jeff quadratic root 1

Percentage Accurate: 72.2% → 90.5%

Time: 4.0s

Alternatives: 10

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: 72.2% 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: 90.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}\\ \mathbf{if}\;b \leq -5.5 \cdot 10^{+77}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{+106}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(\frac{b}{a} + \frac{t\_0}{a}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma b b (* -4.0 (* c a))))))
   (if (<= b -5.5e+77)
     (if (>= b 0.0)
       (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))
       (/ (* 2.0 c) (* (- b) 2.0)))
     (if (<= b 8.5e+106)
       (if (>= b 0.0) (* (+ (/ b a) (/ t_0 a)) -0.5) (/ (+ c c) (- t_0 b)))
       (if (>= b 0.0)
         (* (/ (* 2.0 b) a) -0.5)
         (/ (+ c c) (- (sqrt (* b b)) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma(b, b, (-4.0 * (c * a))));
	double tmp_1;
	if (b <= -5.5e+77) {
		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 <= 8.5e+106) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = ((b / a) + (t_0 / a)) * -0.5;
		} else {
			tmp_3 = (c + c) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = ((2.0 * 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(b, b, Float64(-4.0 * Float64(c * a))))
	tmp_1 = 0.0
	if (b <= -5.5e+77)
		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 <= 8.5e+106)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(b / a) + Float64(t_0 / 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(2.0 * 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[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -5.5e+77], 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, 8.5e+106], If[GreaterEqual[b, 0.0], N[(N[(N[(b / a), $MachinePrecision] + N[(t$95$0 / 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[(2.0 * 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 < -5.50000000000000036e77

   1. Initial program 72.2%

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

   2. 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. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 70.1

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

   4. Applied rewrites 70.1%

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

   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 70.7%

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

   if -5.50000000000000036e77 < b < 8.4999999999999992e106

   1. Initial program 72.2%

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

   2. 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%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. div-add N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-sqrt.f64 72.2

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

   6. Applied rewrites 72.2%

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

   if 8.4999999999999992e106 < b

   1. Initial program 72.2%

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

   2. 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%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 59.9

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

   7. Applied rewrites 59.9%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.6

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

   10. Applied rewrites 48.6%

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

   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. lower-*.f64 58.5

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

   13. Applied rewrites 58.5%

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

Alternative 2: 90.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}\\ \mathbf{if}\;b \leq -5.5 \cdot 10^{+77}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{+106}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + t\_0}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma b b (* -4.0 (* c a))))))
   (if (<= b -5.5e+77)
     (if (>= b 0.0)
       (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))
       (/ (* 2.0 c) (* (- b) 2.0)))
     (if (<= b 8.5e+106)
       (if (>= b 0.0) (* (/ (+ b t_0) a) -0.5) (/ (+ c c) (- t_0 b)))
       (if (>= b 0.0)
         (* (/ (* 2.0 b) a) -0.5)
         (/ (+ c c) (- (sqrt (* b b)) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma(b, b, (-4.0 * (c * a))));
	double tmp_1;
	if (b <= -5.5e+77) {
		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 <= 8.5e+106) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = ((b + t_0) / a) * -0.5;
		} else {
			tmp_3 = (c + c) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = ((2.0 * 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(b, b, Float64(-4.0 * Float64(c * a))))
	tmp_1 = 0.0
	if (b <= -5.5e+77)
		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 <= 8.5e+106)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(b + t_0) / 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(2.0 * 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[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -5.5e+77], 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, 8.5e+106], If[GreaterEqual[b, 0.0], N[(N[(N[(b + t$95$0), $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[(2.0 * 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 < -5.50000000000000036e77

   1. Initial program 72.2%

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

   2. 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. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 70.1

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

   4. Applied rewrites 70.1%

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

   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 70.7%

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

   if -5.50000000000000036e77 < b < 8.4999999999999992e106

   1. Initial program 72.2%

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

   2. 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%

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

   if 8.4999999999999992e106 < b

   1. Initial program 72.2%

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

   2. 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%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 59.9

