jeff quadratic root 2

Percentage Accurate: 71.9% → 88.9%

Time: 5.4s

Alternatives: 17

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 71.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}\\ t_1 := \frac{\left(-b\right) + t\_0}{2 \cdot a}\\ \mathbf{if}\;b \leq -5 \cdot 10^{+155}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+73}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{-2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* c -4.0) a (* b b))))
        (t_1 (/ (+ (- b) t_0) (* 2.0 a))))
   (if (<= b -5e+155)
     (if (>= b 0.0)
       (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))
       (/ (* -1.0 (* b (+ 2.0 (* -2.0 (/ (* a c) (pow b 2.0)))))) (* 2.0 a)))
     (if (<= b 5e+73)
       (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_0)) t_1)
       (if (>= b 0.0) (/ (* 2.0 c) (* -2.0 b)) t_1)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((c * -4.0), a, (b * b)));
	double t_1 = (-b + t_0) / (2.0 * a);
	double tmp_1;
	if (b <= -5e+155) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (-b - sqrt(((b * b) - ((4.0 * a) * c))));
		} else {
			tmp_2 = (-1.0 * (b * (2.0 + (-2.0 * ((a * c) / pow(b, 2.0)))))) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 5e+73) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_3 = t_1;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-2.0 * b);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(c * -4.0), a, Float64(b * b)))
	t_1 = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a))
	tmp_1 = 0.0
	if (b <= -5e+155)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))));
		else
			tmp_2 = Float64(Float64(-1.0 * Float64(b * Float64(2.0 + Float64(-2.0 * Float64(Float64(a * c) / (b ^ 2.0)))))) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 5e+73)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0));
		else
			tmp_3 = t_1;
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(-2.0 * b));
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -5e+155], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * N[(b * N[(2.0 + N[(-2.0 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 5e+73], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(-2.0 * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.9999999999999999e155

   1. Initial program 71.9%

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

   2. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 67.4

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

   4. Applied rewrites 67.4%

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

   if -4.9999999999999999e155 < b < 4.99999999999999976e73

   1. Initial program 71.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-flip N/A

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

      4. distribute-neg-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. sqr-abs-rev N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. sqr-neg-rev N/A

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

      16. sqr-abs-rev N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 71.9

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

   3. Applied rewrites 71.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-flip N/A

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

      4. distribute-neg-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. sqr-abs-rev N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. sqr-neg-rev N/A

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

      16. sqr-abs-rev N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 71.9

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

   5. Applied rewrites 71.9%

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

   if 4.99999999999999976e73 < b

   1. Initial program 71.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-flip N/A

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

      4. distribute-neg-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. sqr-abs-rev N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. sqr-neg-rev N/A

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

      16. sqr-abs-rev N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 71.9

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

   3. Applied rewrites 71.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-flip N/A

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

      4. distribute-neg-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. sqr-abs-rev N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. sqr-neg-rev N/A

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

      16. sqr-abs-rev N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 71.9

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

   5. Applied rewrites 71.9%

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

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 69.8

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

   8. Applied rewrites 69.8%

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

Alternative 2: 82.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \sqrt{\left|\frac{a}{c} \cdot -4\right|}\\ t_1 := \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}\\ t_2 := \frac{\left(-b\right) + t\_1}{2 \cdot a}\\ \mathbf{if}\;b \leq -6.6 \cdot 10^{+163}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + t\_0}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+73}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{-2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (sqrt (fabs (* (/ a c) -4.0)))))
        (t_1 (sqrt (fma (* c -4.0) a (* b b))))
        (t_2 (/ (+ (- b) t_1) (* 2.0 a))))
   (if (<= b -6.6e+163)
     (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_0)) (/ (+ (- b) t_0) (* 2.0 a)))
     (if (<= b 5e+73)
       (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_1)) t_2)
       (if (>= b 0.0) (/ (* 2.0 c) (* -2.0 b)) t_2)))))

C

double code(double a, double b, double c) {
	double t_0 = c * sqrt(fabs(((a / c) * -4.0)));
	double t_1 = sqrt(fma((c * -4.0), a, (b * b)));
	double t_2 = (-b + t_1) / (2.0 * a);
	double tmp_1;
	if (b <= -6.6e+163) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_2 = (-b + t_0) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 5e+73) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * c) / (-b - t_1);
		} else {
			tmp_3 = t_2;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-2.0 * b);
	} else {
		tmp_1 = t_2;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(c * sqrt(abs(Float64(Float64(a / c) * -4.0))))
	t_1 = sqrt(fma(Float64(c * -4.0), a, Float64(b * b)))
	t_2 = Float64(Float64(Float64(-b) + t_1) / Float64(2.0 * a))
	tmp_1 = 0.0
	if (b <= -6.6e+163)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0));
		else
			tmp_2 = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 5e+73)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_1));
		else
			tmp_3 = t_2;
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(-2.0 * b));
	else
		tmp_1 = t_2;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[Sqrt[N[Abs[N[(N[(a / c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[((-b) + t$95$1), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -6.6e+163], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 5e+73], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(-2.0 * b), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.5999999999999999e163

   1. Initial program 71.9%

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

   2. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 49.2

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 27.0

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

   7. Applied rewrites 27.0%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-abs-rev N/A

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

      5. mul-fabs N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. lower-fabs.f64 32.8

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 32.8

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

   9. Applied rewrites 32.8%

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

       1. rem-square-sqrt N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. sqr-abs-rev N/A

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

       5. mul-fabs N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. rem-square-sqrt N/A

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

       9. lower-fabs.f64 37.9

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 37.9

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

   11. Applied rewrites 37.9%

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

   if -6.5999999999999999e163 < b < 4.99999999999999976e73

   1. Initial program 71.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-flip N/A

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

      4. distribute-neg-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. sqr-abs-rev N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. sqr-neg-rev N/A

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

      16. sqr-abs-rev N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 71.9

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

   3. Applied rewrites 71.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-flip N/A

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

      4. distribute-neg-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. sqr-abs-rev N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. sqr-neg-rev N/A

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

      16. sqr-abs-rev N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 71.9

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

   5. Applied rewrites 71.9%

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

   if 4.99999999999999976e73 < b

   1. Initial program 71.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-flip N/A

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

      4. distribute-neg-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. sqr-abs-rev N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. sqr-neg-rev N/A

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

      16. sqr-abs-rev N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 71.9

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

   3. Applied rewrites 71.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-flip N/A

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

      4. distribute-neg-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. sqr-abs-rev N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. sqr-neg-rev N/A

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

      16. sqr-abs-rev N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 71.9

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

   5. Applied rewrites 71.9%

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

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 69.8

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

   8. Applied rewrites 69.8%

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

Alternative 3: 82.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \sqrt{\left|\frac{a}{c} \cdot -4\right|}\\ t_1 := \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}\\ \mathbf{if}\;b \leq -6.6 \cdot 10^{+163}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + t\_0}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{+73}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2}{b + t\_1} \cdot c\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 - b}{a + a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{-2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{b}{a}, 0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (sqrt (fabs (* (/ a c) -4.0)))))
        (t_1 (sqrt (fma (* c -4.0) a (* b b)))))
   (if (<= b -6.6e+163)
     (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_0)) (/ (+ (- b) t_0) (* 2.0 a)))
     (if (<= b 3.5e+73)
       (if (>= b 0.0) (* (/ -2.0 (+ b t_1)) c) (/ (- t_1 b) (+ a a)))
       (if (>= b 0.0)
         (/ (* 2.0 c) (* -2.0 b))
         (fma -0.5 (/ b a) (* 0.5 (sqrt (* -4.0 (/ c a))))))))))

