jeff quadratic root 1

Percentage Accurate: 71.4% → 90.3%

Time: 5.3s

Alternatives: 21

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 71.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}\\ \mathbf{if}\;b \leq -1.9 \cdot 10^{+123}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(b + \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-1}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+73}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_0}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t\_0}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + a \cdot \sqrt{-4 \cdot \frac{c}{a}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* c a) -4.0 (* b b)))))
   (if (<= b -1.9e+123)
     (if (>= b 0.0)
       (* (+ b (sqrt (fma (* c -4.0) a (* b b)))) (/ -0.5 a))
       (* c (/ -1.0 b)))
     (if (<= b 1.3e+73)
       (if (>= b 0.0) (/ (- (- b) t_0) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) t_0)))
       (if (>= b 0.0)
         (/ (* -2.0 b) (* 2.0 a))
         (/ (* 2.0 c) (+ (- b) (* a (sqrt (* -4.0 (/ c a)))))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((c * a), -4.0, (b * b)));
	double tmp_1;
	if (b <= -1.9e+123) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (b + sqrt(fma((c * -4.0), a, (b * b)))) * (-0.5 / a);
		} else {
			tmp_2 = c * (-1.0 / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.3e+73) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_0) / (2.0 * a);
		} else {
			tmp_3 = (2.0 * c) / (-b + t_0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * b) / (2.0 * a);
	} else {
		tmp_1 = (2.0 * c) / (-b + (a * sqrt((-4.0 * (c / a)))));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(c * a), -4.0, Float64(b * b)))
	tmp_1 = 0.0
	if (b <= -1.9e+123)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(b + sqrt(fma(Float64(c * -4.0), a, Float64(b * b)))) * Float64(-0.5 / a));
		else
			tmp_2 = Float64(c * Float64(-1.0 / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.3e+73)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(-b) - t_0) / Float64(2.0 * a));
		else
			tmp_3 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_0));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-2.0 * b) / Float64(2.0 * a));
	else
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + Float64(a * sqrt(Float64(-4.0 * Float64(c / a))))));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.9e+123], If[GreaterEqual[b, 0.0], N[(N[(b + N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.3e+73], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + N[(a * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.89999999999999997e123

   1. Initial program 71.4%

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

   3. Applied rewrites 71.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2-rev N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 71.3

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-fma.f64 71.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 71.3

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

   5. Applied rewrites 71.3%

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

   6. Taylor expanded in b around -inf

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

      1. lower-/.f64 69.4

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

   8. Applied rewrites 69.4%

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

   if -1.89999999999999997e123 < b < 1.3e73

   1. Initial program 71.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 71.4

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

   3. Applied rewrites 71.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 71.4

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

   5. Applied rewrites 71.4%

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

   if 1.3e73 < b

   1. Initial program 71.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 48.3

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

   4. Applied rewrites 48.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 26.2

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

   7. Applied rewrites 26.2%

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

   8. Taylor expanded in b around inf

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

      1. lower-*.f64 47.4

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

   10. Applied rewrites 47.4%

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

Alternative 2: 90.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+123}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(b + \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-1}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+73}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + a \cdot \sqrt{-4 \cdot \frac{c}{a}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.2e+123)
   (if (>= b 0.0)
     (* (+ b (sqrt (fma (* c -4.0) a (* b b)))) (/ -0.5 a))
     (* c (/ -1.0 b)))
   (if (<= b 1.3e+73)
     (if (>= b 0.0)
       (/ (- (- b) (sqrt (fma (* c a) -4.0 (* b b)))) (* 2.0 a))
       (* c (/ 2.0 (- (sqrt (fma (* a c) -4.0 (* b b))) b))))
     (if (>= b 0.0)
       (/ (* -2.0 b) (* 2.0 a))
       (/ (* 2.0 c) (+ (- b) (* a (sqrt (* -4.0 (/ c a))))))))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1.2e+123) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (b + sqrt(fma((c * -4.0), a, (b * b)))) * (-0.5 / a);
		} else {
			tmp_2 = c * (-1.0 / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.3e+73) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - sqrt(fma((c * a), -4.0, (b * b)))) / (2.0 * a);
		} else {
			tmp_3 = c * (2.0 / (sqrt(fma((a * c), -4.0, (b * b))) - b));
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * b) / (2.0 * a);
	} else {
		tmp_1 = (2.0 * c) / (-b + (a * sqrt((-4.0 * (c / a)))));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -1.2e+123)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(b + sqrt(fma(Float64(c * -4.0), a, Float64(b * b)))) * Float64(-0.5 / a));
		else
			tmp_2 = Float64(c * Float64(-1.0 / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.3e+73)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(-b) - sqrt(fma(Float64(c * a), -4.0, Float64(b * b)))) / Float64(2.0 * a));
		else
			tmp_3 = Float64(c * Float64(2.0 / Float64(sqrt(fma(Float64(a * c), -4.0, Float64(b * b))) - b)));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-2.0 * b) / Float64(2.0 * a));
	else
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + Float64(a * sqrt(Float64(-4.0 * Float64(c / a))))));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.2e+123], If[GreaterEqual[b, 0.0], N[(N[(b + N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.3e+73], If[GreaterEqual[b, 0.0], N[(N[((-b) - N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(c * N[(2.0 / N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + N[(a * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.19999999999999994e123

   1. Initial program 71.4%

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

   3. Applied rewrites 71.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2-rev N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 71.3

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-fma.f64 71.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 71.3

