jeff quadratic root 2

Percentage Accurate: 71.7% → 92.4%

Time: 5.3s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 71.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 - \frac{c}{b} \cdot \frac{a \cdot 4}{b}}\\ t_1 := \frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{if}\;b \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{1 - \left(a \cdot 4\right) \cdot \frac{c}{b \cdot b}}, \left|b\right|, -b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 6.7 \cdot 10^{-71}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{\left(-b\right) - \left|b\right| \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left|b\right|, t\_0, -b\right)}{a + a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- 1.0 (* (/ c b) (/ (* a 4.0) b)))))
        (t_1 (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))
   (if (<= b -2e-11)
     (if (>= b 0.0)
       t_1
       (/
        (fma (sqrt (- 1.0 (* (* a 4.0) (/ c (* b b))))) (fabs b) (- b))
        (* 2.0 a)))
     (if (<= b 6.7e-71)
       (if (>= b 0.0)
         t_1
         (* (/ 0.5 a) (- (sqrt (fma -4.0 (* a c) (* b b))) b)))
       (if (>= b 0.0)
         (/ (+ c c) (- (- b) (* (fabs b) t_0)))
         (/ (fma (fabs b) t_0 (- b)) (+ a a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((1.0 - ((c / b) * ((a * 4.0) / b))));
	double t_1 = (2.0 * c) / (-b - sqrt(((b * b) - ((4.0 * a) * c))));
	double tmp_1;
	if (b <= -2e-11) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = fma(sqrt((1.0 - ((a * 4.0) * (c / (b * b))))), fabs(b), -b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 6.7e-71) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = (0.5 / a) * (sqrt(fma(-4.0, (a * c), (b * b))) - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c + c) / (-b - (fabs(b) * t_0));
	} else {
		tmp_1 = fma(fabs(b), t_0, -b) / (a + a);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(1.0 - Float64(Float64(c / b) * Float64(Float64(a * 4.0) / b))))
	t_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))))
	tmp_1 = 0.0
	if (b <= -2e-11)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(fma(sqrt(Float64(1.0 - Float64(Float64(a * 4.0) * Float64(c / Float64(b * b))))), abs(b), Float64(-b)) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 6.7e-71)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = Float64(Float64(0.5 / a) * Float64(sqrt(fma(-4.0, Float64(a * c), Float64(b * b))) - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c + c) / Float64(Float64(-b) - Float64(abs(b) * t_0)));
	else
		tmp_1 = Float64(fma(abs(b), t_0, Float64(-b)) / Float64(a + a));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(1.0 - N[(N[(c / b), $MachinePrecision] * N[(N[(a * 4.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2e-11], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(N[Sqrt[N[(1.0 - N[(N[(a * 4.0), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[b], $MachinePrecision] + (-b)), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 6.7e-71], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c + c), $MachinePrecision] / N[((-b) - N[(N[Abs[b], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[b], $MachinePrecision] * t$95$0 + (-b)), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.99999999999999988e-11

   1. Initial program 71.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-to-mult N/A

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

      6. sqrt-prod N/A

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

      7. lift-*.f64 N/A

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

      8. rem-sqrt-square-rev N/A

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 73.9%

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

   if -1.99999999999999988e-11 < b < 6.6999999999999998e-71

   1. Initial program 71.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 71.7

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. sub-flip-reverse N/A

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

      13. lower--.f64 71.7

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

   3. Applied rewrites 71.7%

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

   if 6.6999999999999998e-71 < b

   1. Initial program 71.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-to-mult N/A

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

      4. sqrt-prod N/A

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

      5. lift-*.f64 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-fabs.f64 74.1

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

   3. Applied rewrites 74.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-to-mult N/A

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

      4. sqrt-prod N/A

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

      5. lift-*.f64 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-fabs.f64 76.3

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

   5. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 79.0

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 79.0

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

   7. Applied rewrites 79.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 81.7

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 81.7

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

   9. Applied rewrites 81.7%

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

       1. lift-*.f64 N/A

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

       2. count-2-rev N/A

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

       3. lower-+.f64 81.7

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

   11. Applied rewrites 81.7%

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

   13. Applied rewrites 81.6%

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

Alternative 2: 92.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ t_1 := \frac{c}{b \cdot b}\\ t_2 := \left|b\right| \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, t\_1, 1\right)}\\ \mathbf{if}\;b \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{1 - \left(a \cdot 4\right) \cdot t\_1}, \left|b\right|, -b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-71}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + t\_2}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))))
        (t_1 (/ c (* b b)))
        (t_2 (* (fabs b) (sqrt (fma (* -4.0 a) t_1 1.0)))))
   (if (<= b -2e-11)
     (if (>= b 0.0)
       t_0
       (/ (fma (sqrt (- 1.0 (* (* a 4.0) t_1))) (fabs b) (- b)) (* 2.0 a)))
     (if (<= b 6e-71)
       (if (>= b 0.0)
         t_0
         (* (/ 0.5 a) (- (sqrt (fma -4.0 (* a c) (* b b))) b)))
       (if (>= b 0.0)
         (/ (* 2.0 c) (- (- b) t_2))
         (/ (+ (- b) t_2) (* 2.0 a)))))))