C

double code(double a, double b, double c) {
	double t_0 = c * sqrt(fabs(((a / c) * -4.0)));
	double t_1 = sqrt(fma((c * -4.0), a, (b * b)));
	double tmp_1;
	if (b <= -6.6e+163) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_2 = (-b + t_0) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 3.5e+73) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-2.0 / (b + t_1)) * c;
		} else {
			tmp_3 = (t_1 - b) / (a + a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-2.0 * b);
	} else {
		tmp_1 = fma(-0.5, (b / a), (0.5 * sqrt((-4.0 * (c / a)))));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(c * sqrt(abs(Float64(Float64(a / c) * -4.0))))
	t_1 = sqrt(fma(Float64(c * -4.0), a, Float64(b * b)))
	tmp_1 = 0.0
	if (b <= -6.6e+163)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0));
		else
			tmp_2 = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 3.5e+73)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-2.0 / Float64(b + t_1)) * c);
		else
			tmp_3 = Float64(Float64(t_1 - b) / Float64(a + a));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(-2.0 * b));
	else
		tmp_1 = fma(-0.5, Float64(b / a), Float64(0.5 * sqrt(Float64(-4.0 * Float64(c / a)))));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[Sqrt[N[Abs[N[(N[(a / c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -6.6e+163], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 3.5e+73], If[GreaterEqual[b, 0.0], N[(N[(-2.0 / N[(b + t$95$1), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], N[(N[(t$95$1 - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(-2.0 * b), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(b / a), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.5999999999999999e163

   1. Initial program 71.9%

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

   2. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 49.2

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 27.0

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

   7. Applied rewrites 27.0%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-abs-rev N/A

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

      5. mul-fabs N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. lower-fabs.f64 32.8

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 32.8

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

   9. Applied rewrites 32.8%

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

       1. rem-square-sqrt N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. sqr-abs-rev N/A

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

       5. mul-fabs N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. rem-square-sqrt N/A

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

       9. lower-fabs.f64 37.9

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 37.9

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

   11. Applied rewrites 37.9%

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

   if -6.5999999999999999e163 < b < 3.50000000000000002e73

   1. Initial program 71.9%

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

   3. Applied rewrites 71.9%

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

   if 3.50000000000000002e73 < b

   1. Initial program 71.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-flip N/A

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

      4. distribute-neg-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. sqr-abs-rev N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. sqr-neg-rev N/A

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

      16. sqr-abs-rev N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 71.9

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

   3. Applied rewrites 71.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-flip N/A

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

      4. distribute-neg-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. sqr-abs-rev N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. sqr-neg-rev N/A

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

      16. sqr-abs-rev N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 71.9

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

   5. Applied rewrites 71.9%

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

   6. Taylor expanded in a around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 48.5

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

   8. Applied rewrites 48.5%

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

   9. Taylor expanded in b around inf

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

       1. lower-*.f64 46.3

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

   11. Applied rewrites 46.3%

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

Alternative 4: 82.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \sqrt{\left|\frac{a}{c} \cdot -4\right|}\\ t_1 := \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}\\ \mathbf{if}\;b \leq -6.6 \cdot 10^{+163}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + t\_0}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{+73}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2}{b + t\_1} \cdot c\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 - b}{a + a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{-2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + t\_1}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (sqrt (fabs (* (/ a c) -4.0)))))
        (t_1 (sqrt (fma (* c -4.0) a (* b b)))))
   (if (<= b -6.6e+163)
     (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_0)) (/ (+ (- b) t_0) (* 2.0 a)))
     (if (<= b 3.5e+73)
       (if (>= b 0.0) (* (/ -2.0 (+ b t_1)) c) (/ (- t_1 b) (+ a a)))
       (if (>= b 0.0) (/ (* 2.0 c) (* -2.0 b)) (/ (+ (- b) t_1) (* 2.0 a)))))))

C

double code(double a, double b, double c) {
	double t_0 = c * sqrt(fabs(((a / c) * -4.0)));
	double t_1 = sqrt(fma((c * -4.0), a, (b * b)));
	double tmp_1;
	if (b <= -6.6e+163) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_2 = (-b + t_0) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 3.5e+73) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-2.0 / (b + t_1)) * c;
		} else {
			tmp_3 = (t_1 - b) / (a + a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-2.0 * b);
	} else {
		tmp_1 = (-b + t_1) / (2.0 * a);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(c * sqrt(abs(Float64(Float64(a / c) * -4.0))))
	t_1 = sqrt(fma(Float64(c * -4.0), a, Float64(b * b)))
	tmp_1 = 0.0
	if (b <= -6.6e+163)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0));
		else
			tmp_2 = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 3.5e+73)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-2.0 / Float64(b + t_1)) * c);
		else
			tmp_3 = Float64(Float64(t_1 - b) / Float64(a + a));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(-2.0 * b));
	else
		tmp_1 = Float64(Float64(Float64(-b) + t_1) / Float64(2.0 * a));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[Sqrt[N[Abs[N[(N[(a / c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -6.6e+163], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 3.5e+73], If[GreaterEqual[b, 0.0], N[(N[(-2.0 / N[(b + t$95$1), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], N[(N[(t$95$1 - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(-2.0 * b), $MachinePrecision]), $MachinePrecision], N[(N[((-b) + t$95$1), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.5999999999999999e163

   1. Initial program 71.9%

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

   2. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 49.2

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 27.0

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

   7. Applied rewrites 27.0%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-abs-rev N/A

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

      5. mul-fabs N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. lower-fabs.f64 32.8

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 32.8

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

   9. Applied rewrites 32.8%

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

       1. rem-square-sqrt N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. sqr-abs-rev N/A

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

       5. mul-fabs N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. rem-square-sqrt N/A

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

       9. lower-fabs.f64 37.9

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 37.9

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

   11. Applied rewrites 37.9%

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

   if -6.5999999999999999e163 < b < 3.50000000000000002e73

   1. Initial program 71.9%

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

   3. Applied rewrites 71.9%

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

   if 3.50000000000000002e73 < b

   1. Initial program 71.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-flip N/A

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

      4. distribute-neg-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. sqr-abs-rev N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. sqr-neg-rev N/A

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

      16. sqr-abs-rev N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 71.9

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

   3. Applied rewrites 71.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-flip N/A

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

      4. distribute-neg-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. sqr-abs-rev N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. sqr-neg-rev N/A

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

      16. sqr-abs-rev N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 71.9

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

   5. Applied rewrites 71.9%

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

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 69.8

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

   8. Applied rewrites 69.8%

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

Alternative 5: 81.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}\\ \mathbf{if}\;b \leq 3.5 \cdot 10^{+73}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2}{b + t\_0} \cdot c\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a + a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{-2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{b}{a}, 0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* c -4.0) a (* b b)))))
   (if (<= b 3.5e+73)
     (if (>= b 0.0) (* (/ -2.0 (+ b t_0)) c) (/ (- t_0 b) (+ a a)))
     (if (>= b 0.0)
       (/ (* 2.0 c) (* -2.0 b))
       (fma -0.5 (/ b a) (* 0.5 (sqrt (* -4.0 (/ c a)))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((c * -4.0), a, (b * b)));
	double tmp_1;
	if (b <= 3.5e+73) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-2.0 / (b + t_0)) * c;
		} else {
			tmp_2 = (t_0 - b) / (a + a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-2.0 * b);
	} else {
		tmp_1 = fma(-0.5, (b / a), (0.5 * sqrt((-4.0 * (c / a)))));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(c * -4.0), a, Float64(b * b)))
	tmp_1 = 0.0
	if (b <= 3.5e+73)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-2.0 / Float64(b + t_0)) * c);
		else
			tmp_2 = Float64(Float64(t_0 - b) / Float64(a + a));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(-2.0 * b));
	else
		tmp_1 = fma(-0.5, Float64(b / a), Float64(0.5 * sqrt(Float64(-4.0 * Float64(c / a)))));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, 3.5e+73], If[GreaterEqual[b, 0.0], N[(N[(-2.0 / N[(b + t$95$0), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(-2.0 * b), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(b / a), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.50000000000000002e73

