jeff quadratic root 1

Percentage Accurate: 72.1% → 90.7%

Time: 4.9s

Alternatives: 15

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;b \leq -5.8 \cdot 10^{+82}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+125}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(\frac{t\_0}{a} + \frac{b}{a}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 a) c (* b b)))))
   (if (<= b -5.8e+82)
     (if (>= b 0.0)
       (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))
       (/ (* 2.0 c) (* (- b) 2.0)))
     (if (<= b 2e+125)
       (if (>= b 0.0) (* (+ (/ t_0 a) (/ b a)) -0.5) (/ (+ c c) (- t_0 b)))
       (if (>= b 0.0)
         (* (/ (+ b b) a) -0.5)
         (/ (+ c c) (- (sqrt (* b b)) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-4.0 * a), c, (b * b)));
	double tmp_1;
	if (b <= -5.8e+82) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-b - sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
		} else {
			tmp_2 = (2.0 * c) / (-b * 2.0);
		}
		tmp_1 = tmp_2;
	} else if (b <= 2e+125) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = ((t_0 / a) + (b / a)) * -0.5;
		} else {
			tmp_3 = (c + c) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = ((b + b) / a) * -0.5;
	} else {
		tmp_1 = (c + c) / (sqrt((b * b)) - b);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))
	tmp_1 = 0.0
	if (b <= -5.8e+82)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a));
		else
			tmp_2 = Float64(Float64(2.0 * c) / Float64(Float64(-b) * 2.0));
		end
		tmp_1 = tmp_2;
	elseif (b <= 2e+125)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(t_0 / a) + Float64(b / a)) * -0.5);
		else
			tmp_3 = Float64(Float64(c + c) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(b + b) / a) * -0.5);
	else
		tmp_1 = Float64(Float64(c + c) / Float64(sqrt(Float64(b * b)) - b));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -5.8e+82], If[GreaterEqual[b, 0.0], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) * 2.0), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 2e+125], If[GreaterEqual[b, 0.0], N[(N[(N[(t$95$0 / a), $MachinePrecision] + N[(b / a), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.8000000000000003e82

   1. Initial program 72.1%

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

   2. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.4

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

   4. Applied rewrites 69.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 70.0%

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

   if -5.8000000000000003e82 < b < 1.9999999999999998e125

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. div-add N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-/.f64 72.2

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

   6. Applied rewrites 72.2%

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

   if 1.9999999999999998e125 < b

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.4

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

   7. Applied rewrites 60.4%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.3

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

   10. Applied rewrites 48.3%

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

   11. Taylor expanded in b around inf

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

       1. count-2-rev N/A

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

       2. lower-+.f64 57.9

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

   13. Applied rewrites 57.9%

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

Alternative 2: 90.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;b \leq -5.8 \cdot 10^{+82}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+125}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{t\_0 + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 a) c (* b b)))))
   (if (<= b -5.8e+82)
     (if (>= b 0.0)
       (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))
       (/ (* 2.0 c) (* (- b) 2.0)))
     (if (<= b 2e+125)
       (if (>= b 0.0) (* (/ (+ t_0 b) a) -0.5) (/ (+ c c) (- t_0 b)))
       (if (>= b 0.0)
         (* (/ (+ b b) a) -0.5)
         (/ (+ c c) (- (sqrt (* b b)) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-4.0 * a), c, (b * b)));
	double tmp_1;
	if (b <= -5.8e+82) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-b - sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
		} else {
			tmp_2 = (2.0 * c) / (-b * 2.0);
		}
		tmp_1 = tmp_2;
	} else if (b <= 2e+125) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = ((t_0 + b) / a) * -0.5;
		} else {
			tmp_3 = (c + c) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = ((b + b) / a) * -0.5;
	} else {
		tmp_1 = (c + c) / (sqrt((b * b)) - b);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))
	tmp_1 = 0.0
	if (b <= -5.8e+82)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a));
		else
			tmp_2 = Float64(Float64(2.0 * c) / Float64(Float64(-b) * 2.0));
		end
		tmp_1 = tmp_2;
	elseif (b <= 2e+125)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(t_0 + b) / a) * -0.5);
		else
			tmp_3 = Float64(Float64(c + c) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(b + b) / a) * -0.5);
	else
		tmp_1 = Float64(Float64(c + c) / Float64(sqrt(Float64(b * b)) - b));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -5.8e+82], If[GreaterEqual[b, 0.0], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) * 2.0), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 2e+125], If[GreaterEqual[b, 0.0], N[(N[(N[(t$95$0 + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.8000000000000003e82

   1. Initial program 72.1%

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

   2. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.4

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

   4. Applied rewrites 69.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 70.0%

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

   if -5.8000000000000003e82 < b < 1.9999999999999998e125

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   if 1.9999999999999998e125 < b

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.4

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

   7. Applied rewrites 60.4%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.3

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

   10. Applied rewrites 48.3%

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

   11. Taylor expanded in b around inf

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

       1. count-2-rev N/A

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

       2. lower-+.f64 57.9

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

   13. Applied rewrites 57.9%

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

Alternative 3: 82.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ t_1 := \frac{t\_0 + b}{a} \cdot -0.5\\ \mathbf{if}\;b \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{-c \cdot \frac{b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-281}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(\left(-c \cdot \sqrt{\frac{-4}{a \cdot c}}\right) + \frac{b}{a}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+125}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 a) c (* b b)))) (t_1 (* (/ (+ t_0 b) a) -0.5)))
   (if (<= b -1.35e+154)
     (if (>= b 0.0) t_1 (/ (+ c c) (- (* c (/ b c)))))
     (if (<= b 4.2e-281)
       (if (>= b 0.0)
         (* (+ (- (* c (sqrt (/ -4.0 (* a c))))) (/ b a)) -0.5)
         (/ (+ c c) (- t_0 b)))
       (if (<= b 2e+125)
         (if (>= b 0.0) t_1 (/ -2.0 (sqrt (* -4.0 (/ a c)))))
         (if (>= b 0.0)
           (* (/ (+ b b) a) -0.5)
           (/ (+ c c) (- (sqrt (* b b)) b))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-4.0 * a), c, (b * b)));
	double t_1 = ((t_0 + b) / a) * -0.5;
	double tmp_1;
	if (b <= -1.35e+154) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = (c + c) / -(c * (b / c));
		}
		tmp_1 = tmp_2;
	} else if (b <= 4.2e-281) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-(c * sqrt((-4.0 / (a * c)))) + (b / a)) * -0.5;
		} else {
			tmp_3 = (c + c) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b <= 2e+125) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = t_1;
		} else {
			tmp_4 = -2.0 / sqrt((-4.0 * (a / c)));
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = ((b + b) / a) * -0.5;
	} else {
		tmp_1 = (c + c) / (sqrt((b * b)) - b);
	}
	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(Float64(t_0 + b) / a) * -0.5)
	tmp_1 = 0.0
	if (b <= -1.35e+154)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(Float64(c + c) / Float64(-Float64(c * Float64(b / c))));
		end
		tmp_1 = tmp_2;
	elseif (b <= 4.2e-281)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(-Float64(c * sqrt(Float64(-4.0 / Float64(a * c))))) + Float64(b / a)) * -0.5);
		else
			tmp_3 = Float64(Float64(c + c) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	elseif (b <= 2e+125)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = t_1;
		else
			tmp_4 = Float64(-2.0 / sqrt(Float64(-4.0 * Float64(a / c))));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(b + b) / a) * -0.5);
	else
		tmp_1 = Float64(Float64(c + c) / Float64(sqrt(Float64(b * b)) - b));
	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[(N[(t$95$0 + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[b, -1.35e+154], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(c + c), $MachinePrecision] / (-N[(c * N[(b / c), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]], If[LessEqual[b, 4.2e-281], If[GreaterEqual[b, 0.0], N[(N[((-N[(c * N[Sqrt[N[(-4.0 / N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]) + N[(b / a), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 2e+125], If[GreaterEqual[b, 0.0], t$95$1, N[(-2.0 / N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.35000000000000003e154