C

double code(double a, double b, double c) {
	double t_0 = (2.0 * c) / (-b - sqrt(((b * b) - ((4.0 * a) * c))));
	double t_1 = c / (b * b);
	double t_2 = fabs(b) * sqrt(fma((-4.0 * a), t_1, 1.0));
	double tmp_1;
	if (b <= -2e-11) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = fma(sqrt((1.0 - ((a * 4.0) * t_1))), fabs(b), -b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 6e-71) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_0;
		} else {
			tmp_3 = (0.5 / a) * (sqrt(fma(-4.0, (a * c), (b * b))) - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-b - t_2);
	} else {
		tmp_1 = (-b + t_2) / (2.0 * a);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))))
	t_1 = Float64(c / Float64(b * b))
	t_2 = Float64(abs(b) * sqrt(fma(Float64(-4.0 * a), t_1, 1.0)))
	tmp_1 = 0.0
	if (b <= -2e-11)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(fma(sqrt(Float64(1.0 - Float64(Float64(a * 4.0) * t_1))), abs(b), Float64(-b)) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 6e-71)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_0;
		else
			tmp_3 = Float64(Float64(0.5 / a) * Float64(sqrt(fma(-4.0, Float64(a * c), Float64(b * b))) - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_2));
	else
		tmp_1 = Float64(Float64(Float64(-b) + t_2) / Float64(2.0 * a));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[b], $MachinePrecision] * N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2e-11], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(N[Sqrt[N[(1.0 - N[(N[(a * 4.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[b], $MachinePrecision] + (-b)), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 6e-71], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[((-b) + t$95$2), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.99999999999999988e-11

   1. Initial program 71.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-to-mult N/A

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

      6. sqrt-prod N/A

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

      7. lift-*.f64 N/A

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

      8. rem-sqrt-square-rev N/A

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 73.9%

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

   if -1.99999999999999988e-11 < b < 6.0000000000000003e-71

   1. Initial program 71.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 71.7

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. sub-flip-reverse N/A

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

      13. lower--.f64 71.7

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

   3. Applied rewrites 71.7%

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

   if 6.0000000000000003e-71 < b

   1. Initial program 71.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-to-mult N/A

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

      4. sqrt-prod N/A

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

      5. lift-*.f64 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-fabs.f64 74.1

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

   3. Applied rewrites 74.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-to-mult N/A

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

      4. sqrt-prod N/A

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

      5. lift-*.f64 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-fabs.f64 76.3

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

   5. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 76.3

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 76.3

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

   7. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 76.3

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 76.3

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

   9. Applied rewrites 76.3%

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

Alternative 3: 92.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ t_1 := \frac{c}{b \cdot b}\\ \mathbf{if}\;b \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{1 - \left(a \cdot 4\right) \cdot t\_1}, \left|b\right|, -b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{+35}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2}{\mathsf{fma}\left(\left|b\right|, \sqrt{\mathsf{fma}\left(-4 \cdot a, t\_1, 1\right)}, b\right)} \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{a} + \sqrt{\frac{c}{a} \cdot -4}\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))))
        (t_1 (/ c (* b b))))
   (if (<= b -2e-11)
     (if (>= b 0.0)
       t_0
       (/ (fma (sqrt (- 1.0 (* (* a 4.0) t_1))) (fabs b) (- b)) (* 2.0 a)))
     (if (<= b 1e+35)
       (if (>= b 0.0)
         t_0
         (* (/ 0.5 a) (- (sqrt (fma -4.0 (* a c) (* b b))) b)))
       (if (>= b 0.0)
         (* (/ -2.0 (fma (fabs b) (sqrt (fma (* -4.0 a) t_1 1.0)) b)) c)
         (* (+ (/ b a) (sqrt (* (/ c a) -4.0))) -0.5))))))

C

double code(double a, double b, double c) {
	double t_0 = (2.0 * c) / (-b - sqrt(((b * b) - ((4.0 * a) * c))));
	double t_1 = c / (b * b);
	double tmp_1;
	if (b <= -2e-11) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = fma(sqrt((1.0 - ((a * 4.0) * t_1))), fabs(b), -b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1e+35) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_0;
		} else {
			tmp_3 = (0.5 / a) * (sqrt(fma(-4.0, (a * c), (b * b))) - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 / fma(fabs(b), sqrt(fma((-4.0 * a), t_1, 1.0)), b)) * c;
	} else {
		tmp_1 = ((b / a) + sqrt(((c / a) * -4.0))) * -0.5;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))))
	t_1 = Float64(c / Float64(b * b))
	tmp_1 = 0.0
	if (b <= -2e-11)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(fma(sqrt(Float64(1.0 - Float64(Float64(a * 4.0) * t_1))), abs(b), Float64(-b)) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1e+35)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_0;
		else
			tmp_3 = Float64(Float64(0.5 / a) * Float64(sqrt(fma(-4.0, Float64(a * c), Float64(b * b))) - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-2.0 / fma(abs(b), sqrt(fma(Float64(-4.0 * a), t_1, 1.0)), b)) * c);
	else
		tmp_1 = Float64(Float64(Float64(b / a) + sqrt(Float64(Float64(c / a) * -4.0))) * -0.5);
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2e-11], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(N[Sqrt[N[(1.0 - N[(N[(a * 4.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[b], $MachinePrecision] + (-b)), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1e+35], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(-2.0 / N[(N[Abs[b], $MachinePrecision] * N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], N[(N[(N[(b / a), $MachinePrecision] + N[Sqrt[N[(N[(c / a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.99999999999999988e-11

