jeff quadratic root 1

Percentage Accurate: 72.4% → 90.8%

Time: 5.1s

Alternatives: 14

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array}\\ t_1 := \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}\\ \mathbf{if}\;b \leq -5 \cdot 10^{+142}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{+122}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t\_1}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (if (>= b 0.0) (* -1.0 (/ b a)) (/ (* 2.0 c) (* -1.0 (* b 2.0)))))
        (t_1 (sqrt (fma (* c a) -4.0 (* b b)))))
   (if (<= b -5e+142)
     t_0
     (if (<= b 3.2e+122)
       (if (>= b 0.0) (/ (- (- b) t_1) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) t_1)))
       t_0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -1.0 * (b / a);
	} else {
		tmp = (2.0 * c) / (-1.0 * (b * 2.0));
	}
	double t_0 = tmp;
	double t_1 = sqrt(fma((c * a), -4.0, (b * b)));
	double tmp_1;
	if (b <= -5e+142) {
		tmp_1 = t_0;
	} else if (b <= 3.2e+122) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-b - t_1) / (2.0 * a);
		} else {
			tmp_2 = (2.0 * c) / (-b + t_1);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(-1.0 * Float64(b / a));
	else
		tmp = Float64(Float64(2.0 * c) / Float64(-1.0 * Float64(b * 2.0)));
	end
	t_0 = tmp
	t_1 = sqrt(fma(Float64(c * a), -4.0, Float64(b * b)))
	tmp_1 = 0.0
	if (b <= -5e+142)
		tmp_1 = t_0;
	elseif (b <= 3.2e+122)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(Float64(-b) - t_1) / Float64(2.0 * a));
		else
			tmp_2 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_1));
		end
		tmp_1 = tmp_2;
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = If[GreaterEqual[b, 0.0], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[(-1.0 * N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]}, Block[{t$95$1 = N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -5e+142], t$95$0, If[LessEqual[b, 3.2e+122], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$1), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$1), $MachinePrecision]), $MachinePrecision]], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if b < -5.0000000000000001e142 or 3.2000000000000001e122 < b

   1. Initial program 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 72.4%

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

   3. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 72.4%

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 67.9%

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

   8. Applied rewrites 67.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 69.7%

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

   12. Taylor expanded in b around inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 67.7%

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

   14. Applied rewrites 67.7%

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

   if -5.0000000000000001e142 < b < 3.2000000000000001e122

   1. Initial program 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 72.4%

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

   3. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 72.4%

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

   5. Applied rewrites 72.4%

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

Alternative 2: 90.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array}\\ t_1 := \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}\\ \mathbf{if}\;b \leq -5 \cdot 10^{+142}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{+122}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(b + t\_1\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_1 - b}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (if (>= b 0.0) (* -1.0 (/ b a)) (/ (* 2.0 c) (* -1.0 (* b 2.0)))))
        (t_1 (sqrt (fma (* c -4.0) a (* b b)))))
   (if (<= b -5e+142)
     t_0
     (if (<= b 3.2e+122)
       (if (>= b 0.0) (* (+ b t_1) (/ -0.5 a)) (/ (+ c c) (- t_1 b)))
       t_0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -1.0 * (b / a);
	} else {
		tmp = (2.0 * c) / (-1.0 * (b * 2.0));
	}
	double t_0 = tmp;
	double t_1 = sqrt(fma((c * -4.0), a, (b * b)));
	double tmp_1;
	if (b <= -5e+142) {
		tmp_1 = t_0;
	} else if (b <= 3.2e+122) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (b + t_1) * (-0.5 / a);
		} else {
			tmp_2 = (c + c) / (t_1 - b);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(-1.0 * Float64(b / a));
	else
		tmp = Float64(Float64(2.0 * c) / Float64(-1.0 * Float64(b * 2.0)));
	end
	t_0 = tmp
	t_1 = sqrt(fma(Float64(c * -4.0), a, Float64(b * b)))
	tmp_1 = 0.0
	if (b <= -5e+142)
		tmp_1 = t_0;
	elseif (b <= 3.2e+122)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(b + t_1) * Float64(-0.5 / a));
		else
			tmp_2 = Float64(Float64(c + c) / Float64(t_1 - b));
		end
		tmp_1 = tmp_2;
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = If[GreaterEqual[b, 0.0], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[(-1.0 * N[(b * 2.0), $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, -5e+142], t$95$0, If[LessEqual[b, 3.2e+122], If[GreaterEqual[b, 0.0], N[(N[(b + t$95$1), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(t$95$1 - b), $MachinePrecision]), $MachinePrecision]], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if b < -5.0000000000000001e142 or 3.2000000000000001e122 < b

   1. Initial program 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 72.4%

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

   3. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 72.4%

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 67.9%

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

   8. Applied rewrites 67.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 69.7%

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

   12. Taylor expanded in b around inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 67.7%

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

   14. Applied rewrites 67.7%

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

   if -5.0000000000000001e142 < b < 3.2000000000000001e122

   1. Initial program 72.4%

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

   3. Applied rewrites 72.3%

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

Alternative 3: 86.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\left|\left(a \cdot c\right) \cdot -4\right|}\\ \mathbf{if}\;b \leq -5 \cdot 10^{+142}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq -2.4 \cdot 10^{-172}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \left(\frac{b}{a} - \sqrt{\frac{c}{a} \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{-54}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + t\_0}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + c \cdot \sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fabs (* (* a c) -4.0)))))
   (if (<= b -5e+142)
     (if (>= b 0.0) (* -1.0 (/ b a)) (/ (* 2.0 c) (* -1.0 (* b 2.0))))
     (if (<= b -2.4e-172)
       (if (>= b 0.0)
         (* -0.5 (- (/ b a) (sqrt (* (/ c a) -4.0))))
         (/ (+ c c) (- (sqrt (fma (* a c) -4.0 (* b b))) b)))
       (if (<= b 5.4e-54)
         (if (>= b 0.0) (/ (+ b t_0) (* -2.0 a)) (/ (+ c c) (- t_0 b)))
         (if (>= b 0.0)
           (/ (* -2.0 b) (* 2.0 a))
           (/ (* 2.0 c) (+ (- b) (* c (sqrt (* -4.0 (/ a c))))))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fabs(((a * c) * -4.0)));
	double tmp_1;
	if (b <= -5e+142) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -1.0 * (b / a);
		} else {
			tmp_2 = (2.0 * c) / (-1.0 * (b * 2.0));
		}
		tmp_1 = tmp_2;
	} else if (b <= -2.4e-172) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -0.5 * ((b / a) - sqrt(((c / a) * -4.0)));
		} else {
			tmp_3 = (c + c) / (sqrt(fma((a * c), -4.0, (b * b))) - b);
		}
		tmp_1 = tmp_3;
	} else if (b <= 5.4e-54) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (b + t_0) / (-2.0 * a);
		} else {
			tmp_4 = (c + c) / (t_0 - b);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * b) / (2.0 * a);
	} else {
		tmp_1 = (2.0 * c) / (-b + (c * sqrt((-4.0 * (a / c)))));
	}
	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 <= -5e+142)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-1.0 * Float64(b / a));
		else
			tmp_2 = Float64(Float64(2.0 * c) / Float64(-1.0 * Float64(b * 2.0)));
		end
		tmp_1 = tmp_2;
	elseif (b <= -2.4e-172)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(-0.5 * Float64(Float64(b / a) - sqrt(Float64(Float64(c / a) * -4.0))));
		else
			tmp_3 = Float64(Float64(c + c) / Float64(sqrt(fma(Float64(a * c), -4.0, Float64(b * b))) - b));
		end
		tmp_1 = tmp_3;
	elseif (b <= 5.4e-54)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(b + t_0) / Float64(-2.0 * a));
		else
			tmp_4 = Float64(Float64(c + c) / Float64(t_0 - b));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-2.0 * b) / Float64(2.0 * a));
	else
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + Float64(c * sqrt(Float64(-4.0 * Float64(a / c))))));
	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, -5e+142], If[GreaterEqual[b, 0.0], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[(-1.0 * N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -2.4e-172], If[GreaterEqual[b, 0.0], N[(-0.5 * N[(N[(b / a), $MachinePrecision] - N[Sqrt[N[(N[(c / a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 5.4e-54], If[GreaterEqual[b, 0.0], N[(N[(b + t$95$0), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + N[(c * N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -5.0000000000000001e142

   1. Initial program 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 72.4%

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

   3. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 72.4%

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 67.9%

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

   8. Applied rewrites 67.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 69.7%

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

   12. Taylor expanded in b around inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 67.7%

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

   14. Applied rewrites 67.7%

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

   if -5.0000000000000001e142 < b < -2.4000000000000001e-172

   1. Initial program 72.4%

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

   2. Taylor expanded in a around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 49.8%

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

   4. Applied rewrites 49.8%

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

   6. Applied rewrites 49.8%

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

   if -2.4000000000000001e-172 < b < 5.4000000000000005e-54

   1. Initial program 72.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 57.3%

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

   4. Applied rewrites 57.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 41.7%

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

   7. Applied rewrites 41.7%

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

   9. Applied rewrites 41.7%

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

       1. rem-square-sqrt N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. sqr-abs-rev N/A

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

       5. mul-fabs N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. rem-square-sqrt N/A

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

       9. lower-fabs.f64 46.2%

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

   11. Applied rewrites 46.2%

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

       1. rem-square-sqrt N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. sqr-abs-rev N/A

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

       5. mul-fabs N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. rem-square-sqrt N/A