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lower-/.f64 48.3

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

   7. Applied rewrites 48.3%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 50.7

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

   10. Applied rewrites 50.7%

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

   11. Taylor expanded in b around inf

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

       1. lower-/.f64 49.8

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

   13. Applied rewrites 49.8%

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

   if -1.35000000000000003e154 < b < 4.1999999999999998e-281

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in a around -inf

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

      1. lower-+.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 48.5

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

   7. Applied rewrites 48.5%

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

   8. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 54.8

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

   10. Applied rewrites 54.8%

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

   if 4.1999999999999998e-281 < b < 1.9999999999999998e125

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   if 1.9999999999999998e125 < b

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.4

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

   7. Applied rewrites 60.4%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.3

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

   10. Applied rewrites 48.3%

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

   11. Taylor expanded in b around inf

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

       1. count-2-rev N/A

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

       2. lower-+.f64 57.9

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

   13. Applied rewrites 57.9%

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

Alternative 4: 82.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ t_1 := \frac{t\_0 + b}{a} \cdot -0.5\\ \mathbf{if}\;b \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{-c \cdot \frac{b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+125}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 a) c (* b b)))) (t_1 (* (/ (+ t_0 b) a) -0.5)))
   (if (<= b -1.35e+154)
     (if (>= b 0.0) t_1 (/ (+ c c) (- (* c (/ b c)))))
     (if (<= b 2e+125)
       (if (>= b 0.0) t_1 (/ (+ c c) (- t_0 b)))
       (if (>= b 0.0)
         (* (/ (+ b b) a) -0.5)
         (/ (+ c c) (- (sqrt (* b b)) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-4.0 * a), c, (b * b)));
	double t_1 = ((t_0 + b) / a) * -0.5;
	double tmp_1;
	if (b <= -1.35e+154) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = (c + c) / -(c * (b / c));
		}
		tmp_1 = tmp_2;
	} else if (b <= 2e+125) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = (c + c) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = ((b + b) / a) * -0.5;
	} else {
		tmp_1 = (c + c) / (sqrt((b * b)) - b);
	}
	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(Float64(t_0 + b) / a) * -0.5)
	tmp_1 = 0.0
	if (b <= -1.35e+154)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(Float64(c + c) / Float64(-Float64(c * Float64(b / c))));
		end
		tmp_1 = tmp_2;
	elseif (b <= 2e+125)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = Float64(Float64(c + c) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(b + b) / a) * -0.5);
	else
		tmp_1 = Float64(Float64(c + c) / Float64(sqrt(Float64(b * b)) - b));
	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[(N[(t$95$0 + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[b, -1.35e+154], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(c + c), $MachinePrecision] / (-N[(c * N[(b / c), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]], If[LessEqual[b, 2e+125], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(c + c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.35000000000000003e154

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lower-/.f64 48.3

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

   7. Applied rewrites 48.3%

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

   8. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 50.7

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

   10. Applied rewrites 50.7%

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

   11. Taylor expanded in b around inf

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

       1. lower-/.f64 49.8

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

   13. Applied rewrites 49.8%

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

   if -1.35000000000000003e154 < b < 1.9999999999999998e125

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   if 1.9999999999999998e125 < b

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.4

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

   7. Applied rewrites 60.4%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.3

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

   10. Applied rewrites 48.3%

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

   11. Taylor expanded in b around inf

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

       1. count-2-rev N/A

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

       2. lower-+.f64 57.9

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

   13. Applied rewrites 57.9%

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

Alternative 5: 81.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;b \leq 4.2 \cdot 10^{-281}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(\left(-c \cdot \sqrt{\frac{-4}{a \cdot c}}\right) + \frac{b}{a}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+125}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{t\_0 + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 a) c (* b b)))))
   (if (<= b 4.2e-281)
     (if (>= b 0.0)
       (* (+ (- (* c (sqrt (/ -4.0 (* a c))))) (/ b a)) -0.5)
       (/ (+ c c) (- t_0 b)))
     (if (<= b 2e+125)
       (if (>= b 0.0)
         (* (/ (+ t_0 b) a) -0.5)
         (/ -2.0 (sqrt (* -4.0 (/ a c)))))
       (if (>= b 0.0)
         (* (/ (+ b b) a) -0.5)
         (/ (+ c c) (- (sqrt (* b b)) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-4.0 * a), c, (b * b)));
	double tmp_1;
	if (b <= 4.2e-281) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-(c * sqrt((-4.0 / (a * c)))) + (b / a)) * -0.5;
		} else {
			tmp_2 = (c + c) / (t_0 - b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 2e+125) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = ((t_0 + b) / a) * -0.5;
		} else {
			tmp_3 = -2.0 / sqrt((-4.0 * (a / c)));
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = ((b + b) / a) * -0.5;
	} else {
		tmp_1 = (c + c) / (sqrt((b * b)) - b);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))
	tmp_1 = 0.0
	if (b <= 4.2e-281)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(Float64(-Float64(c * sqrt(Float64(-4.0 / Float64(a * c))))) + Float64(b / a)) * -0.5);
		else
			tmp_2 = Float64(Float64(c + c) / Float64(t_0 - b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 2e+125)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(t_0 + b) / a) * -0.5);
		else
			tmp_3 = Float64(-2.0 / sqrt(Float64(-4.0 * Float64(a / c))));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(b + b) / a) * -0.5);
	else
		tmp_1 = Float64(Float64(c + c) / Float64(sqrt(Float64(b * b)) - b));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, 4.2e-281], If[GreaterEqual[b, 0.0], N[(N[((-N[(c * N[Sqrt[N[(-4.0 / N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]) + N[(b / a), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 2e+125], If[GreaterEqual[b, 0.0], N[(N[(N[(t$95$0 + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], N[(-2.0 / N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.1999999999999998e-281

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in a around -inf

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

      1. lower-+.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 48.5

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

   7. Applied rewrites 48.5%

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

   8. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 54.8

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

   10. Applied rewrites 54.8%

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

   if 4.1999999999999998e-281 < b < 1.9999999999999998e125

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in c around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 44.5