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

   7. Applied rewrites 59.9%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.6

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

   10. Applied rewrites 48.6%

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

   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. lower-*.f64 58.5

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

   13. Applied rewrites 58.5%

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

Alternative 3: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}\\ \mathbf{if}\;b \leq -1.2 \cdot 10^{+163}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\frac{a}{c} \cdot -4} \cdot c}{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c + \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{+106}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + t\_0}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma b b (* -4.0 (* c a))))))
   (if (<= b -1.2e+163)
     (if (>= b 0.0)
       (* (/ (* (sqrt (* (/ a c) -4.0)) c) a) 0.5)
       (* -1.0 (/ (+ c (/ (* a (* c c)) (* b b))) b)))
     (if (<= b 8.5e+106)
       (if (>= b 0.0) (* (/ (+ b t_0) a) -0.5) (/ (+ c c) (- t_0 b)))
       (if (>= b 0.0)
         (* (/ (* 2.0 b) a) -0.5)
         (/ (+ c c) (- (sqrt (* b b)) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma(b, b, (-4.0 * (c * a))));
	double tmp_1;
	if (b <= -1.2e+163) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = ((sqrt(((a / c) * -4.0)) * c) / a) * 0.5;
		} else {
			tmp_2 = -1.0 * ((c + ((a * (c * c)) / (b * b))) / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 8.5e+106) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = ((b + t_0) / a) * -0.5;
		} else {
			tmp_3 = (c + c) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = ((2.0 * 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(b, b, Float64(-4.0 * Float64(c * a))))
	tmp_1 = 0.0
	if (b <= -1.2e+163)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(Float64(sqrt(Float64(Float64(a / c) * -4.0)) * c) / a) * 0.5);
		else
			tmp_2 = Float64(-1.0 * Float64(Float64(c + Float64(Float64(a * Float64(c * c)) / Float64(b * b))) / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 8.5e+106)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(b + t_0) / 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(2.0 * 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[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.2e+163], If[GreaterEqual[b, 0.0], N[(N[(N[(N[Sqrt[N[(N[(a / c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision], N[(-1.0 * N[(N[(c + N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 8.5e+106], If[GreaterEqual[b, 0.0], N[(N[(N[(b + t$95$0), $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[(2.0 * 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 < -1.1999999999999999e163

   1. Initial program 72.2%

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

   2. Taylor expanded in c around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 43.0

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

   4. Applied rewrites 43.0%

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

   5. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 33.8

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

   7. Applied rewrites 33.8%

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

   if -1.1999999999999999e163 < b < 8.4999999999999992e106

   1. Initial program 72.2%

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

   2. 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%

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

   if 8.4999999999999992e106 < b

   1. Initial program 72.2%

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

   2. 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%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 59.9

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

   7. Applied rewrites 59.9%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.6

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

   10. Applied rewrites 48.6%

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

   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. lower-*.f64 58.5

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

   13. Applied rewrites 58.5%

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

Alternative 4: 85.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}\\ \mathbf{if}\;b \leq -1.2 \cdot 10^{+163}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\frac{a}{c} \cdot -4} \cdot c}{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c + \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(b \cdot \mathsf{fma}\left(-2, \frac{c}{b \cdot b}, \frac{2}{a}\right)\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{+106}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + t\_0}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{1}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma b b (* -4.0 (* c a))))))
   (if (<= b -1.2e+163)
     (if (>= b 0.0)
       (* (/ (* (sqrt (* (/ a c) -4.0)) c) a) 0.5)
       (* -1.0 (/ (+ c (/ (* a (* c c)) (* b b))) b)))
     (if (<= b -2e-310)
       (if (>= b 0.0)
         (* (* b (fma -2.0 (/ c (* b b)) (/ 2.0 a))) -0.5)
         (/ (+ c c) (- t_0 b)))
       (if (<= b 8.5e+106)
         (if (>= b 0.0)
           (* (/ (+ b t_0) a) -0.5)
           (* -2.0 (/ 1.0 (* a (sqrt (/ -4.0 (* a c)))))))
         (if (>= b 0.0)
           (* (/ (* 2.0 b) a) -0.5)
           (/ (+ c c) (- (sqrt (* b b)) b))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma(b, b, (-4.0 * (c * a))));
	double tmp_1;
	if (b <= -1.2e+163) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = ((sqrt(((a / c) * -4.0)) * c) / a) * 0.5;
		} else {
			tmp_2 = -1.0 * ((c + ((a * (c * c)) / (b * b))) / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (b * fma(-2.0, (c / (b * b)), (2.0 / a))) * -0.5;
		} else {
			tmp_3 = (c + c) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b <= 8.5e+106) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = ((b + t_0) / a) * -0.5;
		} else {
			tmp_4 = -2.0 * (1.0 / (a * sqrt((-4.0 / (a * c)))));
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = ((2.0 * 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(b, b, Float64(-4.0 * Float64(c * a))))
	tmp_1 = 0.0
	if (b <= -1.2e+163)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(Float64(sqrt(Float64(Float64(a / c) * -4.0)) * c) / a) * 0.5);
		else
			tmp_2 = Float64(-1.0 * Float64(Float64(c + Float64(Float64(a * Float64(c * c)) / Float64(b * b))) / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= -2e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(b * fma(-2.0, Float64(c / Float64(b * b)), Float64(2.0 / a))) * -0.5);
		else
			tmp_3 = Float64(Float64(c + c) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	elseif (b <= 8.5e+106)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(Float64(b + t_0) / a) * -0.5);
		else
			tmp_4 = Float64(-2.0 * Float64(1.0 / Float64(a * sqrt(Float64(-4.0 / Float64(a * c))))));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(2.0 * 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[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.2e+163], If[GreaterEqual[b, 0.0], N[(N[(N[(N[Sqrt[N[(N[(a / c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision], N[(-1.0 * N[(N[(c + N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -2e-310], 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[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 8.5e+106], If[GreaterEqual[b, 0.0], N[(N[(N[(b + t$95$0), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], N[(-2.0 * N[(1.0 / N[(a * N[Sqrt[N[(-4.0 / N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(N[(2.0 * 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 < -1.1999999999999999e163