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

   5. Applied rewrites 71.3%

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

   6. Taylor expanded in b around -inf

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

      1. lower-/.f64 69.4

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

   8. Applied rewrites 69.4%

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

   if -1.19999999999999994e123 < b < 1.3e73

   1. Initial program 71.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 71.4

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

   3. Applied rewrites 71.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 71.4

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

   5. Applied rewrites 71.4%

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

   7. Applied rewrites 71.3%

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

   if 1.3e73 < b

   1. Initial program 71.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 48.3

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

   4. Applied rewrites 48.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 26.2

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

   7. Applied rewrites 26.2%

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

   8. Taylor expanded in b around inf

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

      1. lower-*.f64 47.4

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

   10. Applied rewrites 47.4%

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

Alternative 3: 90.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}\\ t_1 := \left(b + t\_0\right) \cdot \frac{-0.5}{a}\\ t_2 := a \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{if}\;b \leq -1.9 \cdot 10^{+123}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-1}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-308}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;0.5 \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+73}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{c}{t\_2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* c -4.0) a (* b b))))
        (t_1 (* (+ b t_0) (/ -0.5 a)))
        (t_2 (* a (sqrt (* -4.0 (/ c a))))))
   (if (<= b -1.9e+123)
     (if (>= b 0.0) t_1 (* c (/ -1.0 b)))
     (if (<= b 2.2e-308)
       (if (>= b 0.0)
         (* 0.5 (* c (sqrt (/ -4.0 (* a c)))))
         (/ (+ c c) (- t_0 b)))
       (if (<= b 1.3e+73)
         (if (>= b 0.0) t_1 (* -2.0 (/ c t_2)))
         (if (>= b 0.0)
           (/ (* -2.0 b) (* 2.0 a))
           (/ (* 2.0 c) (+ (- b) t_2))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((c * -4.0), a, (b * b)));
	double t_1 = (b + t_0) * (-0.5 / a);
	double t_2 = a * sqrt((-4.0 * (c / a)));
	double tmp_1;
	if (b <= -1.9e+123) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = c * (-1.0 / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 2.2e-308) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = 0.5 * (c * sqrt((-4.0 / (a * c))));
		} else {
			tmp_3 = (c + c) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.3e+73) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = t_1;
		} else {
			tmp_4 = -2.0 * (c / t_2);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * b) / (2.0 * a);
	} else {
		tmp_1 = (2.0 * c) / (-b + t_2);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(c * -4.0), a, Float64(b * b)))
	t_1 = Float64(Float64(b + t_0) * Float64(-0.5 / a))
	t_2 = Float64(a * sqrt(Float64(-4.0 * Float64(c / a))))
	tmp_1 = 0.0
	if (b <= -1.9e+123)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(c * Float64(-1.0 / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 2.2e-308)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(0.5 * Float64(c * sqrt(Float64(-4.0 / Float64(a * c)))));
		else
			tmp_3 = Float64(Float64(c + c) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	elseif (b <= 1.3e+73)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = t_1;
		else
			tmp_4 = Float64(-2.0 * Float64(c / t_2));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-2.0 * b) / Float64(2.0 * a));
	else
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_2));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(b + t$95$0), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.9e+123], If[GreaterEqual[b, 0.0], t$95$1, N[(c * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 2.2e-308], If[GreaterEqual[b, 0.0], N[(0.5 * N[(c * N[Sqrt[N[(-4.0 / N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.3e+73], If[GreaterEqual[b, 0.0], t$95$1, N[(-2.0 * N[(c / t$95$2), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.89999999999999997e123

   1. Initial program 71.4%

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

   3. Applied rewrites 71.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2-rev N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 71.3

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-fma.f64 71.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 71.3