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

   7. Applied rewrites 44.5%

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

   if 1.9999999999999998e125 < b

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.4

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

   7. Applied rewrites 60.4%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.3

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

   10. Applied rewrites 48.3%

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

   11. Taylor expanded in b around inf

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

       1. count-2-rev N/A

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

       2. lower-+.f64 57.9

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

   13. Applied rewrites 57.9%

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

Alternative 6: 76.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b \cdot \mathsf{fma}\left(-2, \frac{c}{b \cdot b}, \frac{2}{a}\right)\right) \cdot -0.5\\ t_1 := \sqrt{b \cdot b}\\ \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-55}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{-4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t\_1}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_1 - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* b (fma -2.0 (/ c (* b b)) (/ 2.0 a))) -0.5))
        (t_1 (sqrt (* b b))))
   (if (<= b -2e-310)
     (if (>= b 0.0) t_0 (/ (+ c c) (- (sqrt (fma (* -4.0 a) c (* b b))) b)))
     (if (<= b 2.5e-55)
       (if (>= b 0.0)
         (* -0.5 (/ (+ b (sqrt (* -4.0 (* a c)))) a))
         (/ (* 2.0 c) (+ (- b) t_1)))
       (if (>= b 0.0) t_0 (/ (+ c c) (- t_1 b)))))))

C

double code(double a, double b, double c) {
	double t_0 = (b * fma(-2.0, (c / (b * b)), (2.0 / a))) * -0.5;
	double t_1 = sqrt((b * b));
	double tmp_1;
	if (b <= -2e-310) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = (c + c) / (sqrt(fma((-4.0 * a), c, (b * b))) - b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 2.5e-55) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -0.5 * ((b + sqrt((-4.0 * (a * c)))) / a);
		} else {
			tmp_3 = (2.0 * c) / (-b + t_1);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = (c + c) / (t_1 - b);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * fma(-2.0, Float64(c / Float64(b * b)), Float64(2.0 / a))) * -0.5)
	t_1 = sqrt(Float64(b * b))
	tmp_1 = 0.0
	if (b <= -2e-310)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(Float64(c + c) / Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) - b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 2.5e-55)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(-0.5 * Float64(Float64(b + sqrt(Float64(-4.0 * Float64(a * c)))) / a));
		else
			tmp_3 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_1));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = t_0;
	else
		tmp_1 = Float64(Float64(c + c) / Float64(t_1 - b));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * N[(-2.0 * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(2.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2e-310], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 2.5e-55], If[GreaterEqual[b, 0.0], N[(-0.5 * N[(N[(b + N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$1), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(c + c), $MachinePrecision] / N[(t$95$1 - b), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 69.6

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

   7. Applied rewrites 69.6%

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

   if -1.999999999999994e-310 < b < 2.5000000000000001e-55

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.4

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.3

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

   7. Applied rewrites 48.3%

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

   8. Taylor expanded in b around 0

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

      1. distribute-lft-out N/A

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

      2. distribute-lft-neg-in N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. div-add-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 44.4

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

   10. Applied rewrites 44.4%

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

   if 2.5000000000000001e-55 < b

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.4

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

   7. Applied rewrites 60.4%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.3

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

   10. Applied rewrites 48.3%

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

   11. Taylor expanded in b around inf

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

       1. count-2-rev N/A

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

       2. lower-+.f64 57.9

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

   13. Applied rewrites 57.9%

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

   14. Taylor expanded in b around inf

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

       1. lower-*.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. associate-*r/ N/A

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

       7. metadata-eval N/A

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

       8. lower-/.f64 57.5

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

   16. Applied rewrites 57.5%

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

Alternative 7: 76.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-55}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(\left(-c \cdot \sqrt{\frac{-4}{a \cdot c}}\right) + \frac{b}{a}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\left(b \cdot \mathsf{fma}\left(-2, \frac{c}{b \cdot b}, \frac{2}{a}\right)\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.1e-55)
   (if (>= b 0.0)
     (* (+ (- (* c (sqrt (/ -4.0 (* a c))))) (/ b a)) -0.5)
     (/ (+ c c) (- (sqrt (fma (* -4.0 a) c (* b b))) b)))
   (if (>= b 0.0)
     (* (* b (fma -2.0 (/ c (* b b)) (/ 2.0 a))) -0.5)
     (/ (+ c c) (- (sqrt (* b b)) b)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= 2.1e-55) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-(c * sqrt((-4.0 / (a * c)))) + (b / a)) * -0.5;
		} else {
			tmp_2 = (c + c) / (sqrt(fma((-4.0 * a), c, (b * b))) - b);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (b * fma(-2.0, (c / (b * b)), (2.0 / a))) * -0.5;
	} else {
		tmp_1 = (c + c) / (sqrt((b * b)) - b);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= 2.1e-55)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(Float64(-Float64(c * sqrt(Float64(-4.0 / Float64(a * c))))) + Float64(b / a)) * -0.5);
		else
			tmp_2 = Float64(Float64(c + c) / Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) - b));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(b * fma(-2.0, Float64(c / Float64(b * b)), Float64(2.0 / a))) * -0.5);
	else
		tmp_1 = Float64(Float64(c + c) / Float64(sqrt(Float64(b * b)) - b));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.1e-55], If[GreaterEqual[b, 0.0], N[(N[((-N[(c * N[Sqrt[N[(-4.0 / N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]) + N[(b / a), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(b * N[(-2.0 * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(2.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.1000000000000002e-55

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in a around -inf

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

      1. lower-+.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 48.5

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

   7. Applied rewrites 48.5%

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

   8. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 54.8

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

   10. Applied rewrites 54.8%

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

   if 2.1000000000000002e-55 < b

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.4

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

   7. Applied rewrites 60.4%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.3

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

   10. Applied rewrites 48.3%

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

   11. Taylor expanded in b around inf

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

       1. count-2-rev N/A

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

       2. lower-+.f64 57.9

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

   13. Applied rewrites 57.9%

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

   14. Taylor expanded in b around inf

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

       1. lower-*.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. associate-*r/ N/A

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

       7. metadata-eval N/A

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

       8. lower-/.f64 57.5

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

   16. Applied rewrites 57.5%

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

Alternative 8: 72.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c + c}{\sqrt{b \cdot b} - b}\\ t_1 := \sqrt{-4 \cdot \left(c \cdot a\right)}\\ \mathbf{if}\;b \leq -6.5 \cdot 10^{-77}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{-4 \cdot \frac{c}{a}} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-55}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t\_1}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\left(b \cdot \mathsf{fma}\left(-2, \frac{c}{b \cdot b}, \frac{2}{a}\right)\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (+ c c) (- (sqrt (* b b)) b))) (t_1 (sqrt (* -4.0 (* c a)))))
   (if (<= b -6.5e-77)
     (if (>= b 0.0) (* (sqrt (* -4.0 (/ c a))) -0.5) t_0)
     (if (<= b 2.5e-55)
       (if (>= b 0.0) (/ (- (- b) t_1) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) t_1)))
       (if (>= b 0.0)
         (* (* b (fma -2.0 (/ c (* b b)) (/ 2.0 a))) -0.5)
         t_0)))))