   1. Initial program 71.9%

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

   3. Applied rewrites 71.9%

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

   if 3.50000000000000002e73 < b

   1. Initial program 71.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-flip N/A

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

      4. distribute-neg-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. sqr-abs-rev N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. sqr-neg-rev N/A

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

      16. sqr-abs-rev N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 71.9

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

   3. Applied rewrites 71.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-flip N/A

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

      4. distribute-neg-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. sqr-abs-rev N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. sqr-neg-rev N/A

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

      16. sqr-abs-rev N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 71.9

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

   5. Applied rewrites 71.9%

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

   6. Taylor expanded in a around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 48.5

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

   8. Applied rewrites 48.5%

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

   9. Taylor expanded in b around inf

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

       1. lower-*.f64 46.3

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

   11. Applied rewrites 46.3%

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

Alternative 6: 76.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left(-4 \cdot c\right) \cdot a}\\ \mathbf{if}\;b \leq 4.6 \cdot 10^{-302}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{\sqrt{\frac{c}{a} \cdot -4} \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} - b}{a + a}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-12}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{t\_0 + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a + a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{-2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{b}{a}, 0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* (* -4.0 c) a))))
   (if (<= b 4.6e-302)
     (if (>= b 0.0)
       (/ (+ c c) (* (sqrt (* (/ c a) -4.0)) a))
       (/ (- (sqrt (fma (* a c) -4.0 (* b b))) b) (+ a a)))
     (if (<= b 2.5e-12)
       (if (>= b 0.0) (/ (* -2.0 c) (+ t_0 b)) (/ (- t_0 b) (+ a a)))
       (if (>= b 0.0)
         (/ (* 2.0 c) (* -2.0 b))
         (fma -0.5 (/ b a) (* 0.5 (sqrt (* -4.0 (/ c a))))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((-4.0 * c) * a));
	double tmp_1;
	if (b <= 4.6e-302) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c + c) / (sqrt(((c / a) * -4.0)) * a);
		} else {
			tmp_2 = (sqrt(fma((a * c), -4.0, (b * b))) - b) / (a + a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 2.5e-12) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-2.0 * c) / (t_0 + b);
		} else {
			tmp_3 = (t_0 - b) / (a + a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-2.0 * b);
	} else {
		tmp_1 = fma(-0.5, (b / a), (0.5 * sqrt((-4.0 * (c / a)))));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(-4.0 * c) * a))
	tmp_1 = 0.0
	if (b <= 4.6e-302)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(c + c) / Float64(sqrt(Float64(Float64(c / a) * -4.0)) * a));
		else
			tmp_2 = Float64(Float64(sqrt(fma(Float64(a * c), -4.0, Float64(b * b))) - b) / Float64(a + a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 2.5e-12)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-2.0 * c) / Float64(t_0 + b));
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(a + a));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(-2.0 * b));
	else
		tmp_1 = fma(-0.5, Float64(b / a), Float64(0.5 * sqrt(Float64(-4.0 * Float64(c / a)))));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, 4.6e-302], If[GreaterEqual[b, 0.0], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(N[(c / a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 2.5e-12], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * c), $MachinePrecision] / N[(t$95$0 + b), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(-2.0 * b), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(b / a), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.60000000000000004e-302

   1. Initial program 71.9%

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

   2. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 43.1

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

   4. Applied rewrites 43.1%

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

   6. Applied rewrites 43.1%

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

   if 4.60000000000000004e-302 < b < 2.49999999999999985e-12

   1. Initial program 71.9%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.3

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

   4. Applied rewrites 56.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 41.1

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

   7. Applied rewrites 41.1%

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

   9. Applied rewrites 41.1%

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

   11. Applied rewrites 41.1%

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

   if 2.49999999999999985e-12 < b

   1. Initial program 71.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-flip N/A

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

      4. distribute-neg-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. sqr-abs-rev N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. sqr-neg-rev N/A

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

      16. sqr-abs-rev N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 71.9

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

   3. Applied rewrites 71.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-flip N/A

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

      4. distribute-neg-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. sqr-abs-rev N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. sqr-neg-rev N/A

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

      16. sqr-abs-rev N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 71.9

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

   5. Applied rewrites 71.9%

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

   6. Taylor expanded in a around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 48.5

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

   8. Applied rewrites 48.5%

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

   9. Taylor expanded in b around inf

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

       1. lower-*.f64 46.3

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

   11. Applied rewrites 46.3%

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

Alternative 7: 66.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left|\left(a \cdot c\right) \cdot -4\right|}\\ \mathbf{if}\;b \leq 2.5 \cdot 10^{-12}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + t\_0}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{-2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{b}{a}, 0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fabs (* (* a c) -4.0)))))
   (if (<= b 2.5e-12)
     (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_0)) (/ (+ (- b) t_0) (* 2.0 a)))
     (if (>= b 0.0)
       (/ (* 2.0 c) (* -2.0 b))
       (fma -0.5 (/ b a) (* 0.5 (sqrt (* -4.0 (/ c a)))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fabs(((a * c) * -4.0)));
	double tmp_1;
	if (b <= 2.5e-12) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (-b - t_0);
		} else {
			tmp_2 = (-b + t_0) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-2.0 * b);
	} else {
		tmp_1 = fma(-0.5, (b / a), (0.5 * sqrt((-4.0 * (c / a)))));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(abs(Float64(Float64(a * c) * -4.0)))
	tmp_1 = 0.0
	if (b <= 2.5e-12)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0));
		else
			tmp_2 = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(-2.0 * b));
	else
		tmp_1 = fma(-0.5, Float64(b / a), Float64(0.5 * sqrt(Float64(-4.0 * Float64(c / a)))));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[Abs[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, 2.5e-12], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(-2.0 * b), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(b / a), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.49999999999999985e-12

   1. Initial program 71.9%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.3

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

   4. Applied rewrites 56.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 41.1

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

   7. Applied rewrites 41.1%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-abs-rev N/A

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

      5. mul-fabs N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. lower-fabs.f64 45.7

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

   9. Applied rewrites 45.7%

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

       1. rem-square-sqrt N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. sqr-abs-rev N/A

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

       5. mul-fabs N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. rem-square-sqrt N/A

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

       9. lower-fabs.f64 50.6

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

   11. Applied rewrites 50.6%

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

   if 2.49999999999999985e-12 < b

   1. Initial program 71.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-flip N/A

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

      4. distribute-neg-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. sqr-abs-rev N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. sqr-neg-rev N/A

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

      16. sqr-abs-rev N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 71.9

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

   3. Applied rewrites 71.9%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-flip N/A

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

      4. distribute-neg-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. sqr-abs-rev N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. distribute-lft-neg-out N/A