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

   5. Applied rewrites 71.3%

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

   6. Taylor expanded in b around -inf

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

      1. lower-/.f64 69.4

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

   8. Applied rewrites 69.4%

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

   if -1.89999999999999997e123 < b < 2.2000000000000002e-308

   1. Initial program 71.4%

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

   3. Applied rewrites 71.4%

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

   4. Taylor expanded in c around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 42.8

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

   6. Applied rewrites 42.8%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 50.5

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

   9. Applied rewrites 50.5%

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

   if 2.2000000000000002e-308 < b < 1.3e73

   1. Initial program 71.4%

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

   3. Applied rewrites 71.4%

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

   4. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 42.3

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

   6. Applied rewrites 42.3%

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

   if 1.3e73 < b

   1. Initial program 71.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 48.3

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

   4. Applied rewrites 48.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 26.2

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

   7. Applied rewrites 26.2%

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

   8. Taylor expanded in b around inf

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

      1. lower-*.f64 47.4

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

   10. Applied rewrites 47.4%

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

Alternative 4: 90.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}\\ t_1 := \left(b + t\_0\right) \cdot \frac{-0.5}{a}\\ \mathbf{if}\;b \leq -1.9 \cdot 10^{+123}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-1}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+73}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + a \cdot \sqrt{-4 \cdot \frac{c}{a}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* c -4.0) a (* b b)))) (t_1 (* (+ b t_0) (/ -0.5 a))))
   (if (<= b -1.9e+123)
     (if (>= b 0.0) t_1 (* c (/ -1.0 b)))
     (if (<= b 1.3e+73)
       (if (>= b 0.0) t_1 (/ (+ c c) (- t_0 b)))
       (if (>= b 0.0)
         (/ (* -2.0 b) (* 2.0 a))
         (/ (* 2.0 c) (+ (- b) (* a (sqrt (* -4.0 (/ c a)))))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((c * -4.0), a, (b * b)));
	double t_1 = (b + t_0) * (-0.5 / a);
	double tmp_1;
	if (b <= -1.9e+123) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = c * (-1.0 / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.3e+73) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = (c + c) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * b) / (2.0 * a);
	} else {
		tmp_1 = (2.0 * c) / (-b + (a * sqrt((-4.0 * (c / a)))));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(c * -4.0), a, Float64(b * b)))
	t_1 = Float64(Float64(b + t_0) * Float64(-0.5 / a))
	tmp_1 = 0.0
	if (b <= -1.9e+123)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(c * Float64(-1.0 / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.3e+73)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = Float64(Float64(c + c) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-2.0 * b) / Float64(2.0 * a));
	else
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + Float64(a * sqrt(Float64(-4.0 * Float64(c / a))))));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(b + t$95$0), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.9e+123], If[GreaterEqual[b, 0.0], t$95$1, N[(c * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.3e+73], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(c + c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + N[(a * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.89999999999999997e123

   1. Initial program 71.4%

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

   3. Applied rewrites 71.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2-rev N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 71.3

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-fma.f64 71.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 71.3

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

   5. Applied rewrites 71.3%

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

   6. Taylor expanded in b around -inf

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

      1. lower-/.f64 69.4

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

   8. Applied rewrites 69.4%

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

   if -1.89999999999999997e123 < b < 1.3e73

   1. Initial program 71.4%

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

   3. Applied rewrites 71.4%

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

   if 1.3e73 < b

   1. Initial program 71.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 48.3

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

   4. Applied rewrites 48.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 26.2

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

   7. Applied rewrites 26.2%

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

   8. Taylor expanded in b around inf

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

      1. lower-*.f64 47.4

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

   10. Applied rewrites 47.4%

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

Alternative 5: 90.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}\\ t_1 := a \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{if}\;b \leq -2.5 \cdot 10^{+145}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-1 \cdot \frac{b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-308}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;0.5 \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+73}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(b + t\_0\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{c}{t\_1}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* c -4.0) a (* b b))))
        (t_1 (* a (sqrt (* -4.0 (/ c a))))))
   (if (<= b -2.5e+145)
     (if (>= b 0.0) (/ (- (- b) t_1) (* 2.0 a)) (/ 1.0 (* -1.0 (/ b c))))
     (if (<= b 2.2e-308)
       (if (>= b 0.0)
         (* 0.5 (* c (sqrt (/ -4.0 (* a c)))))
         (/ (+ c c) (- t_0 b)))
       (if (<= b 1.3e+73)
         (if (>= b 0.0) (* (+ b t_0) (/ -0.5 a)) (* -2.0 (/ c t_1)))
         (if (>= b 0.0)
           (/ (* -2.0 b) (* 2.0 a))
           (/ (* 2.0 c) (+ (- b) t_1))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((c * -4.0), a, (b * b)));
	double t_1 = a * sqrt((-4.0 * (c / a)));
	double tmp_1;
	if (b <= -2.5e+145) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-b - t_1) / (2.0 * a);
		} else {
			tmp_2 = 1.0 / (-1.0 * (b / c));
		}
		tmp_1 = tmp_2;
	} else if (b <= 2.2e-308) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = 0.5 * (c * sqrt((-4.0 / (a * c))));
		} else {
			tmp_3 = (c + c) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.3e+73) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (b + t_0) * (-0.5 / a);
		} else {
			tmp_4 = -2.0 * (c / t_1);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * b) / (2.0 * a);
	} else {
		tmp_1 = (2.0 * c) / (-b + t_1);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(c * -4.0), a, Float64(b * b)))
	t_1 = Float64(a * sqrt(Float64(-4.0 * Float64(c / a))))
	tmp_1 = 0.0
	if (b <= -2.5e+145)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(Float64(-b) - t_1) / Float64(2.0 * a));
		else
			tmp_2 = Float64(1.0 / Float64(-1.0 * Float64(b / c)));
		end
		tmp_1 = tmp_2;
	elseif (b <= 2.2e-308)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(0.5 * Float64(c * sqrt(Float64(-4.0 / Float64(a * c)))));
		else
			tmp_3 = Float64(Float64(c + c) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	elseif (b <= 1.3e+73)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(b + t_0) * Float64(-0.5 / a));
		else
			tmp_4 = Float64(-2.0 * Float64(c / t_1));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-2.0 * b) / Float64(2.0 * a));
	else
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_1));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(a * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.5e+145], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$1), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(-1.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 2.2e-308], If[GreaterEqual[b, 0.0], N[(0.5 * N[(c * N[Sqrt[N[(-4.0 / N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.3e+73], If[GreaterEqual[b, 0.0], N[(N[(b + t$95$0), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(c / t$95$1), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.49999999999999983e145