C

double code(double a, double b, double c) {
	double t_0 = (c + c) / (sqrt((b * b)) - b);
	double t_1 = sqrt((-4.0 * (c * a)));
	double tmp_1;
	if (b <= -6.5e-77) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = sqrt((-4.0 * (c / a))) * -0.5;
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= 2.5e-55) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_1) / (2.0 * a);
		} else {
			tmp_3 = (2.0 * c) / (-b + t_1);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (b * fma(-2.0, (c / (b * b)), (2.0 / a))) * -0.5;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c + c) / Float64(sqrt(Float64(b * b)) - b))
	t_1 = sqrt(Float64(-4.0 * Float64(c * a)))
	tmp_1 = 0.0
	if (b <= -6.5e-77)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(sqrt(Float64(-4.0 * Float64(c / a))) * -0.5);
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b <= 2.5e-55)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(-b) - t_1) / Float64(2.0 * a));
		else
			tmp_3 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_1));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(b * fma(-2.0, Float64(c / Float64(b * b)), Float64(2.0 / a))) * -0.5);
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -6.5e-77], If[GreaterEqual[b, 0.0], N[(N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision], t$95$0], If[LessEqual[b, 2.5e-55], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$1), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$1), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(b * N[(-2.0 * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(2.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.4999999999999999e-77

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.4

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

   7. Applied rewrites 60.4%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.3

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

   10. Applied rewrites 48.3%

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

   11. Taylor expanded in b around inf

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

       1. count-2-rev N/A

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

       2. lower-+.f64 57.9

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

   13. Applied rewrites 57.9%

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

   14. Taylor expanded in a around inf

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-sqrt.f64 31.2

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

   16. Applied rewrites 31.2%

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

   if -6.4999999999999999e-77 < b < 2.5000000000000001e-55

   1. Initial program 72.1%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 56.5

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

   4. Applied rewrites 56.5%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 40.9

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

   7. Applied rewrites 40.9%

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

   if 2.5000000000000001e-55 < b

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.4

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

   7. Applied rewrites 60.4%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.3

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

   10. Applied rewrites 48.3%

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

   11. Taylor expanded in b around inf

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

       1. count-2-rev N/A

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

       2. lower-+.f64 57.9

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

   13. Applied rewrites 57.9%

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

   14. Taylor expanded in b around inf

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

       1. lower-*.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. associate-*r/ N/A

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

       7. metadata-eval N/A

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

       8. lower-/.f64 57.5

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

   16. Applied rewrites 57.5%

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

Alternative 9: 72.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c + c}{\sqrt{b \cdot b} - b}\\ t_1 := \sqrt{-4 \cdot \left(c \cdot a\right)}\\ \mathbf{if}\;b \leq -6.5 \cdot 10^{-77}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{-4 \cdot \frac{c}{a}} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-55}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t\_1}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (+ c c) (- (sqrt (* b b)) b))) (t_1 (sqrt (* -4.0 (* c a)))))
   (if (<= b -6.5e-77)
     (if (>= b 0.0) (* (sqrt (* -4.0 (/ c a))) -0.5) t_0)
     (if (<= b 2.5e-55)
       (if (>= b 0.0) (/ (- (- b) t_1) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) t_1)))
       (if (>= b 0.0) (* (/ (+ b b) a) -0.5) t_0)))))

C

double code(double a, double b, double c) {
	double t_0 = (c + c) / (sqrt((b * b)) - b);
	double t_1 = sqrt((-4.0 * (c * a)));
	double tmp_1;
	if (b <= -6.5e-77) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = sqrt((-4.0 * (c / a))) * -0.5;
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= 2.5e-55) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_1) / (2.0 * a);
		} else {
			tmp_3 = (2.0 * c) / (-b + t_1);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = ((b + b) / a) * -0.5;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = (c + c) / (sqrt((b * b)) - b)
    t_1 = sqrt(((-4.0d0) * (c * a)))
    if (b <= (-6.5d-77)) then
        if (b >= 0.0d0) then
            tmp_2 = sqrt(((-4.0d0) * (c / a))) * (-0.5d0)
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b <= 2.5d-55) then
        if (b >= 0.0d0) then
            tmp_3 = (-b - t_1) / (2.0d0 * a)
        else
            tmp_3 = (2.0d0 * c) / (-b + t_1)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = ((b + b) / a) * (-0.5d0)
    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 = (c + c) / (Math.sqrt((b * b)) - b);
	double t_1 = Math.sqrt((-4.0 * (c * a)));
	double tmp_1;
	if (b <= -6.5e-77) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = Math.sqrt((-4.0 * (c / a))) * -0.5;
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= 2.5e-55) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_1) / (2.0 * a);
		} else {
			tmp_3 = (2.0 * c) / (-b + t_1);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = ((b + b) / a) * -0.5;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (c + c) / (math.sqrt((b * b)) - b)
	t_1 = math.sqrt((-4.0 * (c * a)))
	tmp_1 = 0
	if b <= -6.5e-77:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = math.sqrt((-4.0 * (c / a))) * -0.5
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b <= 2.5e-55:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (-b - t_1) / (2.0 * a)
		else:
			tmp_3 = (2.0 * c) / (-b + t_1)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = ((b + b) / a) * -0.5
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c + c) / Float64(sqrt(Float64(b * b)) - b))
	t_1 = sqrt(Float64(-4.0 * Float64(c * a)))
	tmp_1 = 0.0
	if (b <= -6.5e-77)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(sqrt(Float64(-4.0 * Float64(c / a))) * -0.5);
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b <= 2.5e-55)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(-b) - t_1) / Float64(2.0 * a));
		else
			tmp_3 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_1));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(b + b) / a) * -0.5);
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = (c + c) / (sqrt((b * b)) - b);
	t_1 = sqrt((-4.0 * (c * a)));
	tmp_2 = 0.0;
	if (b <= -6.5e-77)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = sqrt((-4.0 * (c / a))) * -0.5;
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b <= 2.5e-55)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (-b - t_1) / (2.0 * a);
		else
			tmp_4 = (2.0 * c) / (-b + t_1);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = ((b + b) / a) * -0.5;
	else
		tmp_2 = t_0;
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -6.5e-77], If[GreaterEqual[b, 0.0], N[(N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision], t$95$0], If[LessEqual[b, 2.5e-55], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$1), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$1), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.4999999999999999e-77

