, 4.6e-66]], $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * c), $MachinePrecision] / N[(b + b), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[((-N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision]) / (-a)), $MachinePrecision], (-N[Sqrt[N[(N[(c / a), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if b < -3.75000000000000008e-180 or 4.59999999999999984e-66 < b

   1. Initial program 67.0%

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

   3. 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. Step-by-step derivation

   5. Applied rewrites 67.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 77.2%

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

   9. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. distribute-frac-neg N/A

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

       3. lift-/.f64 N/A

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

       4. lift-neg.f64 78.9

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

   11. Applied rewrites 78.9%

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

   if -3.75000000000000008e-180 < b < 4.59999999999999984e-66

   1. Initial program 79.4%

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

   3. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. sqrt-unprod N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 71.4

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 65.2

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

   8. Applied rewrites 65.2%

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

   9. Taylor expanded in a around inf

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

       1. mul-1-neg N/A

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

       2. sqrt-prod N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-neg.f64 65.2

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

   11. Applied rewrites 65.2%

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

   12. Taylor expanded in a around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lift-*.f64 65.2

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

   14. Applied rewrites 65.2%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.75 \cdot 10^{-180} \lor \neg \left(b \leq 4.6 \cdot 10^{-66}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-\sqrt{\left(-a\right) \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{c}{a} \cdot -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{c}{a} \cdot -1}\\ \mathbf{if}\;b \leq -3.75 \cdot 10^{-180} \lor \neg \left(b \leq 2.35 \cdot 10^{-98}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* (/ c a) -1.0))))
   (if (or (<= b -3.75e-180) (not (<= b 2.35e-98)))
     (if (>= b 0.0) (/ (* -2.0 c) (+ b b)) (/ (- b) a))
     (if (>= b 0.0) t_0 (- t_0)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((c / a) * -1.0));
	double tmp_1;
	if ((b <= -3.75e-180) || !(b <= 2.35e-98)) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-2.0 * c) / (b + b);
		} else {
			tmp_2 = -b / a;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = -t_0;
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = sqrt(((c / a) * (-1.0d0)))
    if ((b <= (-3.75d-180)) .or. (.not. (b <= 2.35d-98))) then
        if (b >= 0.0d0) then
            tmp_2 = ((-2.0d0) * c) / (b + b)
        else
            tmp_2 = -b / a
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = t_0
    else
        tmp_1 = -t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((c / a) * -1.0));
	double tmp_1;
	if ((b <= -3.75e-180) || !(b <= 2.35e-98)) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-2.0 * c) / (b + b);
		} else {
			tmp_2 = -b / a;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = -t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((c / a) * -1.0))
	tmp_1 = 0
	if (b <= -3.75e-180) or not (b <= 2.35e-98):
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (-2.0 * c) / (b + b)
		else:
			tmp_2 = -b / a
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = t_0
	else:
		tmp_1 = -t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(c / a) * -1.0))
	tmp_1 = 0.0
	if ((b <= -3.75e-180) || !(b <= 2.35e-98))
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-2.0 * c) / Float64(b + b));
		else
			tmp_2 = Float64(Float64(-b) / a);
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = t_0;
	else
		tmp_1 = Float64(-t_0);
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = sqrt(((c / a) * -1.0));
	tmp_2 = 0.0;
	if ((b <= -3.75e-180) || ~((b <= 2.35e-98)))
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (-2.0 * c) / (b + b);
		else
			tmp_3 = -b / a;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = t_0;
	else
		tmp_2 = -t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(c / a), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[b, -3.75e-180], N[Not[LessEqual[b, 2.35e-98]], $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * c), $MachinePrecision] / N[(b + b), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$0, (-t$95$0)]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.75000000000000008e-180 or 2.35000000000000003e-98 < b

   1. Initial program 67.5%

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

   3. 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. Step-by-step derivation

   5. Applied rewrites 67.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 76.0%

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

   9. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. distribute-frac-neg N/A

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

       3. lift-/.f64 N/A

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

       4. lift-neg.f64 77.7

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

   11. Applied rewrites 77.7%

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

   if -3.75000000000000008e-180 < b < 2.35000000000000003e-98

   1. Initial program 78.9%

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

   3. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. sqrt-unprod N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 73.6