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

       9. lower-fabs.f64 50.6%

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

   13. Applied rewrites 50.6%

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

   if 5.4000000000000005e-54 < b

   1. Initial program 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 72.4%

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

   3. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 72.4%

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 70.3%

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

   8. Applied rewrites 70.3%

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

   9. Taylor expanded in c around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 46.8%

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

   11. Applied rewrites 46.8%

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

Alternative 4: 81.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array}\\ t_1 := \sqrt{\left|\left(a \cdot c\right) \cdot -4\right|}\\ \mathbf{if}\;b \leq -7.2 \cdot 10^{-22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{-54}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + t\_1}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_1 - b}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (if (>= b 0.0) (* -1.0 (/ b a)) (/ (* 2.0 c) (* -1.0 (* b 2.0)))))
        (t_1 (sqrt (fabs (* (* a c) -4.0)))))
   (if (<= b -7.2e-22)
     t_0
     (if (<= b 5.4e-54)
       (if (>= b 0.0) (/ (+ b t_1) (* -2.0 a)) (/ (+ c c) (- t_1 b)))
       t_0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -1.0 * (b / a);
	} else {
		tmp = (2.0 * c) / (-1.0 * (b * 2.0));
	}
	double t_0 = tmp;
	double t_1 = sqrt(fabs(((a * c) * -4.0)));
	double tmp_1;
	if (b <= -7.2e-22) {
		tmp_1 = t_0;
	} else if (b <= 5.4e-54) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (b + t_1) / (-2.0 * a);
		} else {
			tmp_2 = (c + c) / (t_1 - b);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    if (b >= 0.0d0) then
        tmp = (-1.0d0) * (b / a)
    else
        tmp = (2.0d0 * c) / ((-1.0d0) * (b * 2.0d0))
    end if
    t_0 = tmp
    t_1 = sqrt(abs(((a * c) * (-4.0d0))))
    if (b <= (-7.2d-22)) then
        tmp_1 = t_0
    else if (b <= 5.4d-54) then
        if (b >= 0.0d0) then
            tmp_2 = (b + t_1) / ((-2.0d0) * a)
        else
            tmp_2 = (c + c) / (t_1 - b)
        end if
        tmp_1 = tmp_2
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -1.0 * (b / a);
	} else {
		tmp = (2.0 * c) / (-1.0 * (b * 2.0));
	}
	double t_0 = tmp;
	double t_1 = Math.sqrt(Math.abs(((a * c) * -4.0)));
	double tmp_1;
	if (b <= -7.2e-22) {
		tmp_1 = t_0;
	} else if (b <= 5.4e-54) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (b + t_1) / (-2.0 * a);
		} else {
			tmp_2 = (c + c) / (t_1 - b);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = -1.0 * (b / a)
	else:
		tmp = (2.0 * c) / (-1.0 * (b * 2.0))
	t_0 = tmp
	t_1 = math.sqrt(math.fabs(((a * c) * -4.0)))
	tmp_1 = 0
	if b <= -7.2e-22:
		tmp_1 = t_0
	elif b <= 5.4e-54:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (b + t_1) / (-2.0 * a)
		else:
			tmp_2 = (c + c) / (t_1 - b)
		tmp_1 = tmp_2
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(-1.0 * Float64(b / a));
	else
		tmp = Float64(Float64(2.0 * c) / Float64(-1.0 * Float64(b * 2.0)));
	end
	t_0 = tmp
	t_1 = sqrt(abs(Float64(Float64(a * c) * -4.0)))
	tmp_1 = 0.0
	if (b <= -7.2e-22)
		tmp_1 = t_0;
	elseif (b <= 5.4e-54)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(b + t_1) / Float64(-2.0 * a));
		else
			tmp_2 = Float64(Float64(c + c) / Float64(t_1 - b));
		end
		tmp_1 = tmp_2;
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = -1.0 * (b / a);
	else
		tmp = (2.0 * c) / (-1.0 * (b * 2.0));
	end
	t_0 = tmp;
	t_1 = sqrt(abs(((a * c) * -4.0)));
	tmp_2 = 0.0;
	if (b <= -7.2e-22)
		tmp_2 = t_0;
	elseif (b <= 5.4e-54)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (b + t_1) / (-2.0 * a);
		else
			tmp_3 = (c + c) / (t_1 - b);
		end
		tmp_2 = tmp_3;
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = If[GreaterEqual[b, 0.0], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[(-1.0 * N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]}, Block[{t$95$1 = N[Sqrt[N[Abs[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -7.2e-22], t$95$0, If[LessEqual[b, 5.4e-54], If[GreaterEqual[b, 0.0], N[(N[(b + t$95$1), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(t$95$1 - b), $MachinePrecision]), $MachinePrecision]], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if b < -7.1999999999999996e-22 or 5.4000000000000005e-54 < b

   1. Initial program 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 72.4%

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

   3. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 72.4%

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 67.9%

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

   8. Applied rewrites 67.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 69.7%

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

   12. Taylor expanded in b around inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 67.7%

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

   14. Applied rewrites 67.7%

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

   if -7.1999999999999996e-22 < b < 5.4000000000000005e-54

   1. Initial program 72.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 57.3%

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

   4. Applied rewrites 57.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 41.7%

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

   7. Applied rewrites 41.7%

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

   9. Applied rewrites 41.7%

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

       1. rem-square-sqrt N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. sqr-abs-rev N/A

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

       5. mul-fabs N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. rem-square-sqrt N/A

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

       9. lower-fabs.f64 46.2%

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

   11. Applied rewrites 46.2%

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

       1. rem-square-sqrt N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. sqr-abs-rev N/A

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

       5. mul-fabs N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. rem-square-sqrt N/A

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

       9. lower-fabs.f64 50.6%

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

   13. Applied rewrites 50.6%

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

Alternative 5: 81.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\left|\left(a \cdot c\right) \cdot -4\right|}\\ \mathbf{if}\;b \leq -7.2 \cdot 10^{-22}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{-54}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + t\_0}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + c \cdot \sqrt{-4 \cdot \frac{a}{c}}}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fabs (* (* a c) -4.0)))))
   (if (<= b -7.2e-22)
     (if (>= b 0.0) (* -1.0 (/ b a)) (/ (* 2.0 c) (* -1.0 (* b 2.0))))
     (if (<= b 5.4e-54)
       (if (>= b 0.0) (/ (+ b t_0) (* -2.0 a)) (/ (+ c c) (- t_0 b)))
       (if (>= b 0.0)
         (/ (* -2.0 b) (* 2.0 a))
         (/ (* 2.0 c) (+ (- b) (* c (sqrt (* -4.0 (/ a c)))))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fabs(((a * c) * -4.0)));
	double tmp_1;
	if (b <= -7.2e-22) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -1.0 * (b / a);
		} else {
			tmp_2 = (2.0 * c) / (-1.0 * (b * 2.0));
		}
		tmp_1 = tmp_2;
	} else if (b <= 5.4e-54) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (b + t_0) / (-2.0 * a);
		} else {
			tmp_3 = (c + c) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * b) / (2.0 * a);
	} else {
		tmp_1 = (2.0 * c) / (-b + (c * sqrt((-4.0 * (a / c)))));
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = sqrt(abs(((a * c) * (-4.0d0))))
    if (b <= (-7.2d-22)) then
        if (b >= 0.0d0) then
            tmp_2 = (-1.0d0) * (b / a)
        else
            tmp_2 = (2.0d0 * c) / ((-1.0d0) * (b * 2.0d0))
        end if
        tmp_1 = tmp_2
    else if (b <= 5.4d-54) then
        if (b >= 0.0d0) then
            tmp_3 = (b + t_0) / ((-2.0d0) * a)
        else
            tmp_3 = (c + c) / (t_0 - b)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = ((-2.0d0) * b) / (2.0d0 * a)
    else
        tmp_1 = (2.0d0 * c) / (-b + (c * sqrt(((-4.0d0) * (a / c)))))
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(Math.abs(((a * c) * -4.0)));
	double tmp_1;
	if (b <= -7.2e-22) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -1.0 * (b / a);
		} else {
			tmp_2 = (2.0 * c) / (-1.0 * (b * 2.0));
		}
		tmp_1 = tmp_2;
	} else if (b <= 5.4e-54) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (b + t_0) / (-2.0 * a);
		} else {
			tmp_3 = (c + c) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * b) / (2.0 * a);
	} else {
		tmp_1 = (2.0 * c) / (-b + (c * Math.sqrt((-4.0 * (a / c)))));
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(math.fabs(((a * c) * -4.0)))
	tmp_1 = 0
	if b <= -7.2e-22:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -1.0 * (b / a)
		else:
			tmp_2 = (2.0 * c) / (-1.0 * (b * 2.0))
		tmp_1 = tmp_2
	elif b <= 5.4e-54:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (b + t_0) / (-2.0 * a)
		else:
			tmp_3 = (c + c) / (t_0 - b)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = (-2.0 * b) / (2.0 * a)
	else:
		tmp_1 = (2.0 * c) / (-b + (c * math.sqrt((-4.0 * (a / c)))))
	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 <= -7.2e-22)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-1.0 * Float64(b / a));
		else
			tmp_2 = Float64(Float64(2.0 * c) / Float64(-1.0 * Float64(b * 2.0)));
		end
		tmp_1 = tmp_2;
	elseif (b <= 5.4e-54)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(b + t_0) / Float64(-2.0 * a));
		else
			tmp_3 = Float64(Float64(c + c) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-2.0 * b) / Float64(2.0 * a));
	else
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + Float64(c * sqrt(Float64(-4.0 * Float64(a / c))))));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(abs(((a * c) * -4.0)));
	tmp_2 = 0.0;
	if (b <= -7.2e-22)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -1.0 * (b / a);
		else
			tmp_3 = (2.0 * c) / (-1.0 * (b * 2.0));
		end
		tmp_2 = tmp_3;
	elseif (b <= 5.4e-54)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (b + t_0) / (-2.0 * a);
		else
			tmp_4 = (c + c) / (t_0 - b);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = (-2.0 * b) / (2.0 * a);
	else
		tmp_2 = (2.0 * c) / (-b + (c * sqrt((-4.0 * (a / c)))));
	end
	tmp_5 = tmp_2;
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, -7.2e-22], If[GreaterEqual[b, 0.0], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[(-1.0 * N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 5.4e-54], If[GreaterEqual[b, 0.0], N[(N[(b + t$95$0), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + N[(c * N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.1999999999999996e-22