   1. Initial program 71.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-to-mult N/A

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

      6. sqrt-prod N/A

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

      7. lift-*.f64 N/A

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

      8. rem-sqrt-square-rev N/A

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 73.9%

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

   if -1.99999999999999988e-11 < b < 9.9999999999999997e34

   1. Initial program 71.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 71.7

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. sub-flip-reverse N/A

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

      13. lower--.f64 71.7

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

   3. Applied rewrites 71.7%

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

   if 9.9999999999999997e34 < b

   1. Initial program 71.7%

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

   2. Taylor expanded in a around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 48.3

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

   4. Applied rewrites 48.3%

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

   6. Applied rewrites 48.2%

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

   8. Applied rewrites 50.6%

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

Alternative 4: 91.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}\\ \mathbf{if}\;b \leq -3 \cdot 10^{+110}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{\left|b\right| + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|b\right| - b}{a + a}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{+35}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{t\_0 + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a + a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2}{\mathsf{fma}\left(\left|b\right|, \sqrt{\mathsf{fma}\left(-4 \cdot a, \frac{c}{b \cdot b}, 1\right)}, b\right)} \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{a} + \sqrt{\frac{c}{a} \cdot -4}\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma -4.0 (* a c) (* b b)))))
   (if (<= b -3e+110)
     (if (>= b 0.0) (/ (* -2.0 c) (+ (fabs b) b)) (/ (- (fabs b) b) (+ a a)))
     (if (<= b 1e+35)
       (if (>= b 0.0) (/ (* -2.0 c) (+ t_0 b)) (/ (- t_0 b) (+ a a)))
       (if (>= b 0.0)
         (*
          (/ -2.0 (fma (fabs b) (sqrt (fma (* -4.0 a) (/ c (* b b)) 1.0)) b))
          c)
         (* (+ (/ b a) (sqrt (* (/ c a) -4.0))) -0.5))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma(-4.0, (a * c), (b * b)));
	double tmp_1;
	if (b <= -3e+110) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-2.0 * c) / (fabs(b) + b);
		} else {
			tmp_2 = (fabs(b) - b) / (a + a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1e+35) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-2.0 * c) / (t_0 + b);
		} else {
			tmp_3 = (t_0 - b) / (a + a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 / fma(fabs(b), sqrt(fma((-4.0 * a), (c / (b * b)), 1.0)), b)) * c;
	} else {
		tmp_1 = ((b / a) + sqrt(((c / a) * -4.0))) * -0.5;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(-4.0, Float64(a * c), Float64(b * b)))
	tmp_1 = 0.0
	if (b <= -3e+110)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-2.0 * c) / Float64(abs(b) + b));
		else
			tmp_2 = Float64(Float64(abs(b) - b) / Float64(a + a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1e+35)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-2.0 * c) / Float64(t_0 + b));
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(a + a));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-2.0 / fma(abs(b), sqrt(fma(Float64(-4.0 * a), Float64(c / Float64(b * b)), 1.0)), b)) * c);
	else
		tmp_1 = Float64(Float64(Float64(b / a) + sqrt(Float64(Float64(c / a) * -4.0))) * -0.5);
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -3e+110], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * c), $MachinePrecision] / N[(N[Abs[b], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[b], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1e+35], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * c), $MachinePrecision] / N[(t$95$0 + b), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(-2.0 / N[(N[Abs[b], $MachinePrecision] * N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], N[(N[(N[(b / a), $MachinePrecision] + N[Sqrt[N[(N[(c / a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.00000000000000007e110

   1. Initial program 71.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-to-mult N/A

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

      4. sqrt-prod N/A

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

      5. lift-*.f64 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-fabs.f64 74.1

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

   3. Applied rewrites 74.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-to-mult N/A

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

      4. sqrt-prod N/A

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

      5. lift-*.f64 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-fabs.f64 76.3

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in a around 0

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

      1. lower-fabs.f64 72.8

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

   8. Applied rewrites 72.8%

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

   9. Taylor expanded in a around 0

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

       1. lower-fabs.f64 68.5

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

   11. Applied rewrites 68.5%

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

   13. Applied rewrites 68.5%

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

   if -3.00000000000000007e110 < b < 9.9999999999999997e34

   1. Initial program 71.7%

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

   3. Applied rewrites 71.7%

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

   if 9.9999999999999997e34 < b

   1. Initial program 71.7%

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

   2. Taylor expanded in a around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 48.3

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

   4. Applied rewrites 48.3%

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

   6. Applied rewrites 48.2%

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

   8. Applied rewrites 50.6%

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

Alternative 5: 91.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{\left|b\right| + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|b\right| - b}{a + a}\\ \end{array}\\ t_1 := \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}\\ \mathbf{if}\;b \leq -3 \cdot 10^{+110}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{+102}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{t\_1 + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 - b}{a + a}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0
         (if (>= b 0.0)
           (/ (* -2.0 c) (+ (fabs b) b))
           (/ (- (fabs b) b) (+ a a))))
        (t_1 (sqrt (fma -4.0 (* a c) (* b b)))))
   (if (<= b -3e+110)
     t_0
     (if (<= b 4.1e+102)
       (if (>= b 0.0) (/ (* -2.0 c) (+ t_1 b)) (/ (- t_1 b) (+ a a)))
       t_0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (-2.0 * c) / (fabs(b) + b);
	} else {
		tmp = (fabs(b) - b) / (a + a);
	}
	double t_0 = tmp;
	double t_1 = sqrt(fma(-4.0, (a * c), (b * b)));
	double tmp_1;
	if (b <= -3e+110) {
		tmp_1 = t_0;
	} else if (b <= 4.1e+102) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-2.0 * c) / (t_1 + b);
		} else {
			tmp_2 = (t_1 - b) / (a + a);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(-2.0 * c) / Float64(abs(b) + b));
	else
		tmp = Float64(Float64(abs(b) - b) / Float64(a + a));
	end
	t_0 = tmp
	t_1 = sqrt(fma(-4.0, Float64(a * c), Float64(b * b)))
	tmp_1 = 0.0
	if (b <= -3e+110)
		tmp_1 = t_0;
	elseif (b <= 4.1e+102)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-2.0 * c) / Float64(t_1 + b));
		else
			tmp_2 = Float64(Float64(t_1 - b) / Float64(a + a));
		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[(N[(-2.0 * c), $MachinePrecision] / N[(N[Abs[b], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[b], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]}, Block[{t$95$1 = N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -3e+110], t$95$0, If[LessEqual[b, 4.1e+102], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * c), $MachinePrecision] / N[(t$95$1 + b), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.00000000000000007e110 or 4.1e102 < b