   1. Initial program 72.2%

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

   2. Taylor expanded in c around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 43.0

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

   4. Applied rewrites 43.0%

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

   5. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 33.8

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

   7. Applied rewrites 33.8%

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

   if -1.1999999999999999e163 < b < -1.999999999999994e-310

   1. Initial program 72.2%

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

   2. 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%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. mult-flip-rev N/A

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

      7. lower-/.f64 69.0

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

   7. Applied rewrites 69.0%

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

   if -1.999999999999994e-310 < b < 8.4999999999999992e106

   1. Initial program 72.2%

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

   2. 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%

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

   5. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 43.5

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

   7. Applied rewrites 43.5%

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

   8. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 50.2

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

   10. Applied rewrites 50.2%

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

   if 8.4999999999999992e106 < b

   1. Initial program 72.2%

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

   2. 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%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 59.9

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

   7. Applied rewrites 59.9%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.6

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

   10. Applied rewrites 48.6%

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

   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. lower-*.f64 58.5

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

   13. Applied rewrites 58.5%

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

Alternative 5: 81.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -4 \cdot \left(c \cdot a\right)\\ t_1 := \sqrt{t\_0}\\ t_2 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(b \cdot \mathsf{fma}\left(-2, \frac{c}{b \cdot b}, \frac{2}{a}\right)\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(b, b, t\_0\right)} - b}\\ \end{array}\\ \mathbf{if}\;b \leq -1.2 \cdot 10^{+163}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\frac{a}{c} \cdot -4} \cdot c}{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c + \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-238}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;b \leq 8.8 \cdot 10^{-72}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t\_1}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* -4.0 (* c a)))
        (t_1 (sqrt t_0))
        (t_2
         (if (>= b 0.0)
           (* (* b (fma -2.0 (/ c (* b b)) (/ 2.0 a))) -0.5)
           (/ (+ c c) (- (sqrt (fma b b t_0)) b)))))
   (if (<= b -1.2e+163)
     (if (>= b 0.0)
       (* (/ (* (sqrt (* (/ a c) -4.0)) c) a) 0.5)
       (* -1.0 (/ (+ c (/ (* a (* c c)) (* b b))) b)))
     (if (<= b -5e-238)
       t_2
       (if (<= b 8.8e-72)
         (if (>= b 0.0)
           (/ (- (- b) t_1) (* 2.0 a))
           (/ (* 2.0 c) (+ (- b) t_1)))
         t_2)))))

C

double code(double a, double b, double c) {
	double t_0 = -4.0 * (c * a);
	double t_1 = sqrt(t_0);
	double tmp;
	if (b >= 0.0) {
		tmp = (b * fma(-2.0, (c / (b * b)), (2.0 / a))) * -0.5;
	} else {
		tmp = (c + c) / (sqrt(fma(b, b, t_0)) - b);
	}
	double t_2 = tmp;
	double tmp_2;
	if (b <= -1.2e+163) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = ((sqrt(((a / c) * -4.0)) * c) / a) * 0.5;
		} else {
			tmp_3 = -1.0 * ((c + ((a * (c * c)) / (b * b))) / b);
		}
		tmp_2 = tmp_3;
	} else if (b <= -5e-238) {
		tmp_2 = t_2;
	} else if (b <= 8.8e-72) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (-b - t_1) / (2.0 * a);
		} else {
			tmp_4 = (2.0 * c) / (-b + t_1);
		}
		tmp_2 = tmp_4;
	} else {
		tmp_2 = t_2;
	}
	return tmp_2;
}