   1. Initial program 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 72.4%

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

   3. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 72.4%

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 67.9%

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

   8. Applied rewrites 67.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 69.7%

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

   12. Taylor expanded in b around inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 67.7%

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

   14. Applied rewrites 67.7%

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

   if -7.1999999999999996e-22 < b < 5.4000000000000005e-54

   1. Initial program 72.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 57.3%

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

   4. Applied rewrites 57.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 41.7%

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

   7. Applied rewrites 41.7%

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

   9. Applied rewrites 41.7%

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

       1. rem-square-sqrt N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. sqr-abs-rev N/A

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

       5. mul-fabs N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. rem-square-sqrt N/A

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

       9. lower-fabs.f64 46.2%

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

   11. Applied rewrites 46.2%

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

       1. rem-square-sqrt N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. sqr-abs-rev N/A

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

       5. mul-fabs N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. rem-square-sqrt N/A

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

       9. lower-fabs.f64 50.6%

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

   13. Applied rewrites 50.6%

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

   if 5.4000000000000005e-54 < b

   1. Initial program 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 72.4%

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

   3. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 72.4%

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 70.3%

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

   8. Applied rewrites 70.3%

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

   9. Taylor expanded in c around inf

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 46.8%

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

   11. Applied rewrites 46.8%

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

Alternative 6: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array}\\ t_1 := \sqrt{\left(a \cdot c\right) \cdot -4}\\ \mathbf{if}\;b \leq -4 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{-54}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + t\_1}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_1 - b}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (if (>= b 0.0) (* -1.0 (/ b a)) (/ (* 2.0 c) (* -1.0 (* b 2.0)))))
        (t_1 (sqrt (* (* a c) -4.0))))
   (if (<= b -4e-44)
     t_0
     (if (<= b 5.4e-54)
       (if (>= b 0.0) (/ (+ b t_1) (* -2.0 a)) (/ (+ c c) (- t_1 b)))
       t_0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -1.0 * (b / a);
	} else {
		tmp = (2.0 * c) / (-1.0 * (b * 2.0));
	}
	double t_0 = tmp;
	double t_1 = sqrt(((a * c) * -4.0));
	double tmp_1;
	if (b <= -4e-44) {
		tmp_1 = t_0;
	} else if (b <= 5.4e-54) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (b + t_1) / (-2.0 * a);
		} else {
			tmp_2 = (c + c) / (t_1 - b);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    if (b >= 0.0d0) then
        tmp = (-1.0d0) * (b / a)
    else
        tmp = (2.0d0 * c) / ((-1.0d0) * (b * 2.0d0))
    end if
    t_0 = tmp
    t_1 = sqrt(((a * c) * (-4.0d0)))
    if (b <= (-4d-44)) then
        tmp_1 = t_0
    else if (b <= 5.4d-54) then
        if (b >= 0.0d0) then
            tmp_2 = (b + t_1) / ((-2.0d0) * a)
        else
            tmp_2 = (c + c) / (t_1 - b)
        end if
        tmp_1 = tmp_2
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -1.0 * (b / a);
	} else {
		tmp = (2.0 * c) / (-1.0 * (b * 2.0));
	}
	double t_0 = tmp;
	double t_1 = Math.sqrt(((a * c) * -4.0));
	double tmp_1;
	if (b <= -4e-44) {
		tmp_1 = t_0;
	} else if (b <= 5.4e-54) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (b + t_1) / (-2.0 * a);
		} else {
			tmp_2 = (c + c) / (t_1 - b);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = -1.0 * (b / a)
	else:
		tmp = (2.0 * c) / (-1.0 * (b * 2.0))
	t_0 = tmp
	t_1 = math.sqrt(((a * c) * -4.0))
	tmp_1 = 0
	if b <= -4e-44:
		tmp_1 = t_0
	elif b <= 5.4e-54:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (b + t_1) / (-2.0 * a)
		else:
			tmp_2 = (c + c) / (t_1 - b)
		tmp_1 = tmp_2
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(-1.0 * Float64(b / a));
	else
		tmp = Float64(Float64(2.0 * c) / Float64(-1.0 * Float64(b * 2.0)));
	end
	t_0 = tmp
	t_1 = sqrt(Float64(Float64(a * c) * -4.0))
	tmp_1 = 0.0
	if (b <= -4e-44)
		tmp_1 = t_0;
	elseif (b <= 5.4e-54)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(b + t_1) / Float64(-2.0 * a));
		else
			tmp_2 = Float64(Float64(c + c) / Float64(t_1 - b));
		end
		tmp_1 = tmp_2;
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = -1.0 * (b / a);
	else
		tmp = (2.0 * c) / (-1.0 * (b * 2.0));
	end
	t_0 = tmp;
	t_1 = sqrt(((a * c) * -4.0));
	tmp_2 = 0.0;
	if (b <= -4e-44)
		tmp_2 = t_0;
	elseif (b <= 5.4e-54)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (b + t_1) / (-2.0 * a);
		else
			tmp_3 = (c + c) / (t_1 - b);
		end
		tmp_2 = tmp_3;
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = If[GreaterEqual[b, 0.0], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[(-1.0 * N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]}, Block[{t$95$1 = N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -4e-44], t$95$0, If[LessEqual[b, 5.4e-54], If[GreaterEqual[b, 0.0], N[(N[(b + t$95$1), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(t$95$1 - b), $MachinePrecision]), $MachinePrecision]], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.9999999999999998e-44 or 5.4000000000000005e-54 < b

   1. Initial program 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 72.4%

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

   3. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 72.4%

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 67.9%

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

   8. Applied rewrites 67.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 69.7%

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

   12. Taylor expanded in b around inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 67.7%

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

   14. Applied rewrites 67.7%

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

   if -3.9999999999999998e-44 < b < 5.4000000000000005e-54

   1. Initial program 72.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 57.3%

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

   4. Applied rewrites 57.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 41.7%

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

   7. Applied rewrites 41.7%

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

   9. Applied rewrites 41.7%

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

Alternative 7: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array}\\ t_1 := \sqrt{\left(a \cdot c\right) \cdot -4}\\ \mathbf{if}\;b \leq -4 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{-54}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + t\_1}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-2}{b - t\_1}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (if (>= b 0.0) (* -1.0 (/ b a)) (/ (* 2.0 c) (* -1.0 (* b 2.0)))))
        (t_1 (sqrt (* (* a c) -4.0))))
   (if (<= b -4e-44)
     t_0
     (if (<= b 5.4e-54)
       (if (>= b 0.0) (/ (+ b t_1) (* -2.0 a)) (* c (/ -2.0 (- b t_1))))
       t_0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -1.0 * (b / a);
	} else {
		tmp = (2.0 * c) / (-1.0 * (b * 2.0));
	}
	double t_0 = tmp;
	double t_1 = sqrt(((a * c) * -4.0));
	double tmp_1;
	if (b <= -4e-44) {
		tmp_1 = t_0;
	} else if (b <= 5.4e-54) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (b + t_1) / (-2.0 * a);
		} else {
			tmp_2 = c * (-2.0 / (b - t_1));
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    if (b >= 0.0d0) then
        tmp = (-1.0d0) * (b / a)
    else
        tmp = (2.0d0 * c) / ((-1.0d0) * (b * 2.0d0))
    end if
    t_0 = tmp
    t_1 = sqrt(((a * c) * (-4.0d0)))
    if (b <= (-4d-44)) then
        tmp_1 = t_0
    else if (b <= 5.4d-54) then
        if (b >= 0.0d0) then
            tmp_2 = (b + t_1) / ((-2.0d0) * a)
        else
            tmp_2 = c * ((-2.0d0) / (b - t_1))
        end if
        tmp_1 = tmp_2
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -1.0 * (b / a);
	} else {
		tmp = (2.0 * c) / (-1.0 * (b * 2.0));
	}
	double t_0 = tmp;
	double t_1 = Math.sqrt(((a * c) * -4.0));
	double tmp_1;
	if (b <= -4e-44) {
		tmp_1 = t_0;
	} else if (b <= 5.4e-54) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (b + t_1) / (-2.0 * a);
		} else {
			tmp_2 = c * (-2.0 / (b - t_1));
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = -1.0 * (b / a)
	else:
		tmp = (2.0 * c) / (-1.0 * (b * 2.0))
	t_0 = tmp
	t_1 = math.sqrt(((a * c) * -4.0))
	tmp_1 = 0
	if b <= -4e-44:
		tmp_1 = t_0
	elif b <= 5.4e-54:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (b + t_1) / (-2.0 * a)
		else:
			tmp_2 = c * (-2.0 / (b - t_1))
		tmp_1 = tmp_2
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(-1.0 * Float64(b / a));
	else
		tmp = Float64(Float64(2.0 * c) / Float64(-1.0 * Float64(b * 2.0)));
	end
	t_0 = tmp
	t_1 = sqrt(Float64(Float64(a * c) * -4.0))
	tmp_1 = 0.0
	if (b <= -4e-44)
		tmp_1 = t_0;
	elseif (b <= 5.4e-54)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(b + t_1) / Float64(-2.0 * a));
		else
			tmp_2 = Float64(c * Float64(-2.0 / Float64(b - t_1)));
		end
		tmp_1 = tmp_2;
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = -1.0 * (b / a);
	else
		tmp = (2.0 * c) / (-1.0 * (b * 2.0));
	end
	t_0 = tmp;
	t_1 = sqrt(((a * c) * -4.0));
	tmp_2 = 0.0;
	if (b <= -4e-44)
		tmp_2 = t_0;
	elseif (b <= 5.4e-54)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (b + t_1) / (-2.0 * a);
		else
			tmp_3 = c * (-2.0 / (b - t_1));
		end
		tmp_2 = tmp_3;
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = If[GreaterEqual[b, 0.0], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[(-1.0 * N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]}, Block[{t$95$1 = N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -4e-44], t$95$0, If[LessEqual[b, 5.4e-54], If[GreaterEqual[b, 0.0], N[(N[(b + t$95$1), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-2.0 / N[(b - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.9999999999999998e-44 or 5.4000000000000005e-54 < b