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.4

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

   7. Applied rewrites 60.4%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.3

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

   10. Applied rewrites 48.3%

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

   11. Taylor expanded in b around inf

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

       1. count-2-rev N/A

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

       2. lower-+.f64 57.9

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

   13. Applied rewrites 57.9%

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

   14. Taylor expanded in a around inf

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-sqrt.f64 31.2

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

   16. Applied rewrites 31.2%

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

   if -6.4999999999999999e-77 < b < 2.5000000000000001e-55

   1. Initial program 72.1%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 56.5

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

   4. Applied rewrites 56.5%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 40.9

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

   7. Applied rewrites 40.9%

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

   if 2.5000000000000001e-55 < b

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.4

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

   7. Applied rewrites 60.4%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.3

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

   10. Applied rewrites 48.3%

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

   11. Taylor expanded in b around inf

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

       1. count-2-rev N/A

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

       2. lower-+.f64 57.9

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

   13. Applied rewrites 57.9%

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

Alternative 10: 72.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-4 \cdot \left(a \cdot c\right)}\\ t_1 := \sqrt{b \cdot b}\\ t_2 := \frac{c + c}{t\_1 - b}\\ t_3 := \frac{b + b}{a} \cdot -0.5\\ \mathbf{if}\;b \leq -6.5 \cdot 10^{-77}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{-4 \cdot \frac{c}{a}} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0 + \left(-b\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-55}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + t\_0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t\_1}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* -4.0 (* a c))))
        (t_1 (sqrt (* b b)))
        (t_2 (/ (+ c c) (- t_1 b)))
        (t_3 (* (/ (+ b b) a) -0.5)))
   (if (<= b -6.5e-77)
     (if (>= b 0.0) (* (sqrt (* -4.0 (/ c a))) -0.5) t_2)
     (if (<= b -2e-310)
       (if (>= b 0.0) t_3 (/ (+ c c) (+ t_0 (- b))))
       (if (<= b 2.5e-55)
         (if (>= b 0.0) (* -0.5 (/ (+ b t_0) a)) (/ (* 2.0 c) (+ (- b) t_1)))
         (if (>= b 0.0) t_3 t_2))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((-4.0 * (a * c)));
	double t_1 = sqrt((b * b));
	double t_2 = (c + c) / (t_1 - b);
	double t_3 = ((b + b) / a) * -0.5;
	double tmp_1;
	if (b <= -6.5e-77) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = sqrt((-4.0 * (c / a))) * -0.5;
		} else {
			tmp_2 = t_2;
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_3;
		} else {
			tmp_3 = (c + c) / (t_0 + -b);
		}
		tmp_1 = tmp_3;
	} else if (b <= 2.5e-55) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = -0.5 * ((b + t_0) / a);
		} else {
			tmp_4 = (2.0 * c) / (-b + t_1);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = t_3;
	} else {
		tmp_1 = t_2;
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    real(8) :: tmp_4
    t_0 = sqrt(((-4.0d0) * (a * c)))
    t_1 = sqrt((b * b))
    t_2 = (c + c) / (t_1 - b)
    t_3 = ((b + b) / a) * (-0.5d0)
    if (b <= (-6.5d-77)) then
        if (b >= 0.0d0) then
            tmp_2 = sqrt(((-4.0d0) * (c / a))) * (-0.5d0)
        else
            tmp_2 = t_2
        end if
        tmp_1 = tmp_2
    else if (b <= (-2d-310)) then
        if (b >= 0.0d0) then
            tmp_3 = t_3
        else
            tmp_3 = (c + c) / (t_0 + -b)
        end if
        tmp_1 = tmp_3
    else if (b <= 2.5d-55) then
        if (b >= 0.0d0) then
            tmp_4 = (-0.5d0) * ((b + t_0) / a)
        else
            tmp_4 = (2.0d0 * c) / (-b + t_1)
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = t_3
    else
        tmp_1 = t_2
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt((-4.0 * (a * c)));
	double t_1 = Math.sqrt((b * b));
	double t_2 = (c + c) / (t_1 - b);
	double t_3 = ((b + b) / a) * -0.5;
	double tmp_1;
	if (b <= -6.5e-77) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = Math.sqrt((-4.0 * (c / a))) * -0.5;
		} else {
			tmp_2 = t_2;
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_3;
		} else {
			tmp_3 = (c + c) / (t_0 + -b);
		}
		tmp_1 = tmp_3;
	} else if (b <= 2.5e-55) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = -0.5 * ((b + t_0) / a);
		} else {
			tmp_4 = (2.0 * c) / (-b + t_1);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = t_3;
	} else {
		tmp_1 = t_2;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt((-4.0 * (a * c)))
	t_1 = math.sqrt((b * b))
	t_2 = (c + c) / (t_1 - b)
	t_3 = ((b + b) / a) * -0.5
	tmp_1 = 0
	if b <= -6.5e-77:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = math.sqrt((-4.0 * (c / a))) * -0.5
		else:
			tmp_2 = t_2
		tmp_1 = tmp_2
	elif b <= -2e-310:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = t_3
		else:
			tmp_3 = (c + c) / (t_0 + -b)
		tmp_1 = tmp_3
	elif b <= 2.5e-55:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = -0.5 * ((b + t_0) / a)
		else:
			tmp_4 = (2.0 * c) / (-b + t_1)
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = t_3
	else:
		tmp_1 = t_2
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(-4.0 * Float64(a * c)))
	t_1 = sqrt(Float64(b * b))
	t_2 = Float64(Float64(c + c) / Float64(t_1 - b))
	t_3 = Float64(Float64(Float64(b + b) / a) * -0.5)
	tmp_1 = 0.0
	if (b <= -6.5e-77)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(sqrt(Float64(-4.0 * Float64(c / a))) * -0.5);
		else
			tmp_2 = t_2;
		end
		tmp_1 = tmp_2;
	elseif (b <= -2e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_3;
		else
			tmp_3 = Float64(Float64(c + c) / Float64(t_0 + Float64(-b)));
		end
		tmp_1 = tmp_3;
	elseif (b <= 2.5e-55)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(-0.5 * Float64(Float64(b + t_0) / a));
		else
			tmp_4 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_1));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = t_3;
	else
		tmp_1 = t_2;
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = sqrt((-4.0 * (a * c)));
	t_1 = sqrt((b * b));
	t_2 = (c + c) / (t_1 - b);
	t_3 = ((b + b) / a) * -0.5;
	tmp_2 = 0.0;
	if (b <= -6.5e-77)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = sqrt((-4.0 * (c / a))) * -0.5;
		else
			tmp_3 = t_2;
		end
		tmp_2 = tmp_3;
	elseif (b <= -2e-310)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = t_3;
		else
			tmp_4 = (c + c) / (t_0 + -b);
		end
		tmp_2 = tmp_4;
	elseif (b <= 2.5e-55)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = -0.5 * ((b + t_0) / a);
		else
			tmp_5 = (2.0 * c) / (-b + t_1);
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = t_3;
	else
		tmp_2 = t_2;
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(c + c), $MachinePrecision] / N[(t$95$1 - b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[b, -6.5e-77], If[GreaterEqual[b, 0.0], N[(N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision], t$95$2], If[LessEqual[b, -2e-310], If[GreaterEqual[b, 0.0], t$95$3, N[(N[(c + c), $MachinePrecision] / N[(t$95$0 + (-b)), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 2.5e-55], If[GreaterEqual[b, 0.0], N[(-0.5 * N[(N[(b + t$95$0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$1), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$3, t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -6.4999999999999999e-77

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.4

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

   7. Applied rewrites 60.4%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.3

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

   10. Applied rewrites 48.3%

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

   11. Taylor expanded in b around inf

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

       1. count-2-rev N/A

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

       2. lower-+.f64 57.9

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

   13. Applied rewrites 57.9%

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

   14. Taylor expanded in a around inf

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-sqrt.f64 31.2

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

   16. Applied rewrites 31.2%

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

   if -6.4999999999999999e-77 < b < -1.999999999999994e-310

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.4

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

   7. Applied rewrites 60.4%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.3

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

   10. Applied rewrites 48.3%

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

   11. Taylor expanded in b around inf

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

       1. count-2-rev N/A

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

       2. lower-+.f64 57.9

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

   13. Applied rewrites 57.9%

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

   14. Taylor expanded in b around 0

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

       1. mul-1-neg N/A

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

       2. lift-neg.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lift-*.f64 54.4

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

   16. Applied rewrites 54.4%

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

   if -1.999999999999994e-310 < b < 2.5000000000000001e-55

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.4

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.3

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

   7. Applied rewrites 48.3%

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

   8. Taylor expanded in b around 0

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

      1. distribute-lft-out N/A

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

      2. distribute-lft-neg-in N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. div-add-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 44.4