   1. Initial program 71.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 48.3

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

   4. Applied rewrites 48.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 26.2

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

   7. Applied rewrites 26.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip N/A

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

      4. lower-unsound-/.f64 N/A

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

      5. lower-unsound-/.f64 N/A

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

   9. Applied rewrites 26.2%

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

   10. Taylor expanded in b around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 46.0

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

   12. Applied rewrites 46.0%

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

   if -2.49999999999999983e145 < b < 2.2000000000000002e-308

   1. Initial program 71.4%

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

   3. Applied rewrites 71.4%

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

   4. Taylor expanded in c around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 42.8

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

   6. Applied rewrites 42.8%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 50.5

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

   9. Applied rewrites 50.5%

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

   if 2.2000000000000002e-308 < b < 1.3e73

   1. Initial program 71.4%

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

   3. Applied rewrites 71.4%

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

   4. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 42.3

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

   6. Applied rewrites 42.3%

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

   if 1.3e73 < b

   1. Initial program 71.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 48.3

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

   4. Applied rewrites 48.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 26.2

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

   7. Applied rewrites 26.2%

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

   8. Taylor expanded in b around inf

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

      1. lower-*.f64 47.4

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

   10. Applied rewrites 47.4%

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

Alternative 6: 90.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b + \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{if}\;b \leq -1.2 \cdot 10^{+123}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-1}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+73}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + a \cdot \sqrt{-4 \cdot \frac{c}{a}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (+ b (sqrt (fma (* c -4.0) a (* b b)))) (/ -0.5 a))))
   (if (<= b -1.2e+123)
     (if (>= b 0.0) t_0 (* c (/ -1.0 b)))
     (if (<= b 1.3e+73)
       (if (>= b 0.0)
         t_0
         (* c (/ 2.0 (- (sqrt (fma (* a c) -4.0 (* b b))) b))))
       (if (>= b 0.0)
         (/ (* -2.0 b) (* 2.0 a))
         (/ (* 2.0 c) (+ (- b) (* a (sqrt (* -4.0 (/ c a)))))))))))

C

double code(double a, double b, double c) {
	double t_0 = (b + sqrt(fma((c * -4.0), a, (b * b)))) * (-0.5 / a);
	double tmp_1;
	if (b <= -1.2e+123) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = c * (-1.0 / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.3e+73) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_0;
		} else {
			tmp_3 = c * (2.0 / (sqrt(fma((a * c), -4.0, (b * b))) - b));
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * b) / (2.0 * a);
	} else {
		tmp_1 = (2.0 * c) / (-b + (a * sqrt((-4.0 * (c / a)))));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b + sqrt(fma(Float64(c * -4.0), a, Float64(b * b)))) * Float64(-0.5 / a))
	tmp_1 = 0.0
	if (b <= -1.2e+123)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(c * Float64(-1.0 / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.3e+73)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_0;
		else
			tmp_3 = Float64(c * Float64(2.0 / Float64(sqrt(fma(Float64(a * c), -4.0, Float64(b * b))) - b)));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-2.0 * b) / Float64(2.0 * a));
	else
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + Float64(a * sqrt(Float64(-4.0 * Float64(c / a))))));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b + N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.2e+123], If[GreaterEqual[b, 0.0], t$95$0, N[(c * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.3e+73], If[GreaterEqual[b, 0.0], t$95$0, N[(c * N[(2.0 / N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + N[(a * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.19999999999999994e123

   1. Initial program 71.4%

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

   3. Applied rewrites 71.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2-rev N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 71.3

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-fma.f64 71.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 71.3

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

   5. Applied rewrites 71.3%

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

   6. Taylor expanded in b around -inf

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

      1. lower-/.f64 69.4

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

   8. Applied rewrites 69.4%

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

   if -1.19999999999999994e123 < b < 1.3e73

   1. Initial program 71.4%

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

   3. Applied rewrites 71.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2-rev N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 71.3

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-fma.f64 71.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 71.3

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

   5. Applied rewrites 71.3%

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

   if 1.3e73 < b

   1. Initial program 71.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 48.3

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

   4. Applied rewrites 48.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 26.2

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

   7. Applied rewrites 26.2%

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

   8. Taylor expanded in b around inf

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

      1. lower-*.f64 47.4

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

   10. Applied rewrites 47.4%

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

Alternative 7: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{if}\;b \leq -2.5 \cdot 10^{+145}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_0}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-1 \cdot \frac{b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-9}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(b + \sqrt{-4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (sqrt (* -4.0 (/ c a))))))
   (if (<= b -2.5e+145)
     (if (>= b 0.0) (/ (- (- b) t_0) (* 2.0 a)) (/ 1.0 (* -1.0 (/ b c))))
     (if (<= b 1.2e-9)
       (if (>= b 0.0)
         (* (+ b (sqrt (* -4.0 (* a c)))) (/ -0.5 a))
         (* c (/ 2.0 (- (sqrt (fma (* a c) -4.0 (* b b))) b))))
       (if (>= b 0.0) (/ (* -2.0 b) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) t_0)))))))

C

double code(double a, double b, double c) {
	double t_0 = a * sqrt((-4.0 * (c / a)));
	double tmp_1;
	if (b <= -2.5e+145) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-b - t_0) / (2.0 * a);
		} else {
			tmp_2 = 1.0 / (-1.0 * (b / c));
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.2e-9) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (b + sqrt((-4.0 * (a * c)))) * (-0.5 / a);
		} else {
			tmp_3 = c * (2.0 / (sqrt(fma((a * c), -4.0, (b * b))) - b));
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * b) / (2.0 * a);
	} else {
		tmp_1 = (2.0 * c) / (-b + t_0);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(a * sqrt(Float64(-4.0 * Float64(c / a))))
	tmp_1 = 0.0
	if (b <= -2.5e+145)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(Float64(-b) - t_0) / Float64(2.0 * a));
		else
			tmp_2 = Float64(1.0 / Float64(-1.0 * Float64(b / c)));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.2e-9)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(b + sqrt(Float64(-4.0 * Float64(a * c)))) * Float64(-0.5 / a));
		else
			tmp_3 = Float64(c * Float64(2.0 / Float64(sqrt(fma(Float64(a * c), -4.0, Float64(b * b))) - b)));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-2.0 * b) / Float64(2.0 * a));
	else
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_0));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.5e+145], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(-1.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.2e-9], If[GreaterEqual[b, 0.0], N[(N[(b + N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(2.0 / N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.49999999999999983e145