   1. Initial program 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 72.4%

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

   3. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 72.4%

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 67.9%

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

   8. Applied rewrites 67.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 69.7%

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

   12. Taylor expanded in b around inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 67.7%

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

   14. Applied rewrites 67.7%

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

   if -3.9999999999999998e-44 < b < 5.4000000000000005e-54

   1. Initial program 72.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 57.3%

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

   4. Applied rewrites 57.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 41.7%

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

   7. Applied rewrites 41.7%

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

   9. Applied rewrites 41.7%

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

       1. lift-/.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. count-2-rev N/A

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

       4. *-commutative N/A

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

       5. associate-/l* N/A

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

       6. lower-*.f64 N/A

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

       7. frac-2neg N/A

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

       8. metadata-eval N/A

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

       9. lower-/.f64 N/A

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

       10. lift--.f64 N/A

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

       11. sub-negate-rev N/A

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

       12. lower--.f64 41.6%

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

   11. Applied rewrites 41.6%

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

Alternative 8: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array}\\ t_1 := \sqrt{\left(a \cdot c\right) \cdot -4}\\ \mathbf{if}\;b \leq -4 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{-54}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(t\_1 + b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_1 - b}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (if (>= b 0.0) (* -1.0 (/ b a)) (/ (* 2.0 c) (* -1.0 (* b 2.0)))))
        (t_1 (sqrt (* (* a c) -4.0))))
   (if (<= b -4e-44)
     t_0
     (if (<= b 5.4e-54)
       (if (>= b 0.0) (* (/ -0.5 a) (+ t_1 b)) (/ (+ c c) (- t_1 b)))
       t_0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -1.0 * (b / a);
	} else {
		tmp = (2.0 * c) / (-1.0 * (b * 2.0));
	}
	double t_0 = tmp;
	double t_1 = sqrt(((a * c) * -4.0));
	double tmp_1;
	if (b <= -4e-44) {
		tmp_1 = t_0;
	} else if (b <= 5.4e-54) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-0.5 / a) * (t_1 + b);
		} else {
			tmp_2 = (c + c) / (t_1 - b);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    if (b >= 0.0d0) then
        tmp = (-1.0d0) * (b / a)
    else
        tmp = (2.0d0 * c) / ((-1.0d0) * (b * 2.0d0))
    end if
    t_0 = tmp
    t_1 = sqrt(((a * c) * (-4.0d0)))
    if (b <= (-4d-44)) then
        tmp_1 = t_0
    else if (b <= 5.4d-54) then
        if (b >= 0.0d0) then
            tmp_2 = ((-0.5d0) / a) * (t_1 + b)
        else
            tmp_2 = (c + c) / (t_1 - b)
        end if
        tmp_1 = tmp_2
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -1.0 * (b / a);
	} else {
		tmp = (2.0 * c) / (-1.0 * (b * 2.0));
	}
	double t_0 = tmp;
	double t_1 = Math.sqrt(((a * c) * -4.0));
	double tmp_1;
	if (b <= -4e-44) {
		tmp_1 = t_0;
	} else if (b <= 5.4e-54) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-0.5 / a) * (t_1 + b);
		} else {
			tmp_2 = (c + c) / (t_1 - b);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = -1.0 * (b / a)
	else:
		tmp = (2.0 * c) / (-1.0 * (b * 2.0))
	t_0 = tmp
	t_1 = math.sqrt(((a * c) * -4.0))
	tmp_1 = 0
	if b <= -4e-44:
		tmp_1 = t_0
	elif b <= 5.4e-54:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (-0.5 / a) * (t_1 + b)
		else:
			tmp_2 = (c + c) / (t_1 - b)
		tmp_1 = tmp_2
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(-1.0 * Float64(b / a));
	else
		tmp = Float64(Float64(2.0 * c) / Float64(-1.0 * Float64(b * 2.0)));
	end
	t_0 = tmp
	t_1 = sqrt(Float64(Float64(a * c) * -4.0))
	tmp_1 = 0.0
	if (b <= -4e-44)
		tmp_1 = t_0;
	elseif (b <= 5.4e-54)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-0.5 / a) * Float64(t_1 + b));
		else
			tmp_2 = Float64(Float64(c + c) / Float64(t_1 - b));
		end
		tmp_1 = tmp_2;
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = -1.0 * (b / a);
	else
		tmp = (2.0 * c) / (-1.0 * (b * 2.0));
	end
	t_0 = tmp;
	t_1 = sqrt(((a * c) * -4.0));
	tmp_2 = 0.0;
	if (b <= -4e-44)
		tmp_2 = t_0;
	elseif (b <= 5.4e-54)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (-0.5 / a) * (t_1 + b);
		else
			tmp_3 = (c + c) / (t_1 - b);
		end
		tmp_2 = tmp_3;
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = If[GreaterEqual[b, 0.0], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[(-1.0 * N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]}, Block[{t$95$1 = N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -4e-44], t$95$0, If[LessEqual[b, 5.4e-54], If[GreaterEqual[b, 0.0], N[(N[(-0.5 / a), $MachinePrecision] * N[(t$95$1 + b), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(t$95$1 - b), $MachinePrecision]), $MachinePrecision]], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.9999999999999998e-44 or 5.4000000000000005e-54 < b

   1. Initial program 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 72.4%

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

   3. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 72.4%

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 67.9%

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

   8. Applied rewrites 67.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 69.7%

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

   12. Taylor expanded in b around inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 67.7%

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

   14. Applied rewrites 67.7%

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

   if -3.9999999999999998e-44 < b < 5.4000000000000005e-54

   1. Initial program 72.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 57.3%

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

   4. Applied rewrites 57.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 41.7%

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

   7. Applied rewrites 41.7%

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

   9. Applied rewrites 41.7%

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

       1. lift-/.f64 N/A

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

       2. mult-flip N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-/r* N/A

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

       7. lower-/.f64 N/A

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

       8. metadata-eval 41.6%

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

       9. lift-+.f64 N/A

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

       10. +-commutative N/A

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

       11. lower-+.f64 41.6%

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

   11. Applied rewrites 41.6%

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

Alternative 9: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array}\\ \mathbf{if}\;b \leq -4 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-141}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(a \cdot c\right) \cdot -4} - b}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0
         (if (>= b 0.0) (* -1.0 (/ b a)) (/ (* 2.0 c) (* -1.0 (* b 2.0))))))
   (if (<= b -4e-44)
     t_0
     (if (<= b 3.4e-141)
       (if (>= b 0.0)
         (/ (sqrt (- (* 4.0 (* a c)))) (* -2.0 a))
         (/ (+ c c) (- (sqrt (* (* a c) -4.0)) b)))
       t_0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -1.0 * (b / a);
	} else {
		tmp = (2.0 * c) / (-1.0 * (b * 2.0));
	}
	double t_0 = tmp;
	double tmp_1;
	if (b <= -4e-44) {
		tmp_1 = t_0;
	} else if (b <= 3.4e-141) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = sqrt(-(4.0 * (a * c))) / (-2.0 * a);
		} else {
			tmp_2 = (c + c) / (sqrt(((a * c) * -4.0)) - b);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    if (b >= 0.0d0) then
        tmp = (-1.0d0) * (b / a)
    else
        tmp = (2.0d0 * c) / ((-1.0d0) * (b * 2.0d0))
    end if
    t_0 = tmp
    if (b <= (-4d-44)) then
        tmp_1 = t_0
    else if (b <= 3.4d-141) then
        if (b >= 0.0d0) then
            tmp_2 = sqrt(-(4.0d0 * (a * c))) / ((-2.0d0) * a)
        else
            tmp_2 = (c + c) / (sqrt(((a * c) * (-4.0d0))) - b)
        end if
        tmp_1 = tmp_2
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -1.0 * (b / a);
	} else {
		tmp = (2.0 * c) / (-1.0 * (b * 2.0));
	}
	double t_0 = tmp;
	double tmp_1;
	if (b <= -4e-44) {
		tmp_1 = t_0;
	} else if (b <= 3.4e-141) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = Math.sqrt(-(4.0 * (a * c))) / (-2.0 * a);
		} else {
			tmp_2 = (c + c) / (Math.sqrt(((a * c) * -4.0)) - b);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = -1.0 * (b / a)
	else:
		tmp = (2.0 * c) / (-1.0 * (b * 2.0))
	t_0 = tmp
	tmp_1 = 0
	if b <= -4e-44:
		tmp_1 = t_0
	elif b <= 3.4e-141:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = math.sqrt(-(4.0 * (a * c))) / (-2.0 * a)
		else:
			tmp_2 = (c + c) / (math.sqrt(((a * c) * -4.0)) - b)
		tmp_1 = tmp_2
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(-1.0 * Float64(b / a));
	else
		tmp = Float64(Float64(2.0 * c) / Float64(-1.0 * Float64(b * 2.0)));
	end
	t_0 = tmp
	tmp_1 = 0.0
	if (b <= -4e-44)
		tmp_1 = t_0;
	elseif (b <= 3.4e-141)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(sqrt(Float64(-Float64(4.0 * Float64(a * c)))) / Float64(-2.0 * a));
		else
			tmp_2 = Float64(Float64(c + c) / Float64(sqrt(Float64(Float64(a * c) * -4.0)) - b));
		end
		tmp_1 = tmp_2;
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = -1.0 * (b / a);
	else
		tmp = (2.0 * c) / (-1.0 * (b * 2.0));
	end
	t_0 = tmp;
	tmp_2 = 0.0;
	if (b <= -4e-44)
		tmp_2 = t_0;
	elseif (b <= 3.4e-141)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = sqrt(-(4.0 * (a * c))) / (-2.0 * a);
		else
			tmp_3 = (c + c) / (sqrt(((a * c) * -4.0)) - b);
		end
		tmp_2 = tmp_3;
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = If[GreaterEqual[b, 0.0], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[(-1.0 * N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]}, If[LessEqual[b, -4e-44], t$95$0, If[LessEqual[b, 3.4e-141], If[GreaterEqual[b, 0.0], N[(N[Sqrt[(-N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.9999999999999998e-44 or 3.3999999999999998e-141 < b