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

   10. Applied rewrites 44.4%

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

   if 2.5000000000000001e-55 < b

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.4

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

   7. Applied rewrites 60.4%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.3

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

   10. Applied rewrites 48.3%

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

   11. Taylor expanded in b around inf

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

       1. count-2-rev N/A

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

       2. lower-+.f64 57.9

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

   13. Applied rewrites 57.9%

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

Alternative 11: 71.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b}\\ t_1 := \frac{c + c}{t\_0 - b}\\ t_2 := \frac{b + b}{a} \cdot -0.5\\ t_3 := \sqrt{-4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b \leq -6.5 \cdot 10^{-77}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{-4 \cdot \frac{c}{a}} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_3}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-55}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + t\_3}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t\_0}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* b b)))
        (t_1 (/ (+ c c) (- t_0 b)))
        (t_2 (* (/ (+ b b) a) -0.5))
        (t_3 (sqrt (* -4.0 (* a c)))))
   (if (<= b -6.5e-77)
     (if (>= b 0.0) (* (sqrt (* -4.0 (/ c a))) -0.5) t_1)
     (if (<= b -2e-310)
       (if (>= b 0.0) t_2 (/ (+ c c) t_3))
       (if (<= b 2.5e-55)
         (if (>= b 0.0) (* -0.5 (/ (+ b t_3) a)) (/ (* 2.0 c) (+ (- b) t_0)))
         (if (>= b 0.0) t_2 t_1))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((b * b));
	double t_1 = (c + c) / (t_0 - b);
	double t_2 = ((b + b) / a) * -0.5;
	double t_3 = sqrt((-4.0 * (a * c)));
	double tmp_1;
	if (b <= -6.5e-77) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = sqrt((-4.0 * (c / a))) * -0.5;
		} else {
			tmp_2 = t_1;
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_2;
		} else {
			tmp_3 = (c + c) / t_3;
		}
		tmp_1 = tmp_3;
	} else if (b <= 2.5e-55) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = -0.5 * ((b + t_3) / a);
		} else {
			tmp_4 = (2.0 * c) / (-b + t_0);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = t_2;
	} 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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    real(8) :: tmp_4
    t_0 = sqrt((b * b))
    t_1 = (c + c) / (t_0 - b)
    t_2 = ((b + b) / a) * (-0.5d0)
    t_3 = sqrt(((-4.0d0) * (a * c)))
    if (b <= (-6.5d-77)) then
        if (b >= 0.0d0) then
            tmp_2 = sqrt(((-4.0d0) * (c / a))) * (-0.5d0)
        else
            tmp_2 = t_1
        end if
        tmp_1 = tmp_2
    else if (b <= (-2d-310)) then
        if (b >= 0.0d0) then
            tmp_3 = t_2
        else
            tmp_3 = (c + c) / t_3
        end if
        tmp_1 = tmp_3
    else if (b <= 2.5d-55) then
        if (b >= 0.0d0) then
            tmp_4 = (-0.5d0) * ((b + t_3) / a)
        else
            tmp_4 = (2.0d0 * c) / (-b + t_0)
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = t_2
    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 = Math.sqrt((b * b));
	double t_1 = (c + c) / (t_0 - b);
	double t_2 = ((b + b) / a) * -0.5;
	double t_3 = Math.sqrt((-4.0 * (a * c)));
	double tmp_1;
	if (b <= -6.5e-77) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = Math.sqrt((-4.0 * (c / a))) * -0.5;
		} else {
			tmp_2 = t_1;
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_2;
		} else {
			tmp_3 = (c + c) / t_3;
		}
		tmp_1 = tmp_3;
	} else if (b <= 2.5e-55) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = -0.5 * ((b + t_3) / a);
		} else {
			tmp_4 = (2.0 * c) / (-b + t_0);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = t_2;
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt((b * b))
	t_1 = (c + c) / (t_0 - b)
	t_2 = ((b + b) / a) * -0.5
	t_3 = math.sqrt((-4.0 * (a * c)))
	tmp_1 = 0
	if b <= -6.5e-77:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = math.sqrt((-4.0 * (c / a))) * -0.5
		else:
			tmp_2 = t_1
		tmp_1 = tmp_2
	elif b <= -2e-310:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = t_2
		else:
			tmp_3 = (c + c) / t_3
		tmp_1 = tmp_3
	elif b <= 2.5e-55:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = -0.5 * ((b + t_3) / a)
		else:
			tmp_4 = (2.0 * c) / (-b + t_0)
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = t_2
	else:
		tmp_1 = t_1
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(b * b))
	t_1 = Float64(Float64(c + c) / Float64(t_0 - b))
	t_2 = Float64(Float64(Float64(b + b) / a) * -0.5)
	t_3 = sqrt(Float64(-4.0 * Float64(a * c)))
	tmp_1 = 0.0
	if (b <= -6.5e-77)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(sqrt(Float64(-4.0 * Float64(c / a))) * -0.5);
		else
			tmp_2 = t_1;
		end
		tmp_1 = tmp_2;
	elseif (b <= -2e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_2;
		else
			tmp_3 = Float64(Float64(c + c) / t_3);
		end
		tmp_1 = tmp_3;
	elseif (b <= 2.5e-55)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(-0.5 * Float64(Float64(b + t_3) / a));
		else
			tmp_4 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_0));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = t_2;
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = sqrt((b * b));
	t_1 = (c + c) / (t_0 - b);
	t_2 = ((b + b) / a) * -0.5;
	t_3 = sqrt((-4.0 * (a * c)));
	tmp_2 = 0.0;
	if (b <= -6.5e-77)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = sqrt((-4.0 * (c / a))) * -0.5;
		else
			tmp_3 = t_1;
		end
		tmp_2 = tmp_3;
	elseif (b <= -2e-310)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = t_2;
		else
			tmp_4 = (c + c) / t_3;
		end
		tmp_2 = tmp_4;
	elseif (b <= 2.5e-55)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = -0.5 * ((b + t_3) / a);
		else
			tmp_5 = (2.0 * c) / (-b + t_0);
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = t_2;
	else
		tmp_2 = t_1;
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(c + c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -6.5e-77], If[GreaterEqual[b, 0.0], N[(N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision], t$95$1], If[LessEqual[b, -2e-310], If[GreaterEqual[b, 0.0], t$95$2, N[(N[(c + c), $MachinePrecision] / t$95$3), $MachinePrecision]], If[LessEqual[b, 2.5e-55], If[GreaterEqual[b, 0.0], N[(-0.5 * N[(N[(b + t$95$3), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -6.4999999999999999e-77