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

      15. sqr-neg-rev N/A

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

      16. sqr-abs-rev N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 71.9

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

   5. Applied rewrites 71.9%

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

   6. Taylor expanded in a around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 48.5

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

   8. Applied rewrites 48.5%

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

   9. Taylor expanded in b around inf

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

       1. lower-*.f64 46.3

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

   11. Applied rewrites 46.3%

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

Alternative 8: 66.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot c\right) \cdot -4\\ t_1 := \sqrt{\left|t\_0\right|}\\ \mathbf{if}\;b \leq 2.5 \cdot 10^{-12}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + t\_1}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{t\_0} - b}{a + a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* a c) -4.0)) (t_1 (sqrt (fabs t_0))))
   (if (<= b 2.5e-12)
     (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_1)) (/ (+ (- b) t_1) (* 2.0 a)))
     (if (>= b 0.0) (* c (/ -1.0 b)) (/ (- (sqrt t_0) b) (+ a a))))))

C

double code(double a, double b, double c) {
	double t_0 = (a * c) * -4.0;
	double t_1 = sqrt(fabs(t_0));
	double tmp_1;
	if (b <= 2.5e-12) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (-b - t_1);
		} else {
			tmp_2 = (-b + t_1) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = c * (-1.0 / b);
	} else {
		tmp_1 = (sqrt(t_0) - b) / (a + a);
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = (a * c) * (-4.0d0)
    t_1 = sqrt(abs(t_0))
    if (b <= 2.5d-12) then
        if (b >= 0.0d0) then
            tmp_2 = (2.0d0 * c) / (-b - t_1)
        else
            tmp_2 = (-b + t_1) / (2.0d0 * a)
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = c * ((-1.0d0) / b)
    else
        tmp_1 = (sqrt(t_0) - b) / (a + a)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (a * c) * -4.0;
	double t_1 = Math.sqrt(Math.abs(t_0));
	double tmp_1;
	if (b <= 2.5e-12) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (-b - t_1);
		} else {
			tmp_2 = (-b + t_1) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = c * (-1.0 / b);
	} else {
		tmp_1 = (Math.sqrt(t_0) - b) / (a + a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (a * c) * -4.0
	t_1 = math.sqrt(math.fabs(t_0))
	tmp_1 = 0
	if b <= 2.5e-12:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (2.0 * c) / (-b - t_1)
		else:
			tmp_2 = (-b + t_1) / (2.0 * a)
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = c * (-1.0 / b)
	else:
		tmp_1 = (math.sqrt(t_0) - b) / (a + a)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(a * c) * -4.0)
	t_1 = sqrt(abs(t_0))
	tmp_1 = 0.0
	if (b <= 2.5e-12)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_1));
		else
			tmp_2 = Float64(Float64(Float64(-b) + t_1) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(c * Float64(-1.0 / b));
	else
		tmp_1 = Float64(Float64(sqrt(t_0) - b) / Float64(a + a));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = (a * c) * -4.0;
	t_1 = sqrt(abs(t_0));
	tmp_2 = 0.0;
	if (b <= 2.5e-12)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (2.0 * c) / (-b - t_1);
		else
			tmp_3 = (-b + t_1) / (2.0 * a);
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = c * (-1.0 / b);
	else
		tmp_2 = (sqrt(t_0) - b) / (a + a);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[Abs[t$95$0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, 2.5e-12], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[((-b) + t$95$1), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(c * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.49999999999999985e-12

   1. Initial program 71.9%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.3

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

   4. Applied rewrites 56.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 41.1

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

   7. Applied rewrites 41.1%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-abs-rev N/A

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

      5. mul-fabs N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. lower-fabs.f64 45.7

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

   9. Applied rewrites 45.7%

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

       1. rem-square-sqrt N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. sqr-abs-rev N/A

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

       5. mul-fabs N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. rem-square-sqrt N/A

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

       9. lower-fabs.f64 50.6

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

   11. Applied rewrites 50.6%

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

   if 2.49999999999999985e-12 < b

   1. Initial program 71.9%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.3

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

   4. Applied rewrites 56.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 41.1

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

   7. Applied rewrites 41.1%

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

   9. Applied rewrites 41.1%

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

   10. Taylor expanded in b around inf

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

       1. lower-/.f64 54.5

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

   12. Applied rewrites 54.5%

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

Alternative 9: 66.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left|\left(-4 \cdot c\right) \cdot a\right|}\\ \mathbf{if}\;b \leq 2.5 \cdot 10^{-12}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{t\_0 + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a + a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fabs (* (* -4.0 c) a)))))
   (if (<= b 2.5e-12)
     (if (>= b 0.0) (* c (/ -2.0 (+ t_0 b))) (/ (- t_0 b) (+ a a)))
     (if (>= b 0.0)
       (* c (/ -1.0 b))
       (/ (- (sqrt (* (* a c) -4.0)) b) (+ a a))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fabs(((-4.0 * c) * a)));
	double tmp_1;
	if (b <= 2.5e-12) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / (t_0 + b));
		} else {
			tmp_2 = (t_0 - b) / (a + a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = c * (-1.0 / b);
	} else {
		tmp_1 = (sqrt(((a * c) * -4.0)) - b) / (a + a);
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = sqrt(abs((((-4.0d0) * c) * a)))
    if (b <= 2.5d-12) then
        if (b >= 0.0d0) then
            tmp_2 = c * ((-2.0d0) / (t_0 + b))
        else
            tmp_2 = (t_0 - b) / (a + a)
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = c * ((-1.0d0) / b)
    else
        tmp_1 = (sqrt(((a * c) * (-4.0d0))) - b) / (a + a)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(Math.abs(((-4.0 * c) * a)));
	double tmp_1;
	if (b <= 2.5e-12) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / (t_0 + b));
		} else {
			tmp_2 = (t_0 - b) / (a + a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = c * (-1.0 / b);
	} else {
		tmp_1 = (Math.sqrt(((a * c) * -4.0)) - b) / (a + a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(math.fabs(((-4.0 * c) * a)))
	tmp_1 = 0
	if b <= 2.5e-12:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = c * (-2.0 / (t_0 + b))
		else:
			tmp_2 = (t_0 - b) / (a + a)
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = c * (-1.0 / b)
	else:
		tmp_1 = (math.sqrt(((a * c) * -4.0)) - b) / (a + a)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(abs(Float64(Float64(-4.0 * c) * a)))
	tmp_1 = 0.0
	if (b <= 2.5e-12)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * Float64(-2.0 / Float64(t_0 + b)));
		else
			tmp_2 = Float64(Float64(t_0 - b) / Float64(a + a));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(c * Float64(-1.0 / b));
	else
		tmp_1 = Float64(Float64(sqrt(Float64(Float64(a * c) * -4.0)) - b) / Float64(a + a));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = sqrt(abs(((-4.0 * c) * a)));
	tmp_2 = 0.0;
	if (b <= 2.5e-12)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = c * (-2.0 / (t_0 + b));
		else
			tmp_3 = (t_0 - b) / (a + a);
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = c * (-1.0 / b);
	else
		tmp_2 = (sqrt(((a * c) * -4.0)) - b) / (a + a);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[Abs[N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, 2.5e-12], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[(t$95$0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(c * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.49999999999999985e-12

   1. Initial program 71.9%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.3

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

   4. Applied rewrites 56.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 41.1

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

   7. Applied rewrites 41.1%

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

   9. Applied rewrites 41.1%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 41.1

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

       4. rem-square-sqrt N/A

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

       5. pow1/2 N/A

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

       6. pow1/2 N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lift-*.f64 N/A

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

       10. pow1/2 N/A

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

       11. lift-sqrt.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. *-commutative N/A

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

       14. lift-*.f64 N/A

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

   11. Applied rewrites 45.7%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 45.7

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

       4. rem-square-sqrt N/A

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

       5. pow1/2 N/A

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

       6. pow1/2 N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lift-*.f64 N/A

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

       10. pow1/2 N/A

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

       11. lift-sqrt.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. *-commutative N/A