Julia

function code(a, b, c)
	t_0 = Float64(-4.0 * Float64(c * a))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(b * fma(-2.0, Float64(c / Float64(b * b)), Float64(2.0 / a))) * -0.5);
	else
		tmp = Float64(Float64(c + c) / Float64(sqrt(fma(b, b, t_0)) - b));
	end
	t_2 = tmp
	tmp_2 = 0.0
	if (b <= -1.2e+163)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(sqrt(Float64(Float64(a / c) * -4.0)) * c) / a) * 0.5);
		else
			tmp_3 = Float64(-1.0 * Float64(Float64(c + Float64(Float64(a * Float64(c * c)) / Float64(b * b))) / b));
		end
		tmp_2 = tmp_3;
	elseif (b <= -5e-238)
		tmp_2 = t_2;
	elseif (b <= 8.8e-72)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(Float64(-b) - t_1) / Float64(2.0 * a));
		else
			tmp_4 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_1));
		end
		tmp_2 = tmp_4;
	else
		tmp_2 = t_2;
	end
	return tmp_2
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$2 = 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 + t$95$0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]}, If[LessEqual[b, -1.2e+163], If[GreaterEqual[b, 0.0], N[(N[(N[(N[Sqrt[N[(N[(a / c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision], N[(-1.0 * N[(N[(c + N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -5e-238], t$95$2, If[LessEqual[b, 8.8e-72], 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]], t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.1999999999999999e163

   1. Initial program 72.2%

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

   2. Taylor expanded in c around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 43.0

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

   4. Applied rewrites 43.0%

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

   5. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 33.8

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

   7. Applied rewrites 33.8%

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

   if -1.1999999999999999e163 < b < -5e-238 or 8.8000000000000001e-72 < b

   1. Initial program 72.2%

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

   2. 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%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. mult-flip-rev N/A