   1. Initial program 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 72.4%

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

   3. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 72.4%

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 67.9%

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

   8. Applied rewrites 67.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 69.7%

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

   12. Taylor expanded in b around inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 67.7%

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

   14. Applied rewrites 67.7%

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

   if -3.9999999999999998e-44 < b < 3.3999999999999998e-141

   1. Initial program 72.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 57.3%

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

   4. Applied rewrites 57.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 41.7%

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

   7. Applied rewrites 41.7%

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

   9. Applied rewrites 41.7%

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

   10. Taylor expanded in b around 0

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

       1. lower-sqrt.f64 N/A

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

       2. lower-neg.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 34.1%

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

   12. Applied rewrites 34.1%

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

Alternative 10: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array}\\ t_1 := \sqrt{\left(a \cdot c\right) \cdot -4}\\ \mathbf{if}\;b \leq -4 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-141}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_1 - b}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (if (>= b 0.0) (* -1.0 (/ b a)) (/ (* 2.0 c) (* -1.0 (* b 2.0)))))
        (t_1 (sqrt (* (* a c) -4.0))))
   (if (<= b -4e-44)
     t_0
     (if (<= b 3.4e-141)
       (if (>= b 0.0) (* (/ -0.5 a) t_1) (/ (+ c c) (- t_1 b)))
       t_0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -1.0 * (b / a);
	} else {
		tmp = (2.0 * c) / (-1.0 * (b * 2.0));
	}
	double t_0 = tmp;
	double t_1 = sqrt(((a * c) * -4.0));
	double tmp_1;
	if (b <= -4e-44) {
		tmp_1 = t_0;
	} else if (b <= 3.4e-141) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-0.5 / a) * t_1;
		} else {
			tmp_2 = (c + c) / (t_1 - b);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    if (b >= 0.0d0) then
        tmp = (-1.0d0) * (b / a)
    else
        tmp = (2.0d0 * c) / ((-1.0d0) * (b * 2.0d0))
    end if
    t_0 = tmp
    t_1 = sqrt(((a * c) * (-4.0d0)))
    if (b <= (-4d-44)) then
        tmp_1 = t_0
    else if (b <= 3.4d-141) then
        if (b >= 0.0d0) then
            tmp_2 = ((-0.5d0) / a) * t_1
        else
            tmp_2 = (c + c) / (t_1 - b)
        end if
        tmp_1 = tmp_2
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -1.0 * (b / a);
	} else {
		tmp = (2.0 * c) / (-1.0 * (b * 2.0));
	}
	double t_0 = tmp;
	double t_1 = Math.sqrt(((a * c) * -4.0));
	double tmp_1;
	if (b <= -4e-44) {
		tmp_1 = t_0;
	} else if (b <= 3.4e-141) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-0.5 / a) * t_1;
		} else {
			tmp_2 = (c + c) / (t_1 - b);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = -1.0 * (b / a)
	else:
		tmp = (2.0 * c) / (-1.0 * (b * 2.0))
	t_0 = tmp
	t_1 = math.sqrt(((a * c) * -4.0))
	tmp_1 = 0
	if b <= -4e-44:
		tmp_1 = t_0
	elif b <= 3.4e-141:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (-0.5 / a) * t_1
		else:
			tmp_2 = (c + c) / (t_1 - b)
		tmp_1 = tmp_2
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(-1.0 * Float64(b / a));
	else
		tmp = Float64(Float64(2.0 * c) / Float64(-1.0 * Float64(b * 2.0)));
	end
	t_0 = tmp
	t_1 = sqrt(Float64(Float64(a * c) * -4.0))
	tmp_1 = 0.0
	if (b <= -4e-44)
		tmp_1 = t_0;
	elseif (b <= 3.4e-141)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-0.5 / a) * t_1);
		else
			tmp_2 = Float64(Float64(c + c) / Float64(t_1 - b));
		end
		tmp_1 = tmp_2;
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = -1.0 * (b / a);
	else
		tmp = (2.0 * c) / (-1.0 * (b * 2.0));
	end
	t_0 = tmp;
	t_1 = sqrt(((a * c) * -4.0));
	tmp_2 = 0.0;
	if (b <= -4e-44)
		tmp_2 = t_0;
	elseif (b <= 3.4e-141)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (-0.5 / a) * t_1;
		else
			tmp_3 = (c + c) / (t_1 - b);
		end
		tmp_2 = tmp_3;
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = If[GreaterEqual[b, 0.0], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[(-1.0 * N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]}, Block[{t$95$1 = N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -4e-44], t$95$0, If[LessEqual[b, 3.4e-141], If[GreaterEqual[b, 0.0], N[(N[(-0.5 / a), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(t$95$1 - b), $MachinePrecision]), $MachinePrecision]], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.9999999999999998e-44 or 3.3999999999999998e-141 < b

   1. Initial program 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 72.4%

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

   3. Applied rewrites 72.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 72.4%

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

   5. Applied rewrites 72.4%

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

   6. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 67.9%

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

   8. Applied rewrites 67.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 69.7%

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

   12. Taylor expanded in b around inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 67.7%

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

   14. Applied rewrites 67.7%

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

   if -3.9999999999999998e-44 < b < 3.3999999999999998e-141

   1. Initial program 72.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 57.3%

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

   4. Applied rewrites 57.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 41.7%

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

   7. Applied rewrites 41.7%

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

   9. Applied rewrites 41.7%

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

   10. Taylor expanded in b around 0

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

       1. lower-sqrt.f64 N/A

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

       2. lower-neg.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 34.1%

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

   12. Applied rewrites 34.1%

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

       1. lift-/.f64 N/A

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

       2. mult-flip N/A

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

       3. *-commutative N/A

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

       4. lift-*.f64 N/A

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

       5. associate-/r* N/A

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

       6. metadata-eval N/A

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

       7. lift-/.f64 N/A

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

       8. lower-*.f64 34.1%

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

       9. lift-neg.f64 N/A

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

       10. lift-*.f64 N/A

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

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

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

       12. metadata-eval N/A

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

       13. *-commutative N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{-1}{2}}{a} \cdot \sqrt{\left(a \cdot c\right) \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(a \cdot c\right) \cdot -4} - b}\\ \end{array} \]

       14. lift-*.f64 34.1%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \sqrt{\left(a \cdot c\right) \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(a \cdot c\right) \cdot -4} - b}\\ \end{array} \]