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

       14. lift-*.f64 N/A

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

   13. Applied rewrites 50.6%

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

   if 2.49999999999999985e-12 < b

   1. Initial program 71.9%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.3

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

   4. Applied rewrites 56.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 41.1

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

   7. Applied rewrites 41.1%

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

   9. Applied rewrites 41.1%

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

   10. Taylor expanded in b around inf

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

       1. lower-/.f64 54.5

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

   12. Applied rewrites 54.5%

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

Alternative 10: 65.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot c\right) \cdot -4\\ \mathbf{if}\;b \leq 2.5 \cdot 10^{-12}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left|t\_0\right|} - b}{a + a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{t\_0} - b}{a + a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* a c) -4.0)))
   (if (<= b 2.5e-12)
     (if (>= b 0.0)
       (* -2.0 (/ c (sqrt (- (* 4.0 (* a c))))))
       (/ (- (sqrt (fabs t_0)) b) (+ a a)))
     (if (>= b 0.0) (* c (/ -1.0 b)) (/ (- (sqrt t_0) b) (+ a a))))))

C

double code(double a, double b, double c) {
	double t_0 = (a * c) * -4.0;
	double tmp_1;
	if (b <= 2.5e-12) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -2.0 * (c / sqrt(-(4.0 * (a * c))));
		} else {
			tmp_2 = (sqrt(fabs(t_0)) - b) / (a + a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = c * (-1.0 / b);
	} else {
		tmp_1 = (sqrt(t_0) - b) / (a + a);
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = (a * c) * (-4.0d0)
    if (b <= 2.5d-12) then
        if (b >= 0.0d0) then
            tmp_2 = (-2.0d0) * (c / sqrt(-(4.0d0 * (a * c))))
        else
            tmp_2 = (sqrt(abs(t_0)) - b) / (a + a)
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = c * ((-1.0d0) / b)
    else
        tmp_1 = (sqrt(t_0) - b) / (a + a)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (a * c) * -4.0;
	double tmp_1;
	if (b <= 2.5e-12) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -2.0 * (c / Math.sqrt(-(4.0 * (a * c))));
		} else {
			tmp_2 = (Math.sqrt(Math.abs(t_0)) - b) / (a + a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = c * (-1.0 / b);
	} else {
		tmp_1 = (Math.sqrt(t_0) - b) / (a + a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (a * c) * -4.0
	tmp_1 = 0
	if b <= 2.5e-12:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -2.0 * (c / math.sqrt(-(4.0 * (a * c))))
		else:
			tmp_2 = (math.sqrt(math.fabs(t_0)) - b) / (a + a)
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = c * (-1.0 / b)
	else:
		tmp_1 = (math.sqrt(t_0) - b) / (a + a)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(a * c) * -4.0)
	tmp_1 = 0.0
	if (b <= 2.5e-12)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-2.0 * Float64(c / sqrt(Float64(-Float64(4.0 * Float64(a * c))))));
		else
			tmp_2 = Float64(Float64(sqrt(abs(t_0)) - b) / Float64(a + a));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(c * Float64(-1.0 / b));
	else
		tmp_1 = Float64(Float64(sqrt(t_0) - b) / Float64(a + a));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = (a * c) * -4.0;
	tmp_2 = 0.0;
	if (b <= 2.5e-12)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -2.0 * (c / sqrt(-(4.0 * (a * c))));
		else
			tmp_3 = (sqrt(abs(t_0)) - b) / (a + a);
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = c * (-1.0 / b);
	else
		tmp_2 = (sqrt(t_0) - b) / (a + a);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]}, If[LessEqual[b, 2.5e-12], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(c / N[Sqrt[(-N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[Abs[t$95$0], $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(c * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.49999999999999985e-12

   1. Initial program 71.9%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.3

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

   4. Applied rewrites 56.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 41.1

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

   7. Applied rewrites 41.1%

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

   9. Applied rewrites 41.1%

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

   10. Taylor expanded in b around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-neg.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 34.2

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

   12. Applied rewrites 34.2%

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

       1. rem-square-sqrt N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. sqr-abs-rev N/A

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

       5. mul-fabs N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. rem-square-sqrt N/A

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

       9. lower-fabs.f64 39.1

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

   14. Applied rewrites 39.1%

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

   if 2.49999999999999985e-12 < b

   1. Initial program 71.9%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.3

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

   4. Applied rewrites 56.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 41.1

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

   7. Applied rewrites 41.1%

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

   9. Applied rewrites 41.1%

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

   10. Taylor expanded in b around inf

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

       1. lower-/.f64 54.5

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

   12. Applied rewrites 54.5%

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

Alternative 11: 61.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left(-4 \cdot c\right) \cdot a}\\ \mathbf{if}\;b \leq 2.5 \cdot 10^{-12}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{t\_0 + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a + a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* (* -4.0 c) a))))
   (if (<= b 2.5e-12)
     (if (>= b 0.0) (/ (* -2.0 c) (+ t_0 b)) (/ (- t_0 b) (+ a a)))
     (if (>= b 0.0)
       (* c (/ -1.0 b))
       (/ (- (sqrt (* (* a c) -4.0)) b) (+ a a))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((-4.0 * c) * a));
	double tmp_1;
	if (b <= 2.5e-12) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-2.0 * c) / (t_0 + b);
		} else {
			tmp_2 = (t_0 - b) / (a + a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = c * (-1.0 / b);
	} else {
		tmp_1 = (sqrt(((a * c) * -4.0)) - b) / (a + a);
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = sqrt((((-4.0d0) * c) * a))
    if (b <= 2.5d-12) then
        if (b >= 0.0d0) then
            tmp_2 = ((-2.0d0) * c) / (t_0 + b)
        else
            tmp_2 = (t_0 - b) / (a + a)
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = c * ((-1.0d0) / b)
    else
        tmp_1 = (sqrt(((a * c) * (-4.0d0))) - b) / (a + a)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((-4.0 * c) * a));
	double tmp_1;
	if (b <= 2.5e-12) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-2.0 * c) / (t_0 + b);
		} else {
			tmp_2 = (t_0 - b) / (a + a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = c * (-1.0 / b);
	} else {
		tmp_1 = (Math.sqrt(((a * c) * -4.0)) - b) / (a + a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((-4.0 * c) * a))
	tmp_1 = 0
	if b <= 2.5e-12:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (-2.0 * c) / (t_0 + b)
		else:
			tmp_2 = (t_0 - b) / (a + a)
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = c * (-1.0 / b)
	else:
		tmp_1 = (math.sqrt(((a * c) * -4.0)) - b) / (a + a)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(-4.0 * c) * a))
	tmp_1 = 0.0
	if (b <= 2.5e-12)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-2.0 * c) / Float64(t_0 + b));
		else
			tmp_2 = Float64(Float64(t_0 - b) / Float64(a + a));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(c * Float64(-1.0 / b));
	else
		tmp_1 = Float64(Float64(sqrt(Float64(Float64(a * c) * -4.0)) - b) / Float64(a + a));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = sqrt(((-4.0 * c) * a));
	tmp_2 = 0.0;
	if (b <= 2.5e-12)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (-2.0 * c) / (t_0 + b);
		else
			tmp_3 = (t_0 - b) / (a + a);
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = c * (-1.0 / b);
	else
		tmp_2 = (sqrt(((a * c) * -4.0)) - b) / (a + a);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, 2.5e-12], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * c), $MachinePrecision] / N[(t$95$0 + b), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(c * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.49999999999999985e-12

   1. Initial program 71.9%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 56.3

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Applied rewrites 56.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 41.1