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

      7. lower-/.f64 69.0

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

   7. Applied rewrites 69.0%

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

   if -5e-238 < b < 8.8000000000000001e-72

   1. Initial program 72.2%

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

   2. 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 56.3

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

   4. Applied rewrites 56.3%

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

   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 40.4

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

   7. Applied rewrites 40.4%

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

Alternative 6: 74.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-4 \cdot \left(c \cdot a\right)}\\ \mathbf{if}\;b \leq -0.00088:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\frac{a}{c} \cdot -4} \cdot c}{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c + \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 8.8 \cdot 10^{-72}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_0}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t\_0}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* -4.0 (* c a)))))
   (if (<= b -0.00088)
     (if (>= b 0.0)
       (* (/ (* (sqrt (* (/ a c) -4.0)) c) a) 0.5)
       (* -1.0 (/ (+ c (/ (* a (* c c)) (* b b))) b)))
     (if (<= b 8.8e-72)
       (if (>= b 0.0) (/ (- (- b) t_0) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) t_0)))
       (if (>= b 0.0)
         (* (/ (* 2.0 b) a) -0.5)
         (/ (+ c c) (- (sqrt (* b b)) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((-4.0 * (c * a)));
	double tmp_1;
	if (b <= -0.00088) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = ((sqrt(((a / c) * -4.0)) * c) / a) * 0.5;
		} else {
			tmp_2 = -1.0 * ((c + ((a * (c * c)) / (b * b))) / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 8.8e-72) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_0) / (2.0 * a);
		} else {
			tmp_3 = (2.0 * c) / (-b + t_0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = ((2.0 * b) / a) * -0.5;
	} else {
		tmp_1 = (c + c) / (sqrt((b * b)) - b);
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = sqrt(((-4.0d0) * (c * a)))
    if (b <= (-0.00088d0)) then
        if (b >= 0.0d0) then
            tmp_2 = ((sqrt(((a / c) * (-4.0d0))) * c) / a) * 0.5d0
        else
            tmp_2 = (-1.0d0) * ((c + ((a * (c * c)) / (b * b))) / b)
        end if
        tmp_1 = tmp_2
    else if (b <= 8.8d-72) then
        if (b >= 0.0d0) then
            tmp_3 = (-b - t_0) / (2.0d0 * a)
        else
            tmp_3 = (2.0d0 * c) / (-b + t_0)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = ((2.0d0 * b) / a) * (-0.5d0)
    else
        tmp_1 = (c + c) / (sqrt((b * b)) - b)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt((-4.0 * (c * a)));
	double tmp_1;
	if (b <= -0.00088) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = ((Math.sqrt(((a / c) * -4.0)) * c) / a) * 0.5;
		} else {
			tmp_2 = -1.0 * ((c + ((a * (c * c)) / (b * b))) / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 8.8e-72) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_0) / (2.0 * a);
		} else {
			tmp_3 = (2.0 * c) / (-b + t_0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = ((2.0 * b) / a) * -0.5;
	} else {
		tmp_1 = (c + c) / (Math.sqrt((b * b)) - b);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt((-4.0 * (c * a)))
	tmp_1 = 0
	if b <= -0.00088:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = ((math.sqrt(((a / c) * -4.0)) * c) / a) * 0.5
		else:
			tmp_2 = -1.0 * ((c + ((a * (c * c)) / (b * b))) / b)
		tmp_1 = tmp_2
	elif b <= 8.8e-72:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (-b - t_0) / (2.0 * a)
		else:
			tmp_3 = (2.0 * c) / (-b + t_0)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = ((2.0 * b) / a) * -0.5
	else:
		tmp_1 = (c + c) / (math.sqrt((b * b)) - b)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(-4.0 * Float64(c * a)))
	tmp_1 = 0.0
	if (b <= -0.00088)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(Float64(sqrt(Float64(Float64(a / c) * -4.0)) * c) / a) * 0.5);
		else
			tmp_2 = Float64(-1.0 * Float64(Float64(c + Float64(Float64(a * Float64(c * c)) / Float64(b * b))) / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 8.8e-72)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(-b) - t_0) / Float64(2.0 * a));
		else
			tmp_3 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_0));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(2.0 * b) / a) * -0.5);
	else
		tmp_1 = Float64(Float64(c + c) / Float64(sqrt(Float64(b * b)) - b));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt((-4.0 * (c * a)));
	tmp_2 = 0.0;
	if (b <= -0.00088)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = ((sqrt(((a / c) * -4.0)) * c) / a) * 0.5;
		else
			tmp_3 = -1.0 * ((c + ((a * (c * c)) / (b * b))) / b);
		end
		tmp_2 = tmp_3;
	elseif (b <= 8.8e-72)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (-b - t_0) / (2.0 * a);
		else
			tmp_4 = (2.0 * c) / (-b + t_0);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = ((2.0 * b) / a) * -0.5;
	else
		tmp_2 = (c + c) / (sqrt((b * b)) - b);
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -0.00088], If[GreaterEqual[b, 0.0], N[(N[(N[(N[Sqrt[N[(N[(a / c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision], N[(-1.0 * N[(N[(c + N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 8.8e-72], 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]], If[GreaterEqual[b, 0.0], N[(N[(N[(2.0 * 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 < -8.80000000000000031e-4

   1. Initial program 72.2%

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

   2. Taylor expanded in c around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 43.0

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

   4. Applied rewrites 43.0%

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

   5. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 33.8

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

   7. Applied rewrites 33.8%

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

   if -8.80000000000000031e-4 < b < 8.8000000000000001e-72

   1. Initial program 72.2%

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

   2. 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 56.3

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

   4. Applied rewrites 56.3%

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

   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 40.4

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

   7. Applied rewrites 40.4%

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

   if 8.8000000000000001e-72 < b

   1. Initial program 72.2%

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

   2. 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%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 59.9