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.4

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

   7. Applied rewrites 60.4%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.3

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

   10. Applied rewrites 48.3%

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

   11. Taylor expanded in b around inf

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

       1. count-2-rev N/A

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

       2. lower-+.f64 57.9

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

   13. Applied rewrites 57.9%

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

   14. Taylor expanded in a around inf

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-sqrt.f64 31.2

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

   16. Applied rewrites 31.2%

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

   if -6.4999999999999999e-77 < b < -1.999999999999994e-310

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.4

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

   7. Applied rewrites 60.4%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.3

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

   10. Applied rewrites 48.3%

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

   11. Taylor expanded in b around inf

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

       1. count-2-rev N/A

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

       2. lower-+.f64 57.9

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

   13. Applied rewrites 57.9%

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

   14. Taylor expanded in b around 0

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

       1. associate-*r/ N/A

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

       2. count-2-rev N/A

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

       3. lift-+.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lift-*.f64 47.4

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

   16. Applied rewrites 47.4%

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

   if -1.999999999999994e-310 < b < 2.5000000000000001e-55

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.4

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.3

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

   7. Applied rewrites 48.3%

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

   8. Taylor expanded in b around 0

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

      1. distribute-lft-out N/A

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

      2. distribute-lft-neg-in N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. div-add-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 44.4

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

   10. Applied rewrites 44.4%

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

   if 2.5000000000000001e-55 < b

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.4

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

   7. Applied rewrites 60.4%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.3

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

   10. Applied rewrites 48.3%

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

   11. Taylor expanded in b around inf

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

       1. count-2-rev N/A

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

       2. lower-+.f64 57.9

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

   13. Applied rewrites 57.9%

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

Alternative 12: 71.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c + c}{\sqrt{b \cdot b} - b}\\ t_1 := \frac{b + b}{a} \cdot -0.5\\ t_2 := \sqrt{-4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b \leq -6.5 \cdot 10^{-77}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{-4 \cdot \frac{c}{a}} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_2}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-71}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{t\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (+ c c) (- (sqrt (* b b)) b)))
        (t_1 (* (/ (+ b b) a) -0.5))
        (t_2 (sqrt (* -4.0 (* a c)))))
   (if (<= b -6.5e-77)
     (if (>= b 0.0) (* (sqrt (* -4.0 (/ c a))) -0.5) t_0)
     (if (<= b -2e-310)
       (if (>= b 0.0) t_1 (/ (+ c c) t_2))
       (if (<= b 2.5e-71)
         (if (>= b 0.0) (* -0.5 (/ t_2 a)) t_0)
         (if (>= b 0.0) t_1 t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = (c + c) / (sqrt((b * b)) - b);
	double t_1 = ((b + b) / a) * -0.5;
	double t_2 = sqrt((-4.0 * (a * c)));
	double tmp_1;
	if (b <= -6.5e-77) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = sqrt((-4.0 * (c / a))) * -0.5;
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = (c + c) / t_2;
		}
		tmp_1 = tmp_3;
	} else if (b <= 2.5e-71) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = -0.5 * (t_2 / a);
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = t_1;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    real(8) :: tmp_4
    t_0 = (c + c) / (sqrt((b * b)) - b)
    t_1 = ((b + b) / a) * (-0.5d0)
    t_2 = sqrt(((-4.0d0) * (a * c)))
    if (b <= (-6.5d-77)) then
        if (b >= 0.0d0) then
            tmp_2 = sqrt(((-4.0d0) * (c / a))) * (-0.5d0)
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b <= (-2d-310)) then
        if (b >= 0.0d0) then
            tmp_3 = t_1
        else
            tmp_3 = (c + c) / t_2
        end if
        tmp_1 = tmp_3
    else if (b <= 2.5d-71) then
        if (b >= 0.0d0) then
            tmp_4 = (-0.5d0) * (t_2 / a)
        else
            tmp_4 = t_0
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = t_1
    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 = (c + c) / (Math.sqrt((b * b)) - b);
	double t_1 = ((b + b) / a) * -0.5;
	double t_2 = Math.sqrt((-4.0 * (a * c)));
	double tmp_1;
	if (b <= -6.5e-77) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = Math.sqrt((-4.0 * (c / a))) * -0.5;
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = (c + c) / t_2;
		}
		tmp_1 = tmp_3;
	} else if (b <= 2.5e-71) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = -0.5 * (t_2 / a);
		} else {
			tmp_4 = t_0;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = t_1;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (c + c) / (math.sqrt((b * b)) - b)
	t_1 = ((b + b) / a) * -0.5
	t_2 = math.sqrt((-4.0 * (a * c)))
	tmp_1 = 0
	if b <= -6.5e-77:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = math.sqrt((-4.0 * (c / a))) * -0.5
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b <= -2e-310:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = t_1
		else:
			tmp_3 = (c + c) / t_2
		tmp_1 = tmp_3
	elif b <= 2.5e-71:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = -0.5 * (t_2 / a)
		else:
			tmp_4 = t_0
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = t_1
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c + c) / Float64(sqrt(Float64(b * b)) - b))
	t_1 = Float64(Float64(Float64(b + b) / a) * -0.5)
	t_2 = sqrt(Float64(-4.0 * Float64(a * c)))
	tmp_1 = 0.0
	if (b <= -6.5e-77)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(sqrt(Float64(-4.0 * Float64(c / a))) * -0.5);
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b <= -2e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = Float64(Float64(c + c) / t_2);
		end
		tmp_1 = tmp_3;
	elseif (b <= 2.5e-71)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(-0.5 * Float64(t_2 / a));
		else
			tmp_4 = t_0;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = t_1;
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = (c + c) / (sqrt((b * b)) - b);
	t_1 = ((b + b) / a) * -0.5;
	t_2 = sqrt((-4.0 * (a * c)));
	tmp_2 = 0.0;
	if (b <= -6.5e-77)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = sqrt((-4.0 * (c / a))) * -0.5;
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b <= -2e-310)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = t_1;
		else
			tmp_4 = (c + c) / t_2;
		end
		tmp_2 = tmp_4;
	elseif (b <= 2.5e-71)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = -0.5 * (t_2 / a);
		else
			tmp_5 = t_0;
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = t_1;
	else
		tmp_2 = t_0;
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -6.5e-77], If[GreaterEqual[b, 0.0], N[(N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision], t$95$0], If[LessEqual[b, -2e-310], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(c + c), $MachinePrecision] / t$95$2), $MachinePrecision]], If[LessEqual[b, 2.5e-71], If[GreaterEqual[b, 0.0], N[(-0.5 * N[(t$95$2 / a), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -6.4999999999999999e-77