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   7. Applied rewrites 41.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 41.1%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ } \end{array}} \]
   10. Step-by-step derivation

   11. Applied rewrites 41.1%

       \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{\sqrt{\left(-4 \cdot c\right) \cdot a} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}{a + a}\\ } \end{array}} \]

   if 2.49999999999999985e-12 < b

   1. Initial program 71.9%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 56.3

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Applied rewrites 56.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 41.1

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   7. Applied rewrites 41.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 41.1%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ } \end{array}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-1}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-/.f64 54.5

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-1}{\color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

   12. Applied rewrites 54.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-1}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 61.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left(a \cdot c\right) \cdot -4}\\ t_1 := \frac{t\_0 - b}{a + a}\\ \mathbf{if}\;b \leq 2.5 \cdot 10^{-12}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{t\_0 + b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-1}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* (* a c) -4.0))) (t_1 (/ (- t_0 b) (+ a a))))
   (if (<= b 2.5e-12)
     (if (>= b 0.0) (* c (/ -2.0 (+ t_0 b))) t_1)
     (if (>= b 0.0) (* c (/ -1.0 b)) t_1))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((a * c) * -4.0));
	double t_1 = (t_0 - b) / (a + a);
	double tmp_1;
	if (b <= 2.5e-12) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / (t_0 + b));
		} else {
			tmp_2 = t_1;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = c * (-1.0 / b);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = sqrt(((a * c) * (-4.0d0)))
    t_1 = (t_0 - b) / (a + a)
    if (b <= 2.5d-12) then
        if (b >= 0.0d0) then
            tmp_2 = c * ((-2.0d0) / (t_0 + b))
        else
            tmp_2 = t_1
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = c * ((-1.0d0) / b)
    else
        tmp_1 = t_1
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((a * c) * -4.0));
	double t_1 = (t_0 - b) / (a + a);
	double tmp_1;
	if (b <= 2.5e-12) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / (t_0 + b));
		} else {
			tmp_2 = t_1;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = c * (-1.0 / b);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((a * c) * -4.0))
	t_1 = (t_0 - b) / (a + a)
	tmp_1 = 0
	if b <= 2.5e-12:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = c * (-2.0 / (t_0 + b))
		else:
			tmp_2 = t_1
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = c * (-1.0 / b)
	else:
		tmp_1 = t_1
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(a * c) * -4.0))
	t_1 = Float64(Float64(t_0 - b) / Float64(a + a))
	tmp_1 = 0.0
	if (b <= 2.5e-12)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * Float64(-2.0 / Float64(t_0 + b)));
		else
			tmp_2 = t_1;
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(c * Float64(-1.0 / b));
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = sqrt(((a * c) * -4.0));
	t_1 = (t_0 - b) / (a + a);
	tmp_2 = 0.0;
	if (b <= 2.5e-12)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = c * (-2.0 / (t_0 + b));
		else
			tmp_3 = t_1;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = c * (-1.0 / b);
	else
		tmp_2 = t_1;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 2.5e-12], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[(t$95$0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1], If[GreaterEqual[b, 0.0], N[(c * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.49999999999999985e-12

   1. Initial program 71.9%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 56.3

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Applied rewrites 56.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 41.1

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   7. Applied rewrites 41.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 41.1%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ } \end{array}} \]

   if 2.49999999999999985e-12 < b

   1. Initial program 71.9%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 56.3

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Applied rewrites 56.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 41.1

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   7. Applied rewrites 41.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 41.1%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ } \end{array}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-1}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-/.f64 54.5

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-1}{\color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

   12. Applied rewrites 54.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-1}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 61.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left(a \cdot c\right) \cdot -4}\\ \mathbf{if}\;b \leq 2.5 \cdot 10^{-12}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{t\_0 + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\left(-4 \cdot c\right) \cdot a} - b\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a + a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* (* a c) -4.0))))
   (if (<= b 2.5e-12)
     (if (>= b 0.0)
       (* c (/ -2.0 (+ t_0 b)))
       (* (/ 0.5 a) (- (sqrt (* (* -4.0 c) a)) b)))
     (if (>= b 0.0) (* c (/ -1.0 b)) (/ (- t_0 b) (+ a a))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((a * c) * -4.0));
	double tmp_1;
	if (b <= 2.5e-12) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / (t_0 + b));
		} else {
			tmp_2 = (0.5 / a) * (sqrt(((-4.0 * c) * a)) - b);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = c * (-1.0 / b);
	} else {
		tmp_1 = (t_0 - b) / (a + a);
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = sqrt(((a * c) * (-4.0d0)))
    if (b <= 2.5d-12) then
        if (b >= 0.0d0) then
            tmp_2 = c * ((-2.0d0) / (t_0 + b))
        else
            tmp_2 = (0.5d0 / a) * (sqrt((((-4.0d0) * c) * a)) - b)
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = c * ((-1.0d0) / b)
    else
        tmp_1 = (t_0 - b) / (a + a)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((a * c) * -4.0));
	double tmp_1;
	if (b <= 2.5e-12) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / (t_0 + b));
		} else {
			tmp_2 = (0.5 / a) * (Math.sqrt(((-4.0 * c) * a)) - b);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = c * (-1.0 / b);
	} else {
		tmp_1 = (t_0 - b) / (a + a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((a * c) * -4.0))
	tmp_1 = 0
	if b <= 2.5e-12:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = c * (-2.0 / (t_0 + b))
		else:
			tmp_2 = (0.5 / a) * (math.sqrt(((-4.0 * c) * a)) - b)
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = c * (-1.0 / b)
	else:
		tmp_1 = (t_0 - b) / (a + a)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(a * c) * -4.0))
	tmp_1 = 0.0
	if (b <= 2.5e-12)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * Float64(-2.0 / Float64(t_0 + b)));
		else
			tmp_2 = Float64(Float64(0.5 / a) * Float64(sqrt(Float64(Float64(-4.0 * c) * a)) - b));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(c * Float64(-1.0 / b));
	else
		tmp_1 = Float64(Float64(t_0 - b) / Float64(a + a));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = sqrt(((a * c) * -4.0));
	tmp_2 = 0.0;
	if (b <= 2.5e-12)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = c * (-2.0 / (t_0 + b));
		else
			tmp_3 = (0.5 / a) * (sqrt(((-4.0 * c) * a)) - b);
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = c * (-1.0 / b);
	else
		tmp_2 = (t_0 - b) / (a + a);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, 2.5e-12], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[(t$95$0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(c * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.49999999999999985e-12

   1. Initial program 71.9%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 56.3

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Applied rewrites 56.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 41.1

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   7. Applied rewrites 41.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 41.1%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ } \end{array}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       2. mult-flip N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(a \cdot c\right) \cdot -4} - b\right) \cdot \frac{1}{a + a}\\ \end{array} \]

       3. lift--.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(a \cdot c\right) \cdot -4} - b\right) \cdot \frac{1}{a + a}\\ \end{array} \]

       4. sub-flip N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(a \cdot c\right) \cdot -4} + \left(\mathsf{neg}\left(b\right)\right)\right) \cdot \frac{1}{a + a}\\ \end{array} \]

       5. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(a \cdot c\right) \cdot -4} + \left(\mathsf{neg}\left(b\right)\right)\right) \cdot \frac{1}{a + a}\\ \end{array} \]

       6. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{-4 \cdot \left(a \cdot c\right)} + \left(\mathsf{neg}\left(b\right)\right)\right) \cdot \frac{1}{a + a}\\ \end{array} \]

       7. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{-4 \cdot \left(a \cdot c\right)} + \left(\mathsf{neg}\left(b\right)\right)\right) \cdot \frac{1}{a + a}\\ \end{array} \]