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

   7. Applied rewrites 59.9%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.6

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

   10. Applied rewrites 48.6%

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

   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. lower-*.f64 58.5

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

   13. Applied rewrites 58.5%

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

Alternative 7: 71.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array}\\ t_1 := \sqrt{-4 \cdot \left(c \cdot a\right)}\\ \mathbf{if}\;b \leq -0.00088:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 8.8 \cdot 10^{-72}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t\_1}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0
         (if (>= b 0.0)
           (* (/ (* 2.0 b) a) -0.5)
           (/ (+ c c) (- (sqrt (* b b)) b))))
        (t_1 (sqrt (* -4.0 (* c a)))))
   (if (<= b -0.00088)
     t_0
     (if (<= b 8.8e-72)
       (if (>= b 0.0) (/ (- (- b) t_1) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) t_1)))
       t_0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = ((2.0 * b) / a) * -0.5;
	} else {
		tmp = (c + c) / (sqrt((b * b)) - b);
	}
	double t_0 = tmp;
	double t_1 = sqrt((-4.0 * (c * a)));
	double tmp_1;
	if (b <= -0.00088) {
		tmp_1 = t_0;
	} else if (b <= 8.8e-72) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-b - t_1) / (2.0 * a);
		} else {
			tmp_2 = (2.0 * c) / (-b + t_1);
		}
		tmp_1 = tmp_2;
	} 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
    if (b >= 0.0d0) then
        tmp = ((2.0d0 * b) / a) * (-0.5d0)
    else
        tmp = (c + c) / (sqrt((b * b)) - b)
    end if
    t_0 = tmp
    t_1 = sqrt(((-4.0d0) * (c * a)))
    if (b <= (-0.00088d0)) then
        tmp_1 = t_0
    else if (b <= 8.8d-72) then
        if (b >= 0.0d0) then
            tmp_2 = (-b - t_1) / (2.0d0 * a)
        else
            tmp_2 = (2.0d0 * c) / (-b + t_1)
        end if
        tmp_1 = tmp_2
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = ((2.0 * b) / a) * -0.5;
	} else {
		tmp = (c + c) / (Math.sqrt((b * b)) - b);
	}
	double t_0 = tmp;
	double t_1 = Math.sqrt((-4.0 * (c * a)));
	double tmp_1;
	if (b <= -0.00088) {
		tmp_1 = t_0;
	} else if (b <= 8.8e-72) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-b - t_1) / (2.0 * a);
		} else {
			tmp_2 = (2.0 * c) / (-b + t_1);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = ((2.0 * b) / a) * -0.5
	else:
		tmp = (c + c) / (math.sqrt((b * b)) - b)
	t_0 = tmp
	t_1 = math.sqrt((-4.0 * (c * a)))
	tmp_1 = 0
	if b <= -0.00088:
		tmp_1 = t_0
	elif b <= 8.8e-72:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (-b - t_1) / (2.0 * a)
		else:
			tmp_2 = (2.0 * c) / (-b + t_1)
		tmp_1 = tmp_2
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(Float64(2.0 * b) / a) * -0.5);
	else
		tmp = Float64(Float64(c + c) / Float64(sqrt(Float64(b * b)) - b));
	end
	t_0 = tmp
	t_1 = sqrt(Float64(-4.0 * Float64(c * a)))
	tmp_1 = 0.0
	if (b <= -0.00088)
		tmp_1 = t_0;
	elseif (b <= 8.8e-72)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(Float64(-b) - t_1) / Float64(2.0 * a));
		else
			tmp_2 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_1));
		end
		tmp_1 = tmp_2;
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = ((2.0 * b) / a) * -0.5;
	else
		tmp = (c + c) / (sqrt((b * b)) - b);
	end
	t_0 = tmp;
	t_1 = sqrt((-4.0 * (c * a)));
	tmp_2 = 0.0;
	if (b <= -0.00088)
		tmp_2 = t_0;
	elseif (b <= 8.8e-72)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (-b - t_1) / (2.0 * a);
		else
			tmp_3 = (2.0 * c) / (-b + t_1);
		end
		tmp_2 = tmp_3;
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = If[GreaterEqual[b, 0.0], N[(N[(N[(2.0 * b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], 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, -0.00088], t$95$0, If[LessEqual[b, 8.8e-72], 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]], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if b < -8.80000000000000031e-4 or 8.8000000000000001e-72 < b

   1. Initial program 72.2%

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

   2. 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%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 59.9