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

   5. Applied rewrites 73.6%

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

   6. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 66.7

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

   8. Applied rewrites 66.7%

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

   9. Taylor expanded in a around -inf

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

       1. sqrt-prod N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-sqrt.f64 40.5

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

   11. Applied rewrites 40.5%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.75 \cdot 10^{-180} \lor \neg \left(b \leq 2.35 \cdot 10^{-98}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\sqrt{\frac{c}{a} \cdot -1}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{c}{a} \cdot -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\sqrt{\frac{c}{a} \cdot -1}\\ \mathbf{if}\;b \leq -3.75 \cdot 10^{-180} \lor \neg \left(b \leq 10^{-186}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (sqrt (* (/ c a) -1.0)))))
   (if (or (<= b -3.75e-180) (not (<= b 1e-186)))
     (if (>= b 0.0) (/ (* -2.0 c) (+ b b)) (/ (- b) a))
     (if (>= b 0.0) t_0 t_0))))

C

double code(double a, double b, double c) {
	double t_0 = -sqrt(((c / a) * -1.0));
	double tmp_1;
	if ((b <= -3.75e-180) || !(b <= 1e-186)) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-2.0 * c) / (b + b);
		} else {
			tmp_2 = -b / a;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = -sqrt(((c / a) * (-1.0d0)))
    if ((b <= (-3.75d-180)) .or. (.not. (b <= 1d-186))) then
        if (b >= 0.0d0) then
            tmp_2 = ((-2.0d0) * c) / (b + b)
        else
            tmp_2 = -b / a
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = t_0
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -Math.sqrt(((c / a) * -1.0));
	double tmp_1;
	if ((b <= -3.75e-180) || !(b <= 1e-186)) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-2.0 * c) / (b + b);
		} else {
			tmp_2 = -b / a;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = -math.sqrt(((c / a) * -1.0))
	tmp_1 = 0
	if (b <= -3.75e-180) or not (b <= 1e-186):
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (-2.0 * c) / (b + b)
		else:
			tmp_2 = -b / a
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = t_0
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(-sqrt(Float64(Float64(c / a) * -1.0)))
	tmp_1 = 0.0
	if ((b <= -3.75e-180) || !(b <= 1e-186))
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-2.0 * c) / Float64(b + b));
		else
			tmp_2 = Float64(Float64(-b) / a);
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = t_0;
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = -sqrt(((c / a) * -1.0));
	tmp_2 = 0.0;
	if ((b <= -3.75e-180) || ~((b <= 1e-186)))
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (-2.0 * c) / (b + b);
		else
			tmp_3 = -b / a;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = t_0;
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = (-N[Sqrt[N[(N[(c / a), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision])}, If[Or[LessEqual[b, -3.75e-180], N[Not[LessEqual[b, 1e-186]], $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * c), $MachinePrecision] / N[(b + b), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.75000000000000008e-180 or 9.9999999999999991e-187 < b

   1. Initial program 69.3%

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

   3. 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. Step-by-step derivation

   5. Applied rewrites 69.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 71.3%

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

   9. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. distribute-frac-neg N/A

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

       3. lift-/.f64 N/A

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

       4. lift-neg.f64 72.8

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

   11. Applied rewrites 72.8%

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

   if -3.75000000000000008e-180 < b < 9.9999999999999991e-187

   1. Initial program 74.8%

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

   3. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. sqrt-unprod N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

   6. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 63.3

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

   8. Applied rewrites 63.3%

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

   9. Taylor expanded in a around -inf

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

       1. sqrt-prod N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-sqrt.f64 40.4

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

   11. Applied rewrites 40.4%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.75 \cdot 10^{-180} \lor \neg \left(b \leq 10^{-186}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-\sqrt{\frac{c}{a} \cdot -1}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{c}{a} \cdot -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.9% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.0%

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

3. 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. Step-by-step derivation

5. Applied rewrites 70.1%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 66.9%

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

9. Taylor expanded in b around -inf

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

    1. mul-1-neg N/A

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

    2. distribute-frac-neg N/A

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

    3. lift-/.f64 N/A

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

    4. lift-neg.f64 63.8

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

11. Applied rewrites 63.8%

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

Reproduce

herbie shell --seed 2025061 
(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))))