   1. Initial program 71.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-to-mult N/A

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

      4. sqrt-prod N/A

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

      5. lift-*.f64 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-fabs.f64 74.1

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

   3. Applied rewrites 74.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-to-mult N/A

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

      4. sqrt-prod N/A

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

      5. lift-*.f64 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-fabs.f64 76.3

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in a around 0

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

      1. lower-fabs.f64 72.8

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

   8. Applied rewrites 72.8%

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

   9. Taylor expanded in a around 0

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

       1. lower-fabs.f64 68.5

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

   11. Applied rewrites 68.5%

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

   13. Applied rewrites 68.5%

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

   if -3.00000000000000007e110 < b < 4.1e102

   1. Initial program 71.7%

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

   3. Applied rewrites 71.7%

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

Alternative 6: 91.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}\\ \mathbf{if}\;b \leq -3 \cdot 10^{+110}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{\left|b\right| + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|b\right| - b}{a + a}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{+102}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{t\_0 + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a + a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{-2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma -4.0 (* a c) (* b b)))))
   (if (<= b -3e+110)
     (if (>= b 0.0) (/ (* -2.0 c) (+ (fabs b) b)) (/ (- (fabs b) b) (+ a a)))
     (if (<= b 4.1e+102)
       (if (>= b 0.0) (/ (* -2.0 c) (+ t_0 b)) (/ (- t_0 b) (+ a a)))
       (if (>= b 0.0)
         (/ (* 2.0 c) (* -2.0 b))
         (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma(-4.0, (a * c), (b * b)));
	double tmp_1;
	if (b <= -3e+110) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-2.0 * c) / (fabs(b) + b);
		} else {
			tmp_2 = (fabs(b) - b) / (a + a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 4.1e+102) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-2.0 * c) / (t_0 + b);
		} else {
			tmp_3 = (t_0 - b) / (a + a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-2.0 * b);
	} else {
		tmp_1 = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(-4.0, Float64(a * c), Float64(b * b)))
	tmp_1 = 0.0
	if (b <= -3e+110)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-2.0 * c) / Float64(abs(b) + b));
		else
			tmp_2 = Float64(Float64(abs(b) - b) / Float64(a + a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 4.1e+102)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-2.0 * c) / Float64(t_0 + b));
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(a + a));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(-2.0 * b));
	else
		tmp_1 = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -3e+110], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * c), $MachinePrecision] / N[(N[Abs[b], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[b], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 4.1e+102], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * c), $MachinePrecision] / N[(t$95$0 + b), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(-2.0 * b), $MachinePrecision]), $MachinePrecision], N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.00000000000000007e110

   1. Initial program 71.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-to-mult N/A

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

      4. sqrt-prod N/A

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

      5. lift-*.f64 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-fabs.f64 74.1

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

   3. Applied rewrites 74.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-to-mult N/A

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

      4. sqrt-prod N/A

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

      5. lift-*.f64 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-fabs.f64 76.3