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

   7. Applied rewrites 59.9%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.6

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

   10. Applied rewrites 48.6%

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

   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. lower-*.f64 58.5

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

   13. Applied rewrites 58.5%

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

   if -8.80000000000000031e-4 < b < 8.8000000000000001e-72

   1. Initial program 72.2%

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

   2. 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 56.3

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

   4. Applied rewrites 56.3%

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

   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 40.4

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

   7. Applied rewrites 40.4%

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

Alternative 8: 71.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b} - b}\\ \end{array}\\ t_1 := \sqrt{-4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b \leq -2.6 \cdot 10^{-51}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 8.8 \cdot 10^{-72}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{t\_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{t\_1}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0
         (if (>= b 0.0)
           (* (/ (* 2.0 b) a) -0.5)
           (/ (+ c c) (- (sqrt (* b b)) b))))
        (t_1 (sqrt (- (* 4.0 (* a c))))))
   (if (<= b -2.6e-51)
     t_0
     (if (<= b 8.8e-72)
       (if (>= b 0.0) (* -0.5 (/ t_1 a)) (/ (* 2.0 c) t_1))
       t_0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = ((2.0 * b) / a) * -0.5;
	} else {
		tmp = (c + c) / (sqrt((b * b)) - b);
	}
	double t_0 = tmp;
	double t_1 = sqrt(-(4.0 * (a * c)));
	double tmp_1;
	if (b <= -2.6e-51) {
		tmp_1 = t_0;
	} else if (b <= 8.8e-72) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -0.5 * (t_1 / a);
		} else {
			tmp_2 = (2.0 * c) / t_1;
		}
		tmp_1 = tmp_2;
	} 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
    if (b >= 0.0d0) then
        tmp = ((2.0d0 * b) / a) * (-0.5d0)
    else
        tmp = (c + c) / (sqrt((b * b)) - b)
    end if
    t_0 = tmp
    t_1 = sqrt(-(4.0d0 * (a * c)))
    if (b <= (-2.6d-51)) then
        tmp_1 = t_0
    else if (b <= 8.8d-72) then
        if (b >= 0.0d0) then
            tmp_2 = (-0.5d0) * (t_1 / a)
        else
            tmp_2 = (2.0d0 * c) / t_1
        end if
        tmp_1 = tmp_2
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = ((2.0 * b) / a) * -0.5;
	} else {
		tmp = (c + c) / (Math.sqrt((b * b)) - b);
	}
	double t_0 = tmp;
	double t_1 = Math.sqrt(-(4.0 * (a * c)));
	double tmp_1;
	if (b <= -2.6e-51) {
		tmp_1 = t_0;
	} else if (b <= 8.8e-72) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -0.5 * (t_1 / a);
		} else {
			tmp_2 = (2.0 * c) / t_1;
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = ((2.0 * b) / a) * -0.5
	else:
		tmp = (c + c) / (math.sqrt((b * b)) - b)
	t_0 = tmp
	t_1 = math.sqrt(-(4.0 * (a * c)))
	tmp_1 = 0
	if b <= -2.6e-51:
		tmp_1 = t_0
	elif b <= 8.8e-72:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -0.5 * (t_1 / a)
		else:
			tmp_2 = (2.0 * c) / t_1
		tmp_1 = tmp_2
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(Float64(2.0 * b) / a) * -0.5);
	else
		tmp = Float64(Float64(c + c) / Float64(sqrt(Float64(b * b)) - b));
	end
	t_0 = tmp
	t_1 = sqrt(Float64(-Float64(4.0 * Float64(a * c))))
	tmp_1 = 0.0
	if (b <= -2.6e-51)
		tmp_1 = t_0;
	elseif (b <= 8.8e-72)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-0.5 * Float64(t_1 / a));
		else
			tmp_2 = Float64(Float64(2.0 * c) / t_1);
		end
		tmp_1 = tmp_2;
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = ((2.0 * b) / a) * -0.5;
	else
		tmp = (c + c) / (sqrt((b * b)) - b);
	end
	t_0 = tmp;
	t_1 = sqrt(-(4.0 * (a * c)));
	tmp_2 = 0.0;
	if (b <= -2.6e-51)
		tmp_2 = t_0;
	elseif (b <= 8.8e-72)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -0.5 * (t_1 / a);
		else
			tmp_3 = (2.0 * c) / t_1;
		end
		tmp_2 = tmp_3;
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = If[GreaterEqual[b, 0.0], N[(N[(N[(2.0 * b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], 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[(a * c), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]}, If[LessEqual[b, -2.6e-51], t$95$0, If[LessEqual[b, 8.8e-72], If[GreaterEqual[b, 0.0], N[(-0.5 * N[(t$95$1 / a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / t$95$1), $MachinePrecision]], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.6e-51 or 8.8000000000000001e-72 < b

   1. Initial program 72.2%

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

   2. 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%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 59.9