   1. Initial program 71.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 48.3

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

   4. Applied rewrites 48.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 26.2

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

   7. Applied rewrites 26.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip N/A

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

      4. lower-unsound-/.f64 N/A

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

      5. lower-unsound-/.f64 N/A

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

   9. Applied rewrites 26.2%

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

   10. Taylor expanded in b around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 46.0

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

   12. Applied rewrites 46.0%

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

   if -2.49999999999999983e145 < b < 1.2e-9

   1. Initial program 71.4%

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

   3. Applied rewrites 71.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2-rev N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 71.3

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-fma.f64 71.3

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 71.3

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

   5. Applied rewrites 71.3%

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

   6. Taylor expanded in b around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 56.5

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

   8. Applied rewrites 56.5%

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

   if 1.2e-9 < b

   1. Initial program 71.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 48.3

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

   4. Applied rewrites 48.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 26.2

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

   7. Applied rewrites 26.2%

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

   8. Taylor expanded in b around inf

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

      1. lower-*.f64 47.4

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

   10. Applied rewrites 47.4%

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

Alternative 8: 84.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{if}\;b \leq -2.5 \cdot 10^{+145}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_0}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-1 \cdot \frac{b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-80}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;0.5 \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (sqrt (* -4.0 (/ c a))))))
   (if (<= b -2.5e+145)
     (if (>= b 0.0) (/ (- (- b) t_0) (* 2.0 a)) (/ 1.0 (* -1.0 (/ b c))))
     (if (<= b 8.6e-80)
       (if (>= b 0.0)
         (* 0.5 (* c (sqrt (/ -4.0 (* a c)))))
         (/ (+ c c) (- (sqrt (fma (* c -4.0) a (* b b))) b)))
       (if (>= b 0.0) (/ (* -2.0 b) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) t_0)))))))

C

double code(double a, double b, double c) {
	double t_0 = a * sqrt((-4.0 * (c / a)));
	double tmp_1;
	if (b <= -2.5e+145) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-b - t_0) / (2.0 * a);
		} else {
			tmp_2 = 1.0 / (-1.0 * (b / c));
		}
		tmp_1 = tmp_2;
	} else if (b <= 8.6e-80) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = 0.5 * (c * sqrt((-4.0 / (a * c))));
		} else {
			tmp_3 = (c + c) / (sqrt(fma((c * -4.0), a, (b * b))) - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * b) / (2.0 * a);
	} else {
		tmp_1 = (2.0 * c) / (-b + t_0);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(a * sqrt(Float64(-4.0 * Float64(c / a))))
	tmp_1 = 0.0
	if (b <= -2.5e+145)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(Float64(-b) - t_0) / Float64(2.0 * a));
		else
			tmp_2 = Float64(1.0 / Float64(-1.0 * Float64(b / c)));
		end
		tmp_1 = tmp_2;
	elseif (b <= 8.6e-80)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(0.5 * Float64(c * sqrt(Float64(-4.0 / Float64(a * c)))));
		else
			tmp_3 = Float64(Float64(c + c) / Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-2.0 * b) / Float64(2.0 * a));
	else
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_0));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.5e+145], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(-1.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 8.6e-80], If[GreaterEqual[b, 0.0], N[(0.5 * N[(c * N[Sqrt[N[(-4.0 / N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.49999999999999983e145