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in a around 0

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

      1. lower-fabs.f64 72.8

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

   8. Applied rewrites 72.8%

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

   9. Taylor expanded in a around 0

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

       1. lower-fabs.f64 68.5

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

   11. Applied rewrites 68.5%

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

   13. Applied rewrites 68.5%

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

   if -3.00000000000000007e110 < b < 4.1e102

   1. Initial program 71.7%

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

   3. Applied rewrites 71.7%

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

   if 4.1e102 < b

   1. Initial program 71.7%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 70.6

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

   4. Applied rewrites 70.6%

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

Alternative 7: 91.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{\left|b\right| + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|b\right| - b}{a + a}\\ \end{array}\\ t_1 := \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}\\ \mathbf{if}\;b \leq -3 \cdot 10^{+110}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{+102}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2}{t\_1 + b} \cdot c\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 - b}{a + a}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0
         (if (>= b 0.0)
           (/ (* -2.0 c) (+ (fabs b) b))
           (/ (- (fabs b) b) (+ a a))))
        (t_1 (sqrt (fma (* a c) -4.0 (* b b)))))
   (if (<= b -3e+110)
     t_0
     (if (<= b 4.1e+102)
       (if (>= b 0.0) (* (/ -2.0 (+ t_1 b)) c) (/ (- t_1 b) (+ a a)))
       t_0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (-2.0 * c) / (fabs(b) + b);
	} else {
		tmp = (fabs(b) - b) / (a + a);
	}
	double t_0 = tmp;
	double t_1 = sqrt(fma((a * c), -4.0, (b * b)));
	double tmp_1;
	if (b <= -3e+110) {
		tmp_1 = t_0;
	} else if (b <= 4.1e+102) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-2.0 / (t_1 + b)) * c;
		} else {
			tmp_2 = (t_1 - b) / (a + a);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(-2.0 * c) / Float64(abs(b) + b));
	else
		tmp = Float64(Float64(abs(b) - b) / Float64(a + a));
	end
	t_0 = tmp
	t_1 = sqrt(fma(Float64(a * c), -4.0, Float64(b * b)))
	tmp_1 = 0.0
	if (b <= -3e+110)
		tmp_1 = t_0;
	elseif (b <= 4.1e+102)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-2.0 / Float64(t_1 + b)) * c);
		else
			tmp_2 = Float64(Float64(t_1 - b) / Float64(a + a));
		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[(N[(-2.0 * c), $MachinePrecision] / N[(N[Abs[b], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[b], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]}, Block[{t$95$1 = N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -3e+110], t$95$0, If[LessEqual[b, 4.1e+102], If[GreaterEqual[b, 0.0], N[(N[(-2.0 / N[(t$95$1 + b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], N[(N[(t$95$1 - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.00000000000000007e110 or 4.1e102 < b

   1. Initial program 71.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-to-mult N/A

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

      4. sqrt-prod N/A

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

      5. lift-*.f64 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-fabs.f64 74.1

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

   3. Applied rewrites 74.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-to-mult N/A

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

      4. sqrt-prod N/A

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

      5. lift-*.f64 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-fabs.f64 76.3

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in a around 0

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

      1. lower-fabs.f64 72.8

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

   8. Applied rewrites 72.8%

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

   9. Taylor expanded in a around 0

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

       1. lower-fabs.f64 68.5

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

   11. Applied rewrites 68.5%

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

   13. Applied rewrites 68.5%

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

   if -3.00000000000000007e110 < b < 4.1e102

   1. Initial program 71.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-to-mult N/A

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

      4. sqrt-prod N/A

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

      5. lift-*.f64 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-fabs.f64 74.1

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

   3. Applied rewrites 74.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-to-mult N/A

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

      4. sqrt-prod N/A

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

      5. lift-*.f64 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-fabs.f64 76.3

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

   5. Applied rewrites 76.3%

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

   7. Applied rewrites 71.7%

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

Alternative 8: 85.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{\left|b\right| + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|b\right| - b}{a + a}\\ \end{array}\\ \mathbf{if}\;b \leq -2.1 \cdot 10^{-113}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{+102}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b} \cdot c\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0
         (if (>= b 0.0)
           (/ (* -2.0 c) (+ (fabs b) b))
           (/ (- (fabs b) b) (+ a a)))))
   (if (<= b -2.1e-113)
     t_0
     (if (<= b 4.1e+102)
       (if (>= b 0.0)
         (* (/ -2.0 (+ (sqrt (fma (* a c) -4.0 (* b b))) b)) c)
         (* -0.5 (* c (sqrt (/ -4.0 (* a c))))))
       t_0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (-2.0 * c) / (fabs(b) + b);
	} else {
		tmp = (fabs(b) - b) / (a + a);
	}
	double t_0 = tmp;
	double tmp_1;
	if (b <= -2.1e-113) {
		tmp_1 = t_0;
	} else if (b <= 4.1e+102) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-2.0 / (sqrt(fma((a * c), -4.0, (b * b))) + b)) * c;
		} else {
			tmp_2 = -0.5 * (c * sqrt((-4.0 / (a * c))));
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(-2.0 * c) / Float64(abs(b) + b));
	else
		tmp = Float64(Float64(abs(b) - b) / Float64(a + a));
	end
	t_0 = tmp
	tmp_1 = 0.0
	if (b <= -2.1e-113)
		tmp_1 = t_0;
	elseif (b <= 4.1e+102)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-2.0 / Float64(sqrt(fma(Float64(a * c), -4.0, Float64(b * b))) + b)) * c);
		else
			tmp_2 = Float64(-0.5 * Float64(c * sqrt(Float64(-4.0 / Float64(a * c)))));
		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[(N[(-2.0 * c), $MachinePrecision] / N[(N[Abs[b], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[b], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]}, If[LessEqual[b, -2.1e-113], t$95$0, If[LessEqual[b, 4.1e+102], If[GreaterEqual[b, 0.0], N[(N[(-2.0 / N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], N[(-0.5 * N[(c * N[Sqrt[N[(-4.0 / N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.1e-113 or 4.1e102 < b

   1. Initial program 71.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-to-mult N/A

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

      4. sqrt-prod N/A

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

      5. lift-*.f64 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-fabs.f64 74.1

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

   3. Applied rewrites 74.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-to-mult N/A

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

      4. sqrt-prod N/A

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

      5. lift-*.f64 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-fabs.f64 76.3