   14. Applied rewrites 34.1%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-0.5}{a} \cdot \sqrt{\left(a \cdot c\right) \cdot -4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(a \cdot c\right) \cdot -4} - b}\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array}\\ t_1 := \sqrt{-4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b \leq -4 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-141}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{t\_1}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{t\_1}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (if (>= b 0.0) (* -1.0 (/ b a)) (/ (* 2.0 c) (* -1.0 (* b 2.0)))))
        (t_1 (sqrt (- (* 4.0 (* a c))))))
   (if (<= b -4e-44)
     t_0
     (if (<= b 3.4e-141)
       (if (>= b 0.0) (/ t_1 (* -2.0 a)) (* 2.0 (/ c t_1)))
       t_0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -1.0 * (b / a);
	} else {
		tmp = (2.0 * c) / (-1.0 * (b * 2.0));
	}
	double t_0 = tmp;
	double t_1 = sqrt(-(4.0 * (a * c)));
	double tmp_1;
	if (b <= -4e-44) {
		tmp_1 = t_0;
	} else if (b <= 3.4e-141) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1 / (-2.0 * a);
		} else {
			tmp_2 = 2.0 * (c / t_1);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    if (b >= 0.0d0) then
        tmp = (-1.0d0) * (b / a)
    else
        tmp = (2.0d0 * c) / ((-1.0d0) * (b * 2.0d0))
    end if
    t_0 = tmp
    t_1 = sqrt(-(4.0d0 * (a * c)))
    if (b <= (-4d-44)) then
        tmp_1 = t_0
    else if (b <= 3.4d-141) then
        if (b >= 0.0d0) then
            tmp_2 = t_1 / ((-2.0d0) * a)
        else
            tmp_2 = 2.0d0 * (c / t_1)
        end if
        tmp_1 = tmp_2
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -1.0 * (b / a);
	} else {
		tmp = (2.0 * c) / (-1.0 * (b * 2.0));
	}
	double t_0 = tmp;
	double t_1 = Math.sqrt(-(4.0 * (a * c)));
	double tmp_1;
	if (b <= -4e-44) {
		tmp_1 = t_0;
	} else if (b <= 3.4e-141) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1 / (-2.0 * a);
		} else {
			tmp_2 = 2.0 * (c / t_1);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = -1.0 * (b / a)
	else:
		tmp = (2.0 * c) / (-1.0 * (b * 2.0))
	t_0 = tmp
	t_1 = math.sqrt(-(4.0 * (a * c)))
	tmp_1 = 0
	if b <= -4e-44:
		tmp_1 = t_0
	elif b <= 3.4e-141:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_1 / (-2.0 * a)
		else:
			tmp_2 = 2.0 * (c / t_1)
		tmp_1 = tmp_2
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(-1.0 * Float64(b / a));
	else
		tmp = Float64(Float64(2.0 * c) / Float64(-1.0 * Float64(b * 2.0)));
	end
	t_0 = tmp
	t_1 = sqrt(Float64(-Float64(4.0 * Float64(a * c))))
	tmp_1 = 0.0
	if (b <= -4e-44)
		tmp_1 = t_0;
	elseif (b <= 3.4e-141)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(t_1 / Float64(-2.0 * a));
		else
			tmp_2 = Float64(2.0 * Float64(c / t_1));
		end
		tmp_1 = tmp_2;
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = -1.0 * (b / a);
	else
		tmp = (2.0 * c) / (-1.0 * (b * 2.0));
	end
	t_0 = tmp;
	t_1 = sqrt(-(4.0 * (a * c)));
	tmp_2 = 0.0;
	if (b <= -4e-44)
		tmp_2 = t_0;
	elseif (b <= 3.4e-141)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_1 / (-2.0 * a);
		else
			tmp_3 = 2.0 * (c / t_1);
		end
		tmp_2 = tmp_3;
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = If[GreaterEqual[b, 0.0], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[(-1.0 * N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]}, Block[{t$95$1 = N[Sqrt[(-N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]}, If[LessEqual[b, -4e-44], t$95$0, If[LessEqual[b, 3.4e-141], If[GreaterEqual[b, 0.0], N[(t$95$1 / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(c / t$95$1), $MachinePrecision]), $MachinePrecision]], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.9999999999999998e-44 or 3.3999999999999998e-141 < b

   1. Initial program 72.4%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      2. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \color{blue}{\left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      4. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c + b \cdot b}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      5. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\left(\mathsf{neg}\left(\color{blue}{4 \cdot a}\right)\right) \cdot c + b \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot a\right)} \cdot c + b \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      7. associate-*l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} + b \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      8. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(\mathsf{neg}\left(4\right)\right)} + b \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      9. lower-fma.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, \mathsf{neg}\left(4\right), b \cdot b\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      10. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\color{blue}{c \cdot a}, \mathsf{neg}\left(4\right), b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      11. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\color{blue}{c \cdot a}, \mathsf{neg}\left(4\right), b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      12. metadata-eval 72.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, \color{blue}{-4}, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Applied rewrites 72.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      2. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{b \cdot b + \left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}}\\ \end{array} \]

      4. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{\left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c + b \cdot b}}\\ \end{array} \]

      5. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{\left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c + b \cdot b}}\\ \end{array} \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot a\right) \cdot c + b \cdot b}}\\ \end{array} \]

      7. associate-*l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-\color{blue}{b}\right) + \sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right) + b \cdot b}}\\ \end{array} \]

      8. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-\color{blue}{b}\right) + \sqrt{\left(a \cdot c\right) \cdot \left(\mathsf{neg}\left(4\right)\right) + b \cdot b}}\\ \end{array} \]

      9. lower-fma.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{\mathsf{fma}\left(a \cdot c, \mathsf{neg}\left(4\right), b \cdot b\right)}}\\ \end{array} \]

      10. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-\color{blue}{b}\right) + \sqrt{\mathsf{fma}\left(c \cdot a, \mathsf{neg}\left(4\right), b \cdot b\right)}}\\ \end{array} \]

      11. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-\color{blue}{b}\right) + \sqrt{\mathsf{fma}\left(c \cdot a, \mathsf{neg}\left(4\right), b \cdot b\right)}}\\ \end{array} \]

      12. metadata-eval 72.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}\\ \end{array} \]

   5. Applied rewrites 72.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}\\ \end{array} \]

   6. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}\\ \end{array} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}\\ \end{array} \]

      2. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \color{blue}{\left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}\\ \end{array} \]

      3. lower-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \color{blue}{\left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)}\right)}\\ \end{array} \]

      4. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + \color{blue}{-2 \cdot \frac{a \cdot c}{{b}^{2}}}\right)\right)}\\ \end{array} \]

      5. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \color{blue}{\frac{a \cdot c}{{b}^{2}}}\right)\right)}\\ \end{array} \]

      6. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{\color{blue}{a \cdot c}}{{b}^{2}}\right)\right)}\\ \end{array} \]

      7. lower-pow.f64 67.9%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{\color{blue}{{b}^{2}}}\right)\right)}\\ \end{array} \]

   8. Applied rewrites 67.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}\\ \end{array} \]

   9. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \color{blue}{2}\right)}\\ \end{array} \]
   10. Step-by-step derivation

   11. Applied rewrites 69.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \color{blue}{2}\right)}\\ \end{array} \]

   12. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \color{blue}{\frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array} \]

       2. lower-/.f64 67.7%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{b}{\color{blue}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array} \]

   14. Applied rewrites 67.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array} \]

   if -3.9999999999999998e-44 < b < 3.3999999999999998e-141

   1. Initial program 72.4%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      2. lower-*.f64 57.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   4. Applied rewrites 57.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]

      2. lower-*.f64 41.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]

   7. Applied rewrites 41.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 41.7%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(a \cdot c\right) \cdot -4}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(a \cdot c\right) \cdot -4} - b}\\ } \end{array}} \]

   10. Taylor expanded in b around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(a \cdot c\right) \cdot -4} - b}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(a \cdot c\right) \cdot -4} - b}\\ \end{array} \]

       2. lower-neg.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(a \cdot c\right) \cdot -4} - b}\\ \end{array} \]

       3. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(a \cdot c\right) \cdot -4} - b}\\ \end{array} \]

       4. lower-*.f64 34.1%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(a \cdot c\right) \cdot -4} - b}\\ \end{array} \]

   12. Applied rewrites 34.1%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right)}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(a \cdot c\right) \cdot -4} - b}\\ \end{array} \]

   13. Taylor expanded in b around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}\\ \end{array} \]
   14. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{2 \cdot \frac{c}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}\\ \end{array} \]

       2. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \color{blue}{\frac{c}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}\\ \end{array} \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}\\ \end{array} \]

       4. lower-neg.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \end{array} \]

       5. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-\color{blue}{4 \cdot \left(a \cdot c\right)}}}\\ \end{array} \]

       6. lower-*.f64 26.6%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \end{array} \]

   15. Applied rewrites 26.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 72.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \mathbf{if}\;b \leq 3.4 \cdot 10^{-141}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* 2.0 c) (* -1.0 (* b 2.0)))))
   (if (<= b 3.4e-141)
     (if (>= b 0.0) (* -0.5 (/ (sqrt (* -4.0 (* a c))) a)) t_0)
     (if (>= b 0.0) (* -1.0 (/ b a)) t_0))))

C

double code(double a, double b, double c) {
	double t_0 = (2.0 * c) / (-1.0 * (b * 2.0));
	double tmp_1;
	if (b <= 3.4e-141) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -0.5 * (sqrt((-4.0 * (a * c))) / a);
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = -1.0 * (b / 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 = (2.0d0 * c) / ((-1.0d0) * (b * 2.0d0))
    if (b <= 3.4d-141) then
        if (b >= 0.0d0) then
            tmp_2 = (-0.5d0) * (sqrt(((-4.0d0) * (a * c))) / a)
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (-1.0d0) * (b / 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 = (2.0 * c) / (-1.0 * (b * 2.0));
	double tmp_1;
	if (b <= 3.4e-141) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -0.5 * (Math.sqrt((-4.0 * (a * c))) / a);
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = -1.0 * (b / a);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (2.0 * c) / (-1.0 * (b * 2.0))
	tmp_1 = 0
	if b <= 3.4e-141:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -0.5 * (math.sqrt((-4.0 * (a * c))) / a)
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = -1.0 * (b / a)
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(2.0 * c) / Float64(-1.0 * Float64(b * 2.0)))
	tmp_1 = 0.0
	if (b <= 3.4e-141)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-0.5 * Float64(sqrt(Float64(-4.0 * Float64(a * c))) / a));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(-1.0 * Float64(b / a));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = (2.0 * c) / (-1.0 * (b * 2.0));
	tmp_2 = 0.0;
	if (b <= 3.4e-141)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -0.5 * (sqrt((-4.0 * (a * c))) / a);
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = -1.0 * (b / a);
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(2.0 * c), $MachinePrecision] / N[(-1.0 * N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 3.4e-141], If[GreaterEqual[b, 0.0], N[(-0.5 * N[(N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.3999999999999998e-141