       8. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{-4 \cdot \left(a \cdot c\right)} + \mathsf{Rewrite<=}\left(lift-neg.f64, \left(-b\right)\right)\right) \cdot \frac{1}{a + a}\\ \end{array} \]

       9. +-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{a + a}\\ \end{array} \]

       10. lift-+.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{a + a}\\ \end{array} \]

       11. lift-+.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}\right) \cdot \color{blue}{\frac{1}{a + a}}\\ \end{array} \]

       12. count-2-rev N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}\right) \cdot \color{blue}{\frac{1}{2 \cdot a}}\\ \end{array} \]

       13. lift-*.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}\right) \cdot \color{blue}{\frac{1}{2 \cdot a}}\\ \end{array} \]

       14. *-commutative N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot a} \cdot \left(\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}\right)\\ \end{array} \]

   11. Applied rewrites 41.1%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\left(-4 \cdot c\right) \cdot a} - b\right)\\ \end{array} \]

   if 2.49999999999999985e-12 < b

   1. Initial program 71.9%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 56.3

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Applied rewrites 56.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 41.1

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   7. Applied rewrites 41.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 41.1%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ } \end{array}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-1}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-/.f64 54.5

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-1}{\color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

   12. Applied rewrites 54.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-1}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 60.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \mathbf{if}\;b \leq 2.5 \cdot 10^{-12}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-1}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (* (* a c) -4.0)) b) (+ a a))))
   (if (<= b 2.5e-12)
     (if (>= b 0.0) (* -2.0 (/ c (sqrt (- (* 4.0 (* a c)))))) t_0)
     (if (>= b 0.0) (* c (/ -1.0 b)) t_0))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((a * c) * -4.0)) - b) / (a + a);
	double tmp_1;
	if (b <= 2.5e-12) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -2.0 * (c / sqrt(-(4.0 * (a * c))));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = c * (-1.0 / b);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = (sqrt(((a * c) * (-4.0d0))) - b) / (a + a)
    if (b <= 2.5d-12) then
        if (b >= 0.0d0) then
            tmp_2 = (-2.0d0) * (c / sqrt(-(4.0d0 * (a * c))))
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = c * ((-1.0d0) / b)
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((a * c) * -4.0)) - b) / (a + a);
	double tmp_1;
	if (b <= 2.5e-12) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -2.0 * (c / Math.sqrt(-(4.0 * (a * c))));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = c * (-1.0 / b);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((a * c) * -4.0)) - b) / (a + a)
	tmp_1 = 0
	if b <= 2.5e-12:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -2.0 * (c / math.sqrt(-(4.0 * (a * c))))
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = c * (-1.0 / b)
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(a * c) * -4.0)) - b) / Float64(a + a))
	tmp_1 = 0.0
	if (b <= 2.5e-12)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-2.0 * Float64(c / sqrt(Float64(-Float64(4.0 * Float64(a * c))))));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(c * Float64(-1.0 / b));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = (sqrt(((a * c) * -4.0)) - b) / (a + a);
	tmp_2 = 0.0;
	if (b <= 2.5e-12)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -2.0 * (c / sqrt(-(4.0 * (a * c))));
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = c * (-1.0 / b);
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 2.5e-12], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(c / N[Sqrt[(-N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(c * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.49999999999999985e-12

   1. Initial program 71.9%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 56.3

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Applied rewrites 56.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 41.1

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   7. Applied rewrites 41.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 41.1%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ } \end{array}} \]

   10. Taylor expanded in b around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-2 \cdot \frac{c}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\frac{c}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       2. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       4. lower-neg.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       5. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       6. lower-*.f64 34.2

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

   12. Applied rewrites 34.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

   if 2.49999999999999985e-12 < b

   1. Initial program 71.9%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 56.3

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Applied rewrites 56.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 41.1

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   7. Applied rewrites 41.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 41.1%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ } \end{array}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-1}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-/.f64 54.5

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-1}{\color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

   12. Applied rewrites 54.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-1}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 38.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \mathbf{if}\;a \leq -6 \cdot 10^{-306}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{a \cdot -4} \cdot \sqrt{c}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot c} \cdot \sqrt{a}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (* (* a c) -4.0)) b) (+ a a))))
   (if (<= a -6e-306)
     (if (>= b 0.0) (* -2.0 (/ c (* (sqrt (* a -4.0)) (sqrt c)))) t_0)
     (if (>= b 0.0) (* -2.0 (/ c (* (sqrt (* -4.0 c)) (sqrt a)))) t_0))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((a * c) * -4.0)) - b) / (a + a);
	double tmp_1;
	if (a <= -6e-306) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -2.0 * (c / (sqrt((a * -4.0)) * sqrt(c)));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = -2.0 * (c / (sqrt((-4.0 * c)) * sqrt(a)));
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = (sqrt(((a * c) * (-4.0d0))) - b) / (a + a)
    if (a <= (-6d-306)) then
        if (b >= 0.0d0) then
            tmp_2 = (-2.0d0) * (c / (sqrt((a * (-4.0d0))) * sqrt(c)))
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (-2.0d0) * (c / (sqrt(((-4.0d0) * c)) * sqrt(a)))
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((a * c) * -4.0)) - b) / (a + a);
	double tmp_1;
	if (a <= -6e-306) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -2.0 * (c / (Math.sqrt((a * -4.0)) * Math.sqrt(c)));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = -2.0 * (c / (Math.sqrt((-4.0 * c)) * Math.sqrt(a)));
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((a * c) * -4.0)) - b) / (a + a)
	tmp_1 = 0
	if a <= -6e-306:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -2.0 * (c / (math.sqrt((a * -4.0)) * math.sqrt(c)))
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = -2.0 * (c / (math.sqrt((-4.0 * c)) * math.sqrt(a)))
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(a * c) * -4.0)) - b) / Float64(a + a))
	tmp_1 = 0.0
	if (a <= -6e-306)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-2.0 * Float64(c / Float64(sqrt(Float64(a * -4.0)) * sqrt(c))));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(-2.0 * Float64(c / Float64(sqrt(Float64(-4.0 * c)) * sqrt(a))));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = (sqrt(((a * c) * -4.0)) - b) / (a + a);
	tmp_2 = 0.0;
	if (a <= -6e-306)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -2.0 * (c / (sqrt((a * -4.0)) * sqrt(c)));
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = -2.0 * (c / (sqrt((-4.0 * c)) * sqrt(a)));
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -6e-306], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(c / N[(N[Sqrt[N[(a * -4.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(c / N[(N[Sqrt[N[(-4.0 * c), $MachinePrecision]], $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.00000000000000048e-306

   1. Initial program 71.9%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 56.3

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Applied rewrites 56.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 41.1

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   7. Applied rewrites 41.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 41.1%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ } \end{array}} \]

   10. Taylor expanded in b around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-2 \cdot \frac{c}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\frac{c}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       2. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       4. lower-neg.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       5. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       6. lower-*.f64 34.2

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

   12. Applied rewrites 34.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]
   13. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       2. lift-neg.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       3. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       4. distribute-lft-neg-in N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       5. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       6. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       7. associate-*r* N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{\left(-4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       8. sqrt-prod N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot a} \cdot \color{blue}{\sqrt{c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       9. lower-unsound-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot a} \cdot \color{blue}{\sqrt{c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       10. lower-unsound-sqrt.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot a} \cdot \sqrt{\color{blue}{c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       11. *-commutative N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{a \cdot -4} \cdot \sqrt{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       12. lower-*.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{a \cdot -4} \cdot \sqrt{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       13. lower-unsound-sqrt.f64 29.4