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in a around 0

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

      1. lower-fabs.f64 72.8

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

   8. Applied rewrites 72.8%

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

   9. Taylor expanded in a around 0

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

       1. lower-fabs.f64 68.5

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

   11. Applied rewrites 68.5%

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

   13. Applied rewrites 68.5%

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

   if -2.1e-113 < b < 4.1e102

   1. Initial program 71.7%

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

   2. Taylor expanded in a around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 48.3

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

   4. Applied rewrites 48.3%

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

   6. Applied rewrites 48.2%

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

   7. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 50.3

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

   9. Applied rewrites 50.3%

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

Alternative 9: 81.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{\left|b\right| + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|b\right| - b}{a + a}\\ \end{array}\\ \mathbf{if}\;b \leq -2.1 \cdot 10^{-113}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-12}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\left(a \cdot c\right) \cdot -4} - b\right)\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0
         (if (>= b 0.0)
           (/ (* -2.0 c) (+ (fabs b) b))
           (/ (- (fabs b) b) (+ a a)))))
   (if (<= b -2.1e-113)
     t_0
     (if (<= b 3.5e-12)
       (if (>= b 0.0)
         (/ (* 2.0 c) (- (- b) (sqrt (* -4.0 (* a c)))))
         (* (/ 0.5 a) (- (sqrt (* (* a c) -4.0)) b)))
       t_0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (-2.0 * c) / (fabs(b) + b);
	} else {
		tmp = (fabs(b) - b) / (a + a);
	}
	double t_0 = tmp;
	double tmp_1;
	if (b <= -2.1e-113) {
		tmp_1 = t_0;
	} else if (b <= 3.5e-12) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (-b - sqrt((-4.0 * (a * c))));
		} else {
			tmp_2 = (0.5 / a) * (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 = ((-2.0d0) * c) / (abs(b) + b)
    else
        tmp = (abs(b) - b) / (a + a)
    end if
    t_0 = tmp
    if (b <= (-2.1d-113)) then
        tmp_1 = t_0
    else if (b <= 3.5d-12) then
        if (b >= 0.0d0) then
            tmp_2 = (2.0d0 * c) / (-b - sqrt(((-4.0d0) * (a * c))))
        else
            tmp_2 = (0.5d0 / a) * (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 = (-2.0 * c) / (Math.abs(b) + b);
	} else {
		tmp = (Math.abs(b) - b) / (a + a);
	}
	double t_0 = tmp;
	double tmp_1;
	if (b <= -2.1e-113) {
		tmp_1 = t_0;
	} else if (b <= 3.5e-12) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (-b - Math.sqrt((-4.0 * (a * c))));
		} else {
			tmp_2 = (0.5 / a) * (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 = (-2.0 * c) / (math.fabs(b) + b)
	else:
		tmp = (math.fabs(b) - b) / (a + a)
	t_0 = tmp
	tmp_1 = 0
	if b <= -2.1e-113:
		tmp_1 = t_0
	elif b <= 3.5e-12:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (2.0 * c) / (-b - math.sqrt((-4.0 * (a * c))))
		else:
			tmp_2 = (0.5 / a) * (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(Float64(-2.0 * c) / Float64(abs(b) + b));
	else
		tmp = Float64(Float64(abs(b) - b) / Float64(a + a));
	end
	t_0 = tmp
	tmp_1 = 0.0
	if (b <= -2.1e-113)
		tmp_1 = t_0;
	elseif (b <= 3.5e-12)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - sqrt(Float64(-4.0 * Float64(a * c)))));
		else
			tmp_2 = Float64(Float64(0.5 / a) * 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 = (-2.0 * c) / (abs(b) + b);
	else
		tmp = (abs(b) - b) / (a + a);
	end
	t_0 = tmp;
	tmp_2 = 0.0;
	if (b <= -2.1e-113)
		tmp_2 = t_0;
	elseif (b <= 3.5e-12)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (2.0 * c) / (-b - sqrt((-4.0 * (a * c))));
		else
			tmp_3 = (0.5 / a) * (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[(N[(-2.0 * c), $MachinePrecision] / N[(N[Abs[b], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[b], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]}, If[LessEqual[b, -2.1e-113], t$95$0, If[LessEqual[b, 3.5e-12], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / a), $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 < -2.1e-113 or 3.5e-12 < b

   1. Initial program 71.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-to-mult N/A

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

      4. sqrt-prod N/A

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

      5. lift-*.f64 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-fabs.f64 74.1

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

   3. Applied rewrites 74.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-to-mult N/A

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

      4. sqrt-prod N/A

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

      5. lift-*.f64 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-fabs.f64 76.3