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

   7. Applied rewrites 59.9%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.6

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

   10. Applied rewrites 48.6%

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

   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. lower-*.f64 58.5

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

   13. Applied rewrites 58.5%

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

   if -2.6e-51 < b < 8.8000000000000001e-72

   1. Initial program 72.2%

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

   2. 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 59.9

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

   4. Applied rewrites 59.9%

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

   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 48.6

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

   7. Applied rewrites 48.6%

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

   8. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 37.8

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

   10. Applied rewrites 37.8%

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

   11. Taylor expanded in b around 0

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-neg.f64 N/A

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

       4. lift-sqrt.f64 26.0

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

   13. Applied rewrites 26.0%

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

Alternative 9: 64.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b}\\ \mathbf{if}\;b \leq 8.8 \cdot 10^{-72}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t\_0}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0 - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* b b))))
   (if (<= b 8.8e-72)
     (if (>= b 0.0)
       (* -0.5 (/ (sqrt (* -4.0 (* a c))) a))
       (/ (* 2.0 c) (+ (- b) t_0)))
     (if (>= b 0.0) (* (/ (* 2.0 b) a) -0.5) (/ (+ c c) (- t_0 b))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((b * b));
	double tmp_1;
	if (b <= 8.8e-72) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -0.5 * (sqrt((-4.0 * (a * c))) / a);
		} else {
			tmp_2 = (2.0 * c) / (-b + t_0);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = ((2.0 * b) / a) * -0.5;
	} else {
		tmp_1 = (c + c) / (t_0 - b);
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = sqrt((b * b))
    if (b <= 8.8d-72) then
        if (b >= 0.0d0) then
            tmp_2 = (-0.5d0) * (sqrt(((-4.0d0) * (a * c))) / a)
        else
            tmp_2 = (2.0d0 * c) / (-b + t_0)
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = ((2.0d0 * b) / a) * (-0.5d0)
    else
        tmp_1 = (c + c) / (t_0 - b)
    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 tmp_1;
	if (b <= 8.8e-72) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -0.5 * (Math.sqrt((-4.0 * (a * c))) / a);
		} else {
			tmp_2 = (2.0 * c) / (-b + t_0);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = ((2.0 * b) / a) * -0.5;
	} else {
		tmp_1 = (c + c) / (t_0 - b);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt((b * b))
	tmp_1 = 0
	if b <= 8.8e-72:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -0.5 * (math.sqrt((-4.0 * (a * c))) / a)
		else:
			tmp_2 = (2.0 * c) / (-b + t_0)
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = ((2.0 * b) / a) * -0.5
	else:
		tmp_1 = (c + c) / (t_0 - b)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(b * b))
	tmp_1 = 0.0
	if (b <= 8.8e-72)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-0.5 * Float64(sqrt(Float64(-4.0 * Float64(a * c))) / a));
		else
			tmp_2 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_0));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(2.0 * b) / a) * -0.5);
	else
		tmp_1 = Float64(Float64(c + c) / Float64(t_0 - b));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = sqrt((b * b));
	tmp_2 = 0.0;
	if (b <= 8.8e-72)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -0.5 * (sqrt((-4.0 * (a * c))) / a);
		else
			tmp_3 = (2.0 * c) / (-b + t_0);
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = ((2.0 * b) / a) * -0.5;
	else
		tmp_2 = (c + c) / (t_0 - b);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(b * b), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, 8.8e-72], If[GreaterEqual[b, 0.0], N[(-0.5 * N[(N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(N[(2.0 * b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 8.8000000000000001e-72

   1. Initial program 72.2%

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

   2. 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 59.9

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

   4. Applied rewrites 59.9%

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

   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 48.6

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

   7. Applied rewrites 48.6%

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

   8. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 37.8

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

   10. Applied rewrites 37.8%

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

   11. Taylor expanded in a around 0

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

       1. lower-sqrt.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 37.8

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

   13. Applied rewrites 37.8%

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

   if 8.8000000000000001e-72 < b

   1. Initial program 72.2%

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

   2. 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%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 59.9

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

   7. Applied rewrites 59.9%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 48.6

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

   10. Applied rewrites 48.6%

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

   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. lower-*.f64 58.5

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

   13. Applied rewrites 58.5%

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

Alternative 10: 58.5% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0) (* (/ (* 2.0 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 = ((2.0 * 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 = ((2.0d0 * 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 = ((2.0 * 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 = ((2.0 * 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(2.0 * 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 = ((2.0 * 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[(2.0 * 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 72.2%

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

2. 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%

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

5. Taylor expanded in a around 0

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

   1. pow2 N/A

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

   2. lift-*.f64 59.9

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

7. Applied rewrites 59.9%

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

8. Taylor expanded in a around 0

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

   1. pow2 N/A

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

   2. lift-*.f64 48.6

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

10. Applied rewrites 48.6%

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

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. lower-*.f64 58.5

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

13. Applied rewrites 58.5%

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

Reproduce

herbie shell --seed 2025140 
(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)))))))