   1. Initial program 71.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 48.3

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

   4. Applied rewrites 48.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 26.2

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

   7. Applied rewrites 26.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip N/A

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

      4. lower-unsound-/.f64 N/A

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

      5. lower-unsound-/.f64 N/A

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

   9. Applied rewrites 26.2%

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

   10. Taylor expanded in b around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 46.0

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

   12. Applied rewrites 46.0%

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

   if -2.49999999999999983e145 < b < 8.6000000000000002e-80

   1. Initial program 71.4%

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

   3. Applied rewrites 71.4%

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

   4. Taylor expanded in c around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 42.8

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

   6. Applied rewrites 42.8%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 50.5

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

   9. Applied rewrites 50.5%

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

   if 8.6000000000000002e-80 < b

   1. Initial program 71.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 48.3

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

   4. Applied rewrites 48.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 26.2

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

   7. Applied rewrites 26.2%

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

   8. Taylor expanded in b around inf

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

      1. lower-*.f64 47.4

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

   10. Applied rewrites 47.4%

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

Alternative 9: 80.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left|\left(a \cdot c\right) \cdot -4\right|}\\ t_1 := a \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{if}\;b \leq -6 \cdot 10^{-99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-1 \cdot \frac{b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-9}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + t\_0}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fabs (* (* a c) -4.0))))
        (t_1 (* a (sqrt (* -4.0 (/ c a))))))
   (if (<= b -6e-99)
     (if (>= b 0.0) (/ (- (- b) t_1) (* 2.0 a)) (/ 1.0 (* -1.0 (/ b c))))
     (if (<= b 1.2e-9)
       (if (>= b 0.0) (/ (+ b t_0) (* -2.0 a)) (/ (+ c c) (- t_0 b)))
       (if (>= b 0.0) (/ (* -2.0 b) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) t_1)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fabs(((a * c) * -4.0)));
	double t_1 = a * sqrt((-4.0 * (c / a)));
	double tmp_1;
	if (b <= -6e-99) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-b - t_1) / (2.0 * a);
		} else {
			tmp_2 = 1.0 / (-1.0 * (b / c));
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.2e-9) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (b + t_0) / (-2.0 * a);
		} else {
			tmp_3 = (c + c) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * b) / (2.0 * a);
	} else {
		tmp_1 = (2.0 * c) / (-b + t_1);
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = sqrt(abs(((a * c) * (-4.0d0))))
    t_1 = a * sqrt(((-4.0d0) * (c / a)))
    if (b <= (-6d-99)) then
        if (b >= 0.0d0) then
            tmp_2 = (-b - t_1) / (2.0d0 * a)
        else
            tmp_2 = 1.0d0 / ((-1.0d0) * (b / c))
        end if
        tmp_1 = tmp_2
    else if (b <= 1.2d-9) then
        if (b >= 0.0d0) then
            tmp_3 = (b + t_0) / ((-2.0d0) * a)
        else
            tmp_3 = (c + c) / (t_0 - b)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = ((-2.0d0) * b) / (2.0d0 * a)
    else
        tmp_1 = (2.0d0 * c) / (-b + t_1)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(Math.abs(((a * c) * -4.0)));
	double t_1 = a * Math.sqrt((-4.0 * (c / a)));
	double tmp_1;
	if (b <= -6e-99) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-b - t_1) / (2.0 * a);
		} else {
			tmp_2 = 1.0 / (-1.0 * (b / c));
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.2e-9) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (b + t_0) / (-2.0 * a);
		} else {
			tmp_3 = (c + c) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (-2.0 * b) / (2.0 * a);
	} else {
		tmp_1 = (2.0 * c) / (-b + t_1);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(math.fabs(((a * c) * -4.0)))
	t_1 = a * math.sqrt((-4.0 * (c / a)))
	tmp_1 = 0
	if b <= -6e-99:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (-b - t_1) / (2.0 * a)
		else:
			tmp_2 = 1.0 / (-1.0 * (b / c))
		tmp_1 = tmp_2
	elif b <= 1.2e-9:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (b + t_0) / (-2.0 * a)
		else:
			tmp_3 = (c + c) / (t_0 - b)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = (-2.0 * b) / (2.0 * a)
	else:
		tmp_1 = (2.0 * c) / (-b + t_1)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(abs(Float64(Float64(a * c) * -4.0)))
	t_1 = Float64(a * sqrt(Float64(-4.0 * Float64(c / a))))
	tmp_1 = 0.0
	if (b <= -6e-99)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(Float64(-b) - t_1) / Float64(2.0 * a));
		else
			tmp_2 = Float64(1.0 / Float64(-1.0 * Float64(b / c)));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.2e-9)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(b + t_0) / Float64(-2.0 * a));
		else
			tmp_3 = Float64(Float64(c + c) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-2.0 * b) / Float64(2.0 * a));
	else
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_1));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(abs(((a * c) * -4.0)));
	t_1 = a * sqrt((-4.0 * (c / a)));
	tmp_2 = 0.0;
	if (b <= -6e-99)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (-b - t_1) / (2.0 * a);
		else
			tmp_3 = 1.0 / (-1.0 * (b / c));
		end
		tmp_2 = tmp_3;
	elseif (b <= 1.2e-9)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (b + t_0) / (-2.0 * a);
		else
			tmp_4 = (c + c) / (t_0 - b);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = (-2.0 * b) / (2.0 * a);
	else
		tmp_2 = (2.0 * c) / (-b + t_1);
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[Abs[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(a * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -6e-99], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$1), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(-1.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.2e-9], If[GreaterEqual[b, 0.0], N[(N[(b + t$95$0), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.00000000000000012e-99

   1. Initial program 71.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 48.3

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

   4. Applied rewrites 48.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 26.2

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

   7. Applied rewrites 26.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip N/A

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

      4. lower-unsound-/.f64 N/A

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

      5. lower-unsound-/.f64 N/A

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

   9. Applied rewrites 26.2%

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

   10. Taylor expanded in b around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 46.0

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

   12. Applied rewrites 46.0%

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

   if -6.00000000000000012e-99 < b < 1.2e-9

   1. Initial program 71.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.6

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

   4. Applied rewrites 56.6%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 40.6

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

   7. Applied rewrites 40.6%

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

   9. Applied rewrites 40.6%

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

       1. rem-square-sqrt N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. sqr-abs-rev N/A

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

       5. mul-fabs N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. rem-square-sqrt N/A

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

       9. lower-fabs.f64 45.3

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

   11. Applied rewrites 45.3%

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

       1. rem-square-sqrt N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. sqr-abs-rev N/A

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

       5. mul-fabs N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. rem-square-sqrt N/A