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{a \cdot -4} \cdot \sqrt{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

   14. Applied rewrites 29.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{a \cdot -4} \cdot \color{blue}{\sqrt{c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

   if -6.00000000000000048e-306 < a

   1. Initial program 71.9%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 56.3

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Applied rewrites 56.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 41.1

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   7. Applied rewrites 41.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 41.1%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ } \end{array}} \]

   10. Taylor expanded in b around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-2 \cdot \frac{c}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\frac{c}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       2. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       4. lower-neg.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       5. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       6. lower-*.f64 34.2

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

   12. Applied rewrites 34.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]
   13. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       2. lift-neg.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       3. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       4. distribute-lft-neg-in N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       5. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       6. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{\left(a \cdot c\right) \cdot -4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       7. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{\left(a \cdot c\right) \cdot -4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       8. associate-*l* N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{a \cdot \left(c \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       9. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{a \cdot \left(c \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       10. *-commutative N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{\left(c \cdot -4\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       11. sqrt-prod N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{c \cdot -4} \cdot \color{blue}{\sqrt{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       12. lower-unsound-*.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{c \cdot -4} \cdot \color{blue}{\sqrt{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       13. lower-unsound-sqrt.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{c \cdot -4} \cdot \sqrt{\color{blue}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       14. lift-*.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{c \cdot -4} \cdot \sqrt{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       15. *-commutative N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot c} \cdot \sqrt{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       16. lower-*.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot c} \cdot \sqrt{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       17. lower-unsound-sqrt.f64 29.5

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot c} \cdot \sqrt{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

   14. Applied rewrites 29.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot c} \cdot \color{blue}{\sqrt{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 35.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \mathbf{if}\;b \leq 5.8:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{1}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (* (* a c) -4.0)) b) (+ a a))))
   (if (<= b 5.8)
     (if (>= b 0.0) (* -2.0 (/ c (sqrt (- (* 4.0 (* a c)))))) t_0)
     (if (>= b 0.0) (* -2.0 (/ 1.0 (sqrt (* -4.0 (/ a c))))) t_0))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((a * c) * -4.0)) - b) / (a + a);
	double tmp_1;
	if (b <= 5.8) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -2.0 * (c / sqrt(-(4.0 * (a * c))));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = -2.0 * (1.0 / sqrt((-4.0 * (a / c))));
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = (sqrt(((a * c) * (-4.0d0))) - b) / (a + a)
    if (b <= 5.8d0) then
        if (b >= 0.0d0) then
            tmp_2 = (-2.0d0) * (c / sqrt(-(4.0d0 * (a * c))))
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (-2.0d0) * (1.0d0 / sqrt(((-4.0d0) * (a / c))))
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((a * c) * -4.0)) - b) / (a + a);
	double tmp_1;
	if (b <= 5.8) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -2.0 * (c / Math.sqrt(-(4.0 * (a * c))));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = -2.0 * (1.0 / Math.sqrt((-4.0 * (a / c))));
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((a * c) * -4.0)) - b) / (a + a)
	tmp_1 = 0
	if b <= 5.8:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -2.0 * (c / math.sqrt(-(4.0 * (a * c))))
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = -2.0 * (1.0 / math.sqrt((-4.0 * (a / c))))
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(a * c) * -4.0)) - b) / Float64(a + a))
	tmp_1 = 0.0
	if (b <= 5.8)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-2.0 * Float64(c / sqrt(Float64(-Float64(4.0 * Float64(a * c))))));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(-2.0 * Float64(1.0 / sqrt(Float64(-4.0 * Float64(a / c)))));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = (sqrt(((a * c) * -4.0)) - b) / (a + a);
	tmp_2 = 0.0;
	if (b <= 5.8)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -2.0 * (c / sqrt(-(4.0 * (a * c))));
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = -2.0 * (1.0 / sqrt((-4.0 * (a / c))));
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 5.8], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(c / N[Sqrt[(-N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(1.0 / N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < 5.79999999999999982

   1. Initial program 71.9%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 56.3

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Applied rewrites 56.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 41.1

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   7. Applied rewrites 41.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 41.1%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ } \end{array}} \]

   10. Taylor expanded in b around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-2 \cdot \frac{c}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\frac{c}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       2. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       4. lower-neg.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       5. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       6. lower-*.f64 34.2

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

   12. Applied rewrites 34.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

   if 5.79999999999999982 < b

   1. Initial program 71.9%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 56.3

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Applied rewrites 56.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 41.1

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   7. Applied rewrites 41.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 41.1%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ } \end{array}} \]

   10. Taylor expanded in b around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-2 \cdot \frac{c}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\frac{c}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       2. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       4. lower-neg.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       5. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       6. lower-*.f64 34.2

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

   12. Applied rewrites 34.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

   13. Taylor expanded in c around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{1}{\color{blue}{\sqrt{-4 \cdot \frac{a}{c}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]
   14. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{1}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{1}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       3. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{1}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

       4. lower-/.f64 29.3

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{1}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

   15. Applied rewrites 29.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{1}{\color{blue}{\sqrt{-4 \cdot \frac{a}{c}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 29.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{1}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0)
   (* -2.0 (/ 1.0 (sqrt (* -4.0 (/ a c)))))
   (/ (- (sqrt (* (* a c) -4.0)) b) (+ a a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -2.0 * (1.0 / sqrt((-4.0 * (a / c))));
	} else {
		tmp = (sqrt(((a * c) * -4.0)) - b) / (a + a);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b >= 0.0d0) then
        tmp = (-2.0d0) * (1.0d0 / sqrt(((-4.0d0) * (a / c))))
    else
        tmp = (sqrt(((a * c) * (-4.0d0))) - b) / (a + a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -2.0 * (1.0 / Math.sqrt((-4.0 * (a / c))));
	} else {
		tmp = (Math.sqrt(((a * c) * -4.0)) - b) / (a + a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = -2.0 * (1.0 / math.sqrt((-4.0 * (a / c))))
	else:
		tmp = (math.sqrt(((a * c) * -4.0)) - b) / (a + a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(-2.0 * Float64(1.0 / sqrt(Float64(-4.0 * Float64(a / c)))));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(a * c) * -4.0)) - b) / Float64(a + a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = -2.0 * (1.0 / sqrt((-4.0 * (a / c))));
	else
		tmp = (sqrt(((a * c) * -4.0)) - b) / (a + a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(-2.0 * N[(1.0 / N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 71.9%

   \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

2. Taylor expanded in a around inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. lower-*.f64 56.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

4. Applied rewrites 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

5. Taylor expanded in a around inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   2. lower-*.f64 41.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

7. Applied rewrites 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
8. Step-by-step derivation

9. Applied rewrites 41.1%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ } \end{array}} \]

10. Taylor expanded in b around 0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-2 \cdot \frac{c}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]
11. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\frac{c}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

    2. lower-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

    3. lower-sqrt.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

    4. lower-neg.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

    5. lower-*.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

    6. lower-*.f64 34.2

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

12. Applied rewrites 34.2%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

13. Taylor expanded in c around inf

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{1}{\color{blue}{\sqrt{-4 \cdot \frac{a}{c}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]
14. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{1}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

    2. lower-sqrt.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{1}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

    3. lower-*.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{1}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

    4. lower-/.f64 29.3

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{1}{\sqrt{-4 \cdot \frac{a}{c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]

15. Applied rewrites 29.3%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{1}{\color{blue}{\sqrt{-4 \cdot \frac{a}{c}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4} - b}{a + a}\\ \end{array} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2025159 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))