   1. Initial program 72.4%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      2. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \color{blue}{\left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      4. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c + b \cdot b}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      5. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\left(\mathsf{neg}\left(\color{blue}{4 \cdot a}\right)\right) \cdot c + b \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot a\right)} \cdot c + b \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      7. associate-*l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} + b \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      8. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(\mathsf{neg}\left(4\right)\right)} + b \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      9. lower-fma.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, \mathsf{neg}\left(4\right), b \cdot b\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      10. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\color{blue}{c \cdot a}, \mathsf{neg}\left(4\right), b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      11. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\color{blue}{c \cdot a}, \mathsf{neg}\left(4\right), b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      12. metadata-eval 72.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, \color{blue}{-4}, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Applied rewrites 72.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      2. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{b \cdot b + \left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}}\\ \end{array} \]

      4. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{\left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c + b \cdot b}}\\ \end{array} \]

      5. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{\left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c + b \cdot b}}\\ \end{array} \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot a\right) \cdot c + b \cdot b}}\\ \end{array} \]

      7. associate-*l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-\color{blue}{b}\right) + \sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right) + b \cdot b}}\\ \end{array} \]

      8. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-\color{blue}{b}\right) + \sqrt{\left(a \cdot c\right) \cdot \left(\mathsf{neg}\left(4\right)\right) + b \cdot b}}\\ \end{array} \]

      9. lower-fma.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{\mathsf{fma}\left(a \cdot c, \mathsf{neg}\left(4\right), b \cdot b\right)}}\\ \end{array} \]

      10. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-\color{blue}{b}\right) + \sqrt{\mathsf{fma}\left(c \cdot a, \mathsf{neg}\left(4\right), b \cdot b\right)}}\\ \end{array} \]

      11. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-\color{blue}{b}\right) + \sqrt{\mathsf{fma}\left(c \cdot a, \mathsf{neg}\left(4\right), b \cdot b\right)}}\\ \end{array} \]

      12. metadata-eval 72.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}\\ \end{array} \]

   5. Applied rewrites 72.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}\\ \end{array} \]

   6. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}\\ \end{array} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}\\ \end{array} \]

      2. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \color{blue}{\left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}\\ \end{array} \]

      3. lower-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \color{blue}{\left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)}\right)}\\ \end{array} \]

      4. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + \color{blue}{-2 \cdot \frac{a \cdot c}{{b}^{2}}}\right)\right)}\\ \end{array} \]

      5. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \color{blue}{\frac{a \cdot c}{{b}^{2}}}\right)\right)}\\ \end{array} \]

      6. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{\color{blue}{a \cdot c}}{{b}^{2}}\right)\right)}\\ \end{array} \]

      7. lower-pow.f64 67.9%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{\color{blue}{{b}^{2}}}\right)\right)}\\ \end{array} \]

   8. Applied rewrites 67.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}\\ \end{array} \]

   9. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \color{blue}{2}\right)}\\ \end{array} \]
   10. Step-by-step derivation

   11. Applied rewrites 69.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \color{blue}{2}\right)}\\ \end{array} \]

   12. Taylor expanded in b around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-1}{2} \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \color{blue}{\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array} \]

       2. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{\color{blue}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array} \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array} \]

       4. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array} \]

       5. lower-*.f64 47.1%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array} \]

   14. Applied rewrites 47.1%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-0.5 \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array} \]

   if 3.3999999999999998e-141 < b

   1. Initial program 72.4%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      2. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \color{blue}{\left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      4. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c + b \cdot b}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      5. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\left(\mathsf{neg}\left(\color{blue}{4 \cdot a}\right)\right) \cdot c + b \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot a\right)} \cdot c + b \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      7. associate-*l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} + b \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      8. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(\mathsf{neg}\left(4\right)\right)} + b \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      9. lower-fma.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, \mathsf{neg}\left(4\right), b \cdot b\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      10. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\color{blue}{c \cdot a}, \mathsf{neg}\left(4\right), b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      11. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\color{blue}{c \cdot a}, \mathsf{neg}\left(4\right), b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      12. metadata-eval 72.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, \color{blue}{-4}, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Applied rewrites 72.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      2. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{b \cdot b + \left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}}\\ \end{array} \]

      4. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{\left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c + b \cdot b}}\\ \end{array} \]

      5. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{\left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c + b \cdot b}}\\ \end{array} \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot a\right) \cdot c + b \cdot b}}\\ \end{array} \]

      7. associate-*l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-\color{blue}{b}\right) + \sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right) + b \cdot b}}\\ \end{array} \]

      8. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-\color{blue}{b}\right) + \sqrt{\left(a \cdot c\right) \cdot \left(\mathsf{neg}\left(4\right)\right) + b \cdot b}}\\ \end{array} \]

      9. lower-fma.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{\mathsf{fma}\left(a \cdot c, \mathsf{neg}\left(4\right), b \cdot b\right)}}\\ \end{array} \]

      10. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-\color{blue}{b}\right) + \sqrt{\mathsf{fma}\left(c \cdot a, \mathsf{neg}\left(4\right), b \cdot b\right)}}\\ \end{array} \]

      11. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-\color{blue}{b}\right) + \sqrt{\mathsf{fma}\left(c \cdot a, \mathsf{neg}\left(4\right), b \cdot b\right)}}\\ \end{array} \]

      12. metadata-eval 72.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}\\ \end{array} \]

   5. Applied rewrites 72.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}\\ \end{array} \]

   6. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}\\ \end{array} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}\\ \end{array} \]

      2. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \color{blue}{\left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}\\ \end{array} \]

      3. lower-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \color{blue}{\left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)}\right)}\\ \end{array} \]

      4. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + \color{blue}{-2 \cdot \frac{a \cdot c}{{b}^{2}}}\right)\right)}\\ \end{array} \]

      5. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \color{blue}{\frac{a \cdot c}{{b}^{2}}}\right)\right)}\\ \end{array} \]

      6. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{\color{blue}{a \cdot c}}{{b}^{2}}\right)\right)}\\ \end{array} \]

      7. lower-pow.f64 67.9%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{\color{blue}{{b}^{2}}}\right)\right)}\\ \end{array} \]

   8. Applied rewrites 67.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}\\ \end{array} \]

   9. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \color{blue}{2}\right)}\\ \end{array} \]
   10. Step-by-step derivation

   11. Applied rewrites 69.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \color{blue}{2}\right)}\\ \end{array} \]

   12. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \color{blue}{\frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array} \]

       2. lower-/.f64 67.7%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{b}{\color{blue}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array} \]

   14. Applied rewrites 67.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 69.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \mathbf{if}\;b \leq 1.2 \cdot 10^{-175}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* 2.0 c) (* -1.0 (* b 2.0)))))
   (if (<= b 1.2e-175)
     (if (>= b 0.0) (* -0.5 (sqrt (* -4.0 (/ c a)))) t_0)
     (if (>= b 0.0) (* -1.0 (/ b a)) t_0))))

C

double code(double a, double b, double c) {
	double t_0 = (2.0 * c) / (-1.0 * (b * 2.0));
	double tmp_1;
	if (b <= 1.2e-175) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -0.5 * sqrt((-4.0 * (c / a)));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = -1.0 * (b / 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 = (2.0d0 * c) / ((-1.0d0) * (b * 2.0d0))
    if (b <= 1.2d-175) then
        if (b >= 0.0d0) then
            tmp_2 = (-0.5d0) * sqrt(((-4.0d0) * (c / a)))
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (-1.0d0) * (b / 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 = (2.0 * c) / (-1.0 * (b * 2.0));
	double tmp_1;
	if (b <= 1.2e-175) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -0.5 * Math.sqrt((-4.0 * (c / a)));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = -1.0 * (b / a);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (2.0 * c) / (-1.0 * (b * 2.0))
	tmp_1 = 0
	if b <= 1.2e-175:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -0.5 * math.sqrt((-4.0 * (c / a)))
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = -1.0 * (b / a)
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(2.0 * c) / Float64(-1.0 * Float64(b * 2.0)))
	tmp_1 = 0.0
	if (b <= 1.2e-175)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-0.5 * sqrt(Float64(-4.0 * Float64(c / a))));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(-1.0 * Float64(b / a));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = (2.0 * c) / (-1.0 * (b * 2.0));
	tmp_2 = 0.0;
	if (b <= 1.2e-175)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -0.5 * sqrt((-4.0 * (c / a)));
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = -1.0 * (b / a);
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(2.0 * c), $MachinePrecision] / N[(-1.0 * N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.2e-175], If[GreaterEqual[b, 0.0], N[(-0.5 * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.2e-175

   1. Initial program 72.4%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      2. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \color{blue}{\left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      4. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c + b \cdot b}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      5. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\left(\mathsf{neg}\left(\color{blue}{4 \cdot a}\right)\right) \cdot c + b \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot a\right)} \cdot c + b \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      7. associate-*l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} + b \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      8. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(\mathsf{neg}\left(4\right)\right)} + b \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      9. lower-fma.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, \mathsf{neg}\left(4\right), b \cdot b\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      10. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\color{blue}{c \cdot a}, \mathsf{neg}\left(4\right), b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      11. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\color{blue}{c \cdot a}, \mathsf{neg}\left(4\right), b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      12. metadata-eval 72.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, \color{blue}{-4}, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Applied rewrites 72.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      2. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{b \cdot b + \left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}}\\ \end{array} \]

      4. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{\left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c + b \cdot b}}\\ \end{array} \]

      5. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{\left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c + b \cdot b}}\\ \end{array} \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot a\right) \cdot c + b \cdot b}}\\ \end{array} \]

      7. associate-*l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-\color{blue}{b}\right) + \sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right) + b \cdot b}}\\ \end{array} \]

      8. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-\color{blue}{b}\right) + \sqrt{\left(a \cdot c\right) \cdot \left(\mathsf{neg}\left(4\right)\right) + b \cdot b}}\\ \end{array} \]

      9. lower-fma.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{\mathsf{fma}\left(a \cdot c, \mathsf{neg}\left(4\right), b \cdot b\right)}}\\ \end{array} \]

      10. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-\color{blue}{b}\right) + \sqrt{\mathsf{fma}\left(c \cdot a, \mathsf{neg}\left(4\right), b \cdot b\right)}}\\ \end{array} \]

      11. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-\color{blue}{b}\right) + \sqrt{\mathsf{fma}\left(c \cdot a, \mathsf{neg}\left(4\right), b \cdot b\right)}}\\ \end{array} \]

      12. metadata-eval 72.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}\\ \end{array} \]

   5. Applied rewrites 72.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}\\ \end{array} \]

   6. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}\\ \end{array} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}\\ \end{array} \]

      2. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \color{blue}{\left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}\\ \end{array} \]

      3. lower-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \color{blue}{\left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)}\right)}\\ \end{array} \]

      4. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + \color{blue}{-2 \cdot \frac{a \cdot c}{{b}^{2}}}\right)\right)}\\ \end{array} \]

      5. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \color{blue}{\frac{a \cdot c}{{b}^{2}}}\right)\right)}\\ \end{array} \]

      6. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{\color{blue}{a \cdot c}}{{b}^{2}}\right)\right)}\\ \end{array} \]

      7. lower-pow.f64 67.9%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{\color{blue}{{b}^{2}}}\right)\right)}\\ \end{array} \]

   8. Applied rewrites 67.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}\\ \end{array} \]

   9. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \color{blue}{2}\right)}\\ \end{array} \]
   10. Step-by-step derivation

   11. Applied rewrites 69.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \color{blue}{2}\right)}\\ \end{array} \]

   12. Taylor expanded in a around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \color{blue}{\sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array} \]

       3. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array} \]

       4. lower-/.f64 41.6%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array} \]

   14. Applied rewrites 41.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array} \]

   if 1.2e-175 < b

   1. Initial program 72.4%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      2. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \color{blue}{\left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      4. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c + b \cdot b}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      5. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\left(\mathsf{neg}\left(\color{blue}{4 \cdot a}\right)\right) \cdot c + b \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot a\right)} \cdot c + b \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      7. associate-*l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} + b \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      8. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(\mathsf{neg}\left(4\right)\right)} + b \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      9. lower-fma.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, \mathsf{neg}\left(4\right), b \cdot b\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      10. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\color{blue}{c \cdot a}, \mathsf{neg}\left(4\right), b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      11. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\color{blue}{c \cdot a}, \mathsf{neg}\left(4\right), b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      12. metadata-eval 72.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, \color{blue}{-4}, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Applied rewrites 72.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      2. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{b \cdot b + \left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}}\\ \end{array} \]

      4. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{\left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c + b \cdot b}}\\ \end{array} \]

      5. lift-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{\left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c + b \cdot b}}\\ \end{array} \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot a\right) \cdot c + b \cdot b}}\\ \end{array} \]

      7. associate-*l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-\color{blue}{b}\right) + \sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right) + b \cdot b}}\\ \end{array} \]

      8. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-\color{blue}{b}\right) + \sqrt{\left(a \cdot c\right) \cdot \left(\mathsf{neg}\left(4\right)\right) + b \cdot b}}\\ \end{array} \]

      9. lower-fma.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{\mathsf{fma}\left(a \cdot c, \mathsf{neg}\left(4\right), b \cdot b\right)}}\\ \end{array} \]

      10. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-\color{blue}{b}\right) + \sqrt{\mathsf{fma}\left(c \cdot a, \mathsf{neg}\left(4\right), b \cdot b\right)}}\\ \end{array} \]

      11. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-\color{blue}{b}\right) + \sqrt{\mathsf{fma}\left(c \cdot a, \mathsf{neg}\left(4\right), b \cdot b\right)}}\\ \end{array} \]

      12. metadata-eval 72.4%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}\\ \end{array} \]

   5. Applied rewrites 72.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}\\ \end{array} \]

   6. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}\\ \end{array} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}\\ \end{array} \]

      2. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \color{blue}{\left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}\\ \end{array} \]

      3. lower-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \color{blue}{\left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)}\right)}\\ \end{array} \]

      4. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + \color{blue}{-2 \cdot \frac{a \cdot c}{{b}^{2}}}\right)\right)}\\ \end{array} \]

      5. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \color{blue}{\frac{a \cdot c}{{b}^{2}}}\right)\right)}\\ \end{array} \]

      6. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{\color{blue}{a \cdot c}}{{b}^{2}}\right)\right)}\\ \end{array} \]

      7. lower-pow.f64 67.9%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{\color{blue}{{b}^{2}}}\right)\right)}\\ \end{array} \]

   8. Applied rewrites 67.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}\\ \end{array} \]

   9. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \color{blue}{2}\right)}\\ \end{array} \]
   10. Step-by-step derivation

   11. Applied rewrites 69.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \color{blue}{2}\right)}\\ \end{array} \]

   12. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \color{blue}{\frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array} \]

       2. lower-/.f64 67.7%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{b}{\color{blue}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array} \]

   14. Applied rewrites 67.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 67.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0) (* -1.0 (/ b a)) (/ (* 2.0 c) (* -1.0 (* b 2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -1.0 * (b / a);
	} else {
		tmp = (2.0 * c) / (-1.0 * (b * 2.0));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b >= 0.0d0) then
        tmp = (-1.0d0) * (b / a)
    else
        tmp = (2.0d0 * c) / ((-1.0d0) * (b * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -1.0 * (b / a);
	} else {
		tmp = (2.0 * c) / (-1.0 * (b * 2.0));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = -1.0 * (b / a)
	else:
		tmp = (2.0 * c) / (-1.0 * (b * 2.0))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(-1.0 * Float64(b / a));
	else
		tmp = Float64(Float64(2.0 * c) / Float64(-1.0 * Float64(b * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = -1.0 * (b / a);
	else
		tmp = (2.0 * c) / (-1.0 * (b * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[(-1.0 * N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 72.4%

   \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. lift-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \color{blue}{\left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. fp-cancel-sub-sign-inv N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   4. +-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c + b \cdot b}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. lift-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\left(\mathsf{neg}\left(\color{blue}{4 \cdot a}\right)\right) \cdot c + b \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   6. distribute-lft-neg-in N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot a\right)} \cdot c + b \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   7. associate-*l* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} + b \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   8. *-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(\mathsf{neg}\left(4\right)\right)} + b \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   9. lower-fma.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, \mathsf{neg}\left(4\right), b \cdot b\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   10. *-commutative N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\color{blue}{c \cdot a}, \mathsf{neg}\left(4\right), b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   11. lower-*.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\color{blue}{c \cdot a}, \mathsf{neg}\left(4\right), b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   12. metadata-eval 72.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, \color{blue}{-4}, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

3. Applied rewrites 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
4. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. lift-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. fp-cancel-sub-sign-inv N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{b \cdot b + \left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}}\\ \end{array} \]

   4. +-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{\left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c + b \cdot b}}\\ \end{array} \]

   5. lift-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{\left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c + b \cdot b}}\\ \end{array} \]

   6. distribute-lft-neg-in N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot a\right) \cdot c + b \cdot b}}\\ \end{array} \]

   7. associate-*l* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-\color{blue}{b}\right) + \sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right) + b \cdot b}}\\ \end{array} \]

   8. *-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-\color{blue}{b}\right) + \sqrt{\left(a \cdot c\right) \cdot \left(\mathsf{neg}\left(4\right)\right) + b \cdot b}}\\ \end{array} \]

   9. lower-fma.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{\mathsf{fma}\left(a \cdot c, \mathsf{neg}\left(4\right), b \cdot b\right)}}\\ \end{array} \]

   10. *-commutative N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-\color{blue}{b}\right) + \sqrt{\mathsf{fma}\left(c \cdot a, \mathsf{neg}\left(4\right), b \cdot b\right)}}\\ \end{array} \]

   11. lower-*.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-\color{blue}{b}\right) + \sqrt{\mathsf{fma}\left(c \cdot a, \mathsf{neg}\left(4\right), b \cdot b\right)}}\\ \end{array} \]

   12. metadata-eval 72.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}\\ \end{array} \]

5. Applied rewrites 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}\\ \end{array} \]

6. Taylor expanded in b around -inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}\\ \end{array} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}\\ \end{array} \]

   2. lower-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \color{blue}{\left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}\\ \end{array} \]

   3. lower-+.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \color{blue}{\left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)}\right)}\\ \end{array} \]

   4. lower-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + \color{blue}{-2 \cdot \frac{a \cdot c}{{b}^{2}}}\right)\right)}\\ \end{array} \]

   5. lower-/.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \color{blue}{\frac{a \cdot c}{{b}^{2}}}\right)\right)}\\ \end{array} \]

   6. lower-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{\color{blue}{a \cdot c}}{{b}^{2}}\right)\right)}\\ \end{array} \]

   7. lower-pow.f64 67.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{\color{blue}{{b}^{2}}}\right)\right)}\\ \end{array} \]

8. Applied rewrites 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{-1 \cdot \left(b \cdot \left(2 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}\\ \end{array} \]

9. Taylor expanded in a around 0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \color{blue}{2}\right)}\\ \end{array} \]
10. Step-by-step derivation

11. Applied rewrites 69.7%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot \color{blue}{2}\right)}\\ \end{array} \]

12. Taylor expanded in b around inf

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array} \]
13. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \color{blue}{\frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array} \]

    2. lower-/.f64 67.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{b}{\color{blue}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array} \]

14. Applied rewrites 67.7%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot \left(b \cdot 2\right)}\\ \end{array} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2025189 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  :precision binary64
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))