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

       9. lower-fabs.f64 50.0

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

   13. Applied rewrites 50.0%

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

   if 1.2e-9 < b

   1. Initial program 71.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 48.3

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

   4. Applied rewrites 48.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 26.2

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

   7. Applied rewrites 26.2%

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

   8. Taylor expanded in b around inf

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

      1. lower-*.f64 47.4

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

   10. Applied rewrites 47.4%

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

Alternative 10: 66.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left|\left(a \cdot c\right) \cdot -4\right|}\\ \mathbf{if}\;b \leq -6 \cdot 10^{-99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - a \cdot \sqrt{-4 \cdot \frac{c}{a}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-1 \cdot \frac{b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b + t\_0}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0 - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fabs (* (* a c) -4.0)))))
   (if (<= b -6e-99)
     (if (>= b 0.0)
       (/ (- (- b) (* a (sqrt (* -4.0 (/ c a))))) (* 2.0 a))
       (/ 1.0 (* -1.0 (/ b c))))
     (if (>= b 0.0) (/ (+ b t_0) (* -2.0 a)) (/ (+ c c) (- t_0 b))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fabs(((a * c) * -4.0)));
	double tmp_1;
	if (b <= -6e-99) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-b - (a * sqrt((-4.0 * (c / a))))) / (2.0 * a);
		} else {
			tmp_2 = 1.0 / (-1.0 * (b / c));
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (b + t_0) / (-2.0 * a);
	} else {
		tmp_1 = (c + c) / (t_0 - b);
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = sqrt(abs(((a * c) * (-4.0d0))))
    if (b <= (-6d-99)) then
        if (b >= 0.0d0) then
            tmp_2 = (-b - (a * sqrt(((-4.0d0) * (c / a))))) / (2.0d0 * a)
        else
            tmp_2 = 1.0d0 / ((-1.0d0) * (b / c))
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (b + t_0) / ((-2.0d0) * a)
    else
        tmp_1 = (c + c) / (t_0 - b)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(Math.abs(((a * c) * -4.0)));
	double tmp_1;
	if (b <= -6e-99) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-b - (a * Math.sqrt((-4.0 * (c / a))))) / (2.0 * a);
		} else {
			tmp_2 = 1.0 / (-1.0 * (b / c));
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (b + t_0) / (-2.0 * a);
	} else {
		tmp_1 = (c + c) / (t_0 - b);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(math.fabs(((a * c) * -4.0)))
	tmp_1 = 0
	if b <= -6e-99:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (-b - (a * math.sqrt((-4.0 * (c / a))))) / (2.0 * a)
		else:
			tmp_2 = 1.0 / (-1.0 * (b / c))
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = (b + t_0) / (-2.0 * a)
	else:
		tmp_1 = (c + c) / (t_0 - b)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(abs(Float64(Float64(a * c) * -4.0)))
	tmp_1 = 0.0
	if (b <= -6e-99)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(Float64(-b) - Float64(a * sqrt(Float64(-4.0 * Float64(c / a))))) / Float64(2.0 * a));
		else
			tmp_2 = Float64(1.0 / Float64(-1.0 * Float64(b / c)));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(b + t_0) / Float64(-2.0 * a));
	else
		tmp_1 = Float64(Float64(c + c) / Float64(t_0 - b));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = sqrt(abs(((a * c) * -4.0)));
	tmp_2 = 0.0;
	if (b <= -6e-99)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (-b - (a * sqrt((-4.0 * (c / a))))) / (2.0 * a);
		else
			tmp_3 = 1.0 / (-1.0 * (b / c));
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = (b + t_0) / (-2.0 * a);
	else
		tmp_2 = (c + c) / (t_0 - b);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[Abs[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -6e-99], If[GreaterEqual[b, 0.0], N[(N[((-b) - N[(a * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(-1.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(b + t$95$0), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < -6.00000000000000012e-99

   1. Initial program 71.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 48.3

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

   4. Applied rewrites 48.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 26.2

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

   7. Applied rewrites 26.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip N/A

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

      4. lower-unsound-/.f64 N/A

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

      5. lower-unsound-/.f64 N/A

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

   9. Applied rewrites 26.2%

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

   10. Taylor expanded in b around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 46.0

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

   12. Applied rewrites 46.0%

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

   if -6.00000000000000012e-99 < b

   1. Initial program 71.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.6

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

   4. Applied rewrites 56.6%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 40.6

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

   7. Applied rewrites 40.6%

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

   9. Applied rewrites 40.6%

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

       1. rem-square-sqrt N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. sqr-abs-rev N/A

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

       5. mul-fabs N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. rem-square-sqrt N/A

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

       9. lower-fabs.f64 45.3

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

   11. Applied rewrites 45.3%

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

       1. rem-square-sqrt N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. sqr-abs-rev N/A

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

       5. mul-fabs N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. rem-square-sqrt N/A

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

       9. lower-fabs.f64 50.0

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

   13. Applied rewrites 50.0%

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

Alternative 11: 50.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left|\left(a \cdot c\right) \cdot -4\right|}\\ \mathbf{if}\;c \leq -1.45 \cdot 10^{+135}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - a \cdot \sqrt{-4 \cdot \frac{c}{a}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{\frac{c}{a} \cdot -4} \cdot a - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b + t\_0}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0 - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fabs (* (* a c) -4.0)))))
   (if (<= c -1.45e+135)
     (if (>= b 0.0)
       (/ (- (- b) (* a (sqrt (* -4.0 (/ c a))))) (* 2.0 a))
       (* c (/ 2.0 (- (* (sqrt (* (/ c a) -4.0)) a) b))))
     (if (>= b 0.0) (/ (+ b t_0) (* -2.0 a)) (/ (+ c c) (- t_0 b))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fabs(((a * c) * -4.0)));
	double tmp_1;
	if (c <= -1.45e+135) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-b - (a * sqrt((-4.0 * (c / a))))) / (2.0 * a);
		} else {
			tmp_2 = c * (2.0 / ((sqrt(((c / a) * -4.0)) * a) - b));
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (b + t_0) / (-2.0 * a);
	} else {
		tmp_1 = (c + c) / (t_0 - b);
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = sqrt(abs(((a * c) * (-4.0d0))))
    if (c <= (-1.45d+135)) then
        if (b >= 0.0d0) then
            tmp_2 = (-b - (a * sqrt(((-4.0d0) * (c / a))))) / (2.0d0 * a)
        else
            tmp_2 = c * (2.0d0 / ((sqrt(((c / a) * (-4.0d0))) * a) - b))
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (b + t_0) / ((-2.0d0) * a)
    else
        tmp_1 = (c + c) / (t_0 - b)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(Math.abs(((a * c) * -4.0)));
	double tmp_1;
	if (c <= -1.45e+135) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-b - (a * Math.sqrt((-4.0 * (c / a))))) / (2.0 * a);
		} else {
			tmp_2 = c * (2.0 / ((Math.sqrt(((c / a) * -4.0)) * a) - b));
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (b + t_0) / (-2.0 * a);
	} else {
		tmp_1 = (c + c) / (t_0 - b);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(math.fabs(((a * c) * -4.0)))
	tmp_1 = 0
	if c <= -1.45e+135:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (-b - (a * math.sqrt((-4.0 * (c / a))))) / (2.0 * a)
		else:
			tmp_2 = c * (2.0 / ((math.sqrt(((c / a) * -4.0)) * a) - b))
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = (b + t_0) / (-2.0 * a)
	else:
		tmp_1 = (c + c) / (t_0 - b)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(abs(Float64(Float64(a * c) * -4.0)))
	tmp_1 = 0.0
	if (c <= -1.45e+135)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(Float64(-b) - Float64(a * sqrt(Float64(-4.0 * Float64(c / a))))) / Float64(2.0 * a));
		else
			tmp_2 = Float64(c * Float64(2.0 / Float64(Float64(sqrt(Float64(Float64(c / a) * -4.0)) * a) - b)));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(b + t_0) / Float64(-2.0 * a));
	else
		tmp_1 = Float64(Float64(c + c) / Float64(t_0 - b));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = sqrt(abs(((a * c) * -4.0)));
	tmp_2 = 0.0;
	if (c <= -1.45e+135)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (-b - (a * sqrt((-4.0 * (c / a))))) / (2.0 * a);
		else
			tmp_3 = c * (2.0 / ((sqrt(((c / a) * -4.0)) * a) - b));
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = (b + t_0) / (-2.0 * a);
	else
		tmp_2 = (c + c) / (t_0 - b);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[Abs[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[c, -1.45e+135], If[GreaterEqual[b, 0.0], N[(N[((-b) - N[(a * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(c * N[(2.0 / N[(N[(N[Sqrt[N[(N[(c / a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(b + t$95$0), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c < -1.4499999999999999e135

   1. Initial program 71.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 48.3

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

   4. Applied rewrites 48.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 26.2

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

   7. Applied rewrites 26.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 26.2

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-neg.f64 N/A

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

      10. sub-flip-reverse N/A

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

   9. Applied rewrites 26.2%

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

   if -1.4499999999999999e135 < c

   1. Initial program 71.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.6

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

   4. Applied rewrites 56.6%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 40.6

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

   7. Applied rewrites 40.6%

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

   9. Applied rewrites 40.6%

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

       1. rem-square-sqrt N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. sqr-abs-rev N/A

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

       5. mul-fabs N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. rem-square-sqrt N/A

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

       9. lower-fabs.f64 45.3

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

   11. Applied rewrites 45.3%

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

       1. rem-square-sqrt N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. sqr-abs-rev N/A

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

       5. mul-fabs N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. rem-square-sqrt N/A

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

       9. lower-fabs.f64 50.0

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

   13. Applied rewrites 50.0%

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

Alternative 12: 50.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left|\left(a \cdot c\right) \cdot -4\right|}\\ t_1 := \sqrt{\frac{c}{a} \cdot -4} \cdot a\\ \mathbf{if}\;c \leq -1.45 \cdot 10^{+135}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(t\_1 + b\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{t\_1 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b + t\_0}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0 - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fabs (* (* a c) -4.0))))
        (t_1 (* (sqrt (* (/ c a) -4.0)) a)))
   (if (<= c -1.45e+135)
     (if (>= b 0.0) (* (+ t_1 b) (/ -0.5 a)) (* 2.0 (/ c (- t_1 b))))
     (if (>= b 0.0) (/ (+ b t_0) (* -2.0 a)) (/ (+ c c) (- t_0 b))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fabs(((a * c) * -4.0)));
	double t_1 = sqrt(((c / a) * -4.0)) * a;
	double tmp_1;
	if (c <= -1.45e+135) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (t_1 + b) * (-0.5 / a);
		} else {
			tmp_2 = 2.0 * (c / (t_1 - b));
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (b + t_0) / (-2.0 * a);
	} else {
		tmp_1 = (c + c) / (t_0 - b);
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = sqrt(abs(((a * c) * (-4.0d0))))
    t_1 = sqrt(((c / a) * (-4.0d0))) * a
    if (c <= (-1.45d+135)) then
        if (b >= 0.0d0) then
            tmp_2 = (t_1 + b) * ((-0.5d0) / a)
        else