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.4

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

   7. Applied rewrites 60.4%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.3

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

   10. Applied rewrites 48.3%

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

   11. Taylor expanded in b around inf

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

       1. count-2-rev N/A

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

       2. lower-+.f64 57.9

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

   13. Applied rewrites 57.9%

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

   14. Taylor expanded in a around inf

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-sqrt.f64 31.2

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

   16. Applied rewrites 31.2%

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

   if -6.4999999999999999e-77 < b < -1.999999999999994e-310

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.4

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

   7. Applied rewrites 60.4%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.3

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

   10. Applied rewrites 48.3%

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

   11. Taylor expanded in b around inf

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

       1. count-2-rev N/A

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

       2. lower-+.f64 57.9

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

   13. Applied rewrites 57.9%

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

   14. Taylor expanded in b around 0

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

       1. associate-*r/ N/A

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

       2. count-2-rev N/A

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

       3. lift-+.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lift-*.f64 47.4

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

   16. Applied rewrites 47.4%

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

   if -1.999999999999994e-310 < b < 2.49999999999999999e-71

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.4

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

   7. Applied rewrites 60.4%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.3

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

   10. Applied rewrites 48.3%

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

   11. Taylor expanded in b around inf

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

       1. count-2-rev N/A

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

       2. lower-+.f64 57.9

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

   13. Applied rewrites 57.9%

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

   14. Taylor expanded in b around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 36.8

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

   16. Applied rewrites 36.8%

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

   if 2.49999999999999999e-71 < b

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.4

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

   7. Applied rewrites 60.4%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.3

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

   10. Applied rewrites 48.3%

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

   11. Taylor expanded in b around inf

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

       1. count-2-rev N/A

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

       2. lower-+.f64 57.9

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

   13. Applied rewrites 57.9%

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

Alternative 13: 64.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-77}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{-4 \cdot \frac{c}{a}} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.5e-77)
   (if (>= b 0.0)
     (* (sqrt (* -4.0 (/ c a))) -0.5)
     (/ (+ c c) (- (sqrt (* b b)) b)))
   (if (>= b 0.0) (* (/ (+ b b) a) -0.5) (/ (+ c c) (sqrt (* -4.0 (* a c)))))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -6.5e-77) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = sqrt((-4.0 * (c / a))) * -0.5;
		} else {
			tmp_2 = (c + c) / (sqrt((b * b)) - b);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = ((b + b) / a) * -0.5;
	} else {
		tmp_1 = (c + c) / sqrt((-4.0 * (a * c)));
	}
	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 <= (-6.5d-77)) then
        if (b >= 0.0d0) then
            tmp_2 = sqrt(((-4.0d0) * (c / a))) * (-0.5d0)
        else
            tmp_2 = (c + c) / (sqrt((b * b)) - b)
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = ((b + b) / a) * (-0.5d0)
    else
        tmp_1 = (c + c) / sqrt(((-4.0d0) * (a * c)))
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -6.5e-77) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = Math.sqrt((-4.0 * (c / a))) * -0.5;
		} else {
			tmp_2 = (c + c) / (Math.sqrt((b * b)) - b);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = ((b + b) / a) * -0.5;
	} else {
		tmp_1 = (c + c) / Math.sqrt((-4.0 * (a * c)));
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if b <= -6.5e-77:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = math.sqrt((-4.0 * (c / a))) * -0.5
		else:
			tmp_2 = (c + c) / (math.sqrt((b * b)) - b)
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = ((b + b) / a) * -0.5
	else:
		tmp_1 = (c + c) / math.sqrt((-4.0 * (a * c)))
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -6.5e-77)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(sqrt(Float64(-4.0 * Float64(c / a))) * -0.5);
		else
			tmp_2 = Float64(Float64(c + c) / Float64(sqrt(Float64(b * b)) - b));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(b + b) / a) * -0.5);
	else
		tmp_1 = Float64(Float64(c + c) / sqrt(Float64(-4.0 * Float64(a * c))));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp_2 = 0.0;
	if (b <= -6.5e-77)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = sqrt((-4.0 * (c / a))) * -0.5;
		else
			tmp_3 = (c + c) / (sqrt((b * b)) - b);
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = ((b + b) / a) * -0.5;
	else
		tmp_2 = (c + c) / sqrt((-4.0 * (a * c)));
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.5e-77], If[GreaterEqual[b, 0.0], N[(N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -6.4999999999999999e-77

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.4

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

   7. Applied rewrites 60.4%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.3

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

   10. Applied rewrites 48.3%

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

   11. Taylor expanded in b around inf

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

       1. count-2-rev N/A

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

       2. lower-+.f64 57.9

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

   13. Applied rewrites 57.9%

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

   14. Taylor expanded in a around inf

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-sqrt.f64 31.2

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

   16. Applied rewrites 31.2%

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

   if -6.4999999999999999e-77 < b

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 60.4

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

   7. Applied rewrites 60.4%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.3

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

   10. Applied rewrites 48.3%

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

   11. Taylor expanded in b around inf

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

       1. count-2-rev N/A

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

       2. lower-+.f64 57.9

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

   13. Applied rewrites 57.9%

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

   14. Taylor expanded in b around 0

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

       1. associate-*r/ N/A

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

       2. count-2-rev N/A

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

       3. lift-+.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lift-*.f64 47.4

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

   16. Applied rewrites 47.4%

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

Alternative 14: 57.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.1%

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

2. Taylor expanded in a around 0

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

4. Applied rewrites 72.2%

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

5. Taylor expanded in a around 0

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

   1. pow2 N/A

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

   2. lift-*.f64 60.4

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

7. Applied rewrites 60.4%

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

8. Taylor expanded in a around 0

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

   1. pow2 N/A

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

   2. lift-*.f64 48.3

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

10. Applied rewrites 48.3%

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

11. Taylor expanded in b around inf

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

    1. count-2-rev N/A

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

    2. lower-+.f64 57.9

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

13. Applied rewrites 57.9%

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

Alternative 15: 31.2% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0)
   (* (sqrt (* -4.0 (/ c a))) -0.5)
   (/ (+ c c) (- (sqrt (* b b)) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = sqrt((-4.0 * (c / a))) * -0.5;
	} else {
		tmp = (c + c) / (sqrt((b * b)) - b);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b >= 0.0d0) then
        tmp = sqrt(((-4.0d0) * (c / a))) * (-0.5d0)
    else
        tmp = (c + c) / (sqrt((b * b)) - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = Math.sqrt((-4.0 * (c / a))) * -0.5;
	} else {
		tmp = (c + c) / (Math.sqrt((b * b)) - b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = math.sqrt((-4.0 * (c / a))) * -0.5
	else:
		tmp = (c + c) / (math.sqrt((b * b)) - b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(sqrt(Float64(-4.0 * Float64(c / a))) * -0.5);
	else
		tmp = Float64(Float64(c + c) / Float64(sqrt(Float64(b * b)) - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = sqrt((-4.0 * (c / a))) * -0.5;
	else
		tmp = (c + c) / (sqrt((b * b)) - b);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Initial program 72.1%

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

2. Taylor expanded in a around 0

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

4. Applied rewrites 72.2%

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

5. Taylor expanded in a around 0

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

   1. pow2 N/A

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

   2. lift-*.f64 60.4

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

7. Applied rewrites 60.4%

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

8. Taylor expanded in a around 0

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

   1. pow2 N/A

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

   2. lift-*.f64 48.3

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

10. Applied rewrites 48.3%

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

11. Taylor expanded in b around inf

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

    1. count-2-rev N/A

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

    2. lower-+.f64 57.9

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

13. Applied rewrites 57.9%

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

14. Taylor expanded in a around inf

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-sqrt.f64 31.2

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

16. Applied rewrites 31.2%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{-4 \cdot \frac{c}{a}} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2025139 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  :precision binary64
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))