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in a around 0

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

      1. lower-fabs.f64 72.8

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

   8. Applied rewrites 72.8%

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

   9. Taylor expanded in a around 0

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

       1. lower-fabs.f64 68.5

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

   11. Applied rewrites 68.5%

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

   13. Applied rewrites 68.5%

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

   if -2.1e-113 < b < 3.5e-12

   1. Initial program 71.7%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 55.8

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

   4. Applied rewrites 55.8%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 40.4

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

   7. Applied rewrites 40.4%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 40.4

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. sub-flip-reverse N/A

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

      13. lower--.f64 40.4

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

   9. Applied rewrites 40.4%

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

Alternative 10: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{\left|b\right| + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|b\right| - b}{a + a}\\ \end{array}\\ t_1 := \sqrt{\left(a \cdot c\right) \cdot -4}\\ \mathbf{if}\;b \leq -2.1 \cdot 10^{-113}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-12}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{t\_1 + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 - b}{a + a}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0
         (if (>= b 0.0)
           (/ (* -2.0 c) (+ (fabs b) b))
           (/ (- (fabs b) b) (+ a a))))
        (t_1 (sqrt (* (* a c) -4.0))))
   (if (<= b -2.1e-113)
     t_0
     (if (<= b 3.5e-12)
       (if (>= b 0.0) (/ (* -2.0 c) (+ t_1 b)) (/ (- t_1 b) (+ a a)))
       t_0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (-2.0 * c) / (fabs(b) + b);
	} else {
		tmp = (fabs(b) - b) / (a + a);
	}
	double t_0 = tmp;
	double t_1 = sqrt(((a * c) * -4.0));
	double tmp_1;
	if (b <= -2.1e-113) {
		tmp_1 = t_0;
	} else if (b <= 3.5e-12) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-2.0 * c) / (t_1 + b);
		} else {
			tmp_2 = (t_1 - b) / (a + a);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    if (b >= 0.0d0) then
        tmp = ((-2.0d0) * c) / (abs(b) + b)
    else
        tmp = (abs(b) - b) / (a + a)
    end if
    t_0 = tmp
    t_1 = sqrt(((a * c) * (-4.0d0)))
    if (b <= (-2.1d-113)) then
        tmp_1 = t_0
    else if (b <= 3.5d-12) then
        if (b >= 0.0d0) then
            tmp_2 = ((-2.0d0) * c) / (t_1 + b)
        else
            tmp_2 = (t_1 - b) / (a + a)
        end if
        tmp_1 = tmp_2
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (-2.0 * c) / (Math.abs(b) + b);
	} else {
		tmp = (Math.abs(b) - b) / (a + a);
	}
	double t_0 = tmp;
	double t_1 = Math.sqrt(((a * c) * -4.0));
	double tmp_1;
	if (b <= -2.1e-113) {
		tmp_1 = t_0;
	} else if (b <= 3.5e-12) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-2.0 * c) / (t_1 + b);
		} else {
			tmp_2 = (t_1 - b) / (a + a);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = (-2.0 * c) / (math.fabs(b) + b)
	else:
		tmp = (math.fabs(b) - b) / (a + a)
	t_0 = tmp
	t_1 = math.sqrt(((a * c) * -4.0))
	tmp_1 = 0
	if b <= -2.1e-113:
		tmp_1 = t_0
	elif b <= 3.5e-12:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (-2.0 * c) / (t_1 + b)
		else:
			tmp_2 = (t_1 - b) / (a + a)
		tmp_1 = tmp_2
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(-2.0 * c) / Float64(abs(b) + b));
	else
		tmp = Float64(Float64(abs(b) - b) / Float64(a + a));
	end
	t_0 = tmp
	t_1 = sqrt(Float64(Float64(a * c) * -4.0))
	tmp_1 = 0.0
	if (b <= -2.1e-113)
		tmp_1 = t_0;
	elseif (b <= 3.5e-12)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-2.0 * c) / Float64(t_1 + b));
		else
			tmp_2 = Float64(Float64(t_1 - b) / Float64(a + a));
		end
		tmp_1 = tmp_2;
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (-2.0 * c) / (abs(b) + b);
	else
		tmp = (abs(b) - b) / (a + a);
	end
	t_0 = tmp;
	t_1 = sqrt(((a * c) * -4.0));
	tmp_2 = 0.0;
	if (b <= -2.1e-113)
		tmp_2 = t_0;
	elseif (b <= 3.5e-12)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (-2.0 * c) / (t_1 + b);
		else
			tmp_3 = (t_1 - b) / (a + a);
		end
		tmp_2 = tmp_3;
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = If[GreaterEqual[b, 0.0], N[(N[(-2.0 * c), $MachinePrecision] / N[(N[Abs[b], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[b], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]}, Block[{t$95$1 = N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2.1e-113], t$95$0, If[LessEqual[b, 3.5e-12], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * c), $MachinePrecision] / N[(t$95$1 + b), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.1e-113 or 3.5e-12 < b

   1. Initial program 71.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-to-mult N/A

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

      4. sqrt-prod N/A

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

      5. lift-*.f64 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-fabs.f64 74.1

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

   3. Applied rewrites 74.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-to-mult N/A

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

      4. sqrt-prod N/A

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

      5. lift-*.f64 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-fabs.f64 76.3

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in a around 0

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

      1. lower-fabs.f64 72.8

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

   8. Applied rewrites 72.8%

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

   9. Taylor expanded in a around 0

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

       1. lower-fabs.f64 68.5

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

   11. Applied rewrites 68.5%

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

   13. Applied rewrites 68.5%

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

   if -2.1e-113 < b < 3.5e-12

   1. Initial program 71.7%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 55.8

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

   4. Applied rewrites 55.8%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 40.4

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

   7. Applied rewrites 40.4%

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

   9. Applied rewrites 40.4%

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

Alternative 11: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{\left|b\right| + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|b\right| - b}{a + a}\\ \end{array}\\ \mathbf{if}\;b \leq -2.1 \cdot 10^{-113}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-69}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\left(a \cdot c\right) \cdot -4} - b\right)\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0
         (if (>= b 0.0)
           (/ (* -2.0 c) (+ (fabs b) b))
           (/ (- (fabs b) b) (+ a a)))))
   (if (<= b -2.1e-113)
     t_0
     (if (<= b 2.4e-69)
       (if (>= b 0.0)
         (/ 2.0 (* a (sqrt (/ -4.0 (* a c)))))
         (* (/ 0.5 a) (- (sqrt (* (* a c) -4.0)) b)))
       t_0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (-2.0 * c) / (fabs(b) + b);
	} else {
		tmp = (fabs(b) - b) / (a + a);
	}
	double t_0 = tmp;
	double tmp_1;
	if (b <= -2.1e-113) {
		tmp_1 = t_0;
	} else if (b <= 2.4e-69) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 2.0 / (a * sqrt((-4.0 / (a * c))));
		} else {
			tmp_2 = (0.5 / a) * (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 = ((-2.0d0) * c) / (abs(b) + b)
    else
        tmp = (abs(b) - b) / (a + a)
    end if
    t_0 = tmp
    if (b <= (-2.1d-113)) then
        tmp_1 = t_0
    else if (b <= 2.4d-69) then
        if (b >= 0.0d0) then
            tmp_2 = 2.0d0 / (a * sqrt(((-4.0d0) / (a * c))))
        else
            tmp_2 = (0.5d0 / a) * (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 = (-2.0 * c) / (Math.abs(b) + b);
	} else {
		tmp = (Math.abs(b) - b) / (a + a);
	}
	double t_0 = tmp;
	double tmp_1;
	if (b <= -2.1e-113) {
		tmp_1 = t_0;
	} else if (b <= 2.4e-69) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 2.0 / (a * Math.sqrt((-4.0 / (a * c))));
		} else {
			tmp_2 = (0.5 / a) * (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 = (-2.0 * c) / (math.fabs(b) + b)
	else:
		tmp = (math.fabs(b) - b) / (a + a)
	t_0 = tmp
	tmp_1 = 0
	if b <= -2.1e-113:
		tmp_1 = t_0
	elif b <= 2.4e-69:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = 2.0 / (a * math.sqrt((-4.0 / (a * c))))
		else:
			tmp_2 = (0.5 / a) * (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(Float64(-2.0 * c) / Float64(abs(b) + b));
	else
		tmp = Float64(Float64(abs(b) - b) / Float64(a + a));
	end
	t_0 = tmp
	tmp_1 = 0.0
	if (b <= -2.1e-113)
		tmp_1 = t_0;
	elseif (b <= 2.4e-69)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(2.0 / Float64(a * sqrt(Float64(-4.0 / Float64(a * c)))));
		else
			tmp_2 = Float64(Float64(0.5 / a) * 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 = (-2.0 * c) / (abs(b) + b);
	else
		tmp = (abs(b) - b) / (a + a);
	end
	t_0 = tmp;
	tmp_2 = 0.0;
	if (b <= -2.1e-113)
		tmp_2 = t_0;
	elseif (b <= 2.4e-69)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = 2.0 / (a * sqrt((-4.0 / (a * c))));
		else
			tmp_3 = (0.5 / a) * (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[(N[(-2.0 * c), $MachinePrecision] / N[(N[Abs[b], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[b], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]}, If[LessEqual[b, -2.1e-113], t$95$0, If[LessEqual[b, 2.4e-69], If[GreaterEqual[b, 0.0], N[(2.0 / N[(a * N[Sqrt[N[(-4.0 / N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / a), $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 < -2.1e-113 or 2.4000000000000001e-69 < b

   1. Initial program 71.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-to-mult N/A

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

      4. sqrt-prod N/A

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

      5. lift-*.f64 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-fabs.f64 74.1

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

   3. Applied rewrites 74.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-to-mult N/A

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

      4. sqrt-prod N/A

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

      5. lift-*.f64 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-fabs.f64 76.3

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in a around 0

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

      1. lower-fabs.f64 72.8

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

   8. Applied rewrites 72.8%

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

   9. Taylor expanded in a around 0

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

       1. lower-fabs.f64 68.5

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

   11. Applied rewrites 68.5%

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

   13. Applied rewrites 68.5%

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

   if -2.1e-113 < b < 2.4000000000000001e-69

   1. Initial program 71.7%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 55.8

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

   4. Applied rewrites 55.8%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 40.4

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

   7. Applied rewrites 40.4%

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

   8. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 27.2

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

   10. Applied rewrites 27.2%

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

       1. lift-/.f64 N/A

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

       2. div-flip N/A

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

       3. associate-/r/ N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-/r* N/A

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

       7. metadata-eval N/A

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

       8. lower-/.f64 27.2

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

       9. lift-+.f64 N/A

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

       10. +-commutative N/A

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

       11. lift-neg.f64 N/A

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

       12. sub-flip-reverse N/A

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

       13. lower--.f64 27.2

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

   12. Applied rewrites 27.2%

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

   13. Taylor expanded in c around inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 34.6

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

   15. Applied rewrites 34.6%

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

Alternative 12: 68.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.7%

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

   1. lift-sqrt.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-to-mult N/A

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

   4. sqrt-prod N/A

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

   5. lift-*.f64 N/A

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

   6. rem-sqrt-square-rev N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-/l* N/A

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

   12. lower-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lower-fabs.f64 74.1

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

3. Applied rewrites 74.1%

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

   1. lift-sqrt.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-to-mult N/A

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

   4. sqrt-prod N/A

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

   5. lift-*.f64 N/A

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

   6. rem-sqrt-square-rev N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-/l* N/A

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

   12. lower-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lower-fabs.f64 76.3

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

5. Applied rewrites 76.3%

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

6. Taylor expanded in a around 0

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

   1. lower-fabs.f64 72.8

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

8. Applied rewrites 72.8%

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

9. Taylor expanded in a around 0

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

    1. lower-fabs.f64 68.5

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

11. Applied rewrites 68.5%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left|b\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left|b\right|}{2 \cdot a}\\ \end{array} \]
12. Step-by-step derivation

13. Applied rewrites 68.5%

    \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{\left|b\right| + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|b\right| - b}{a + a}\\ } \end{array}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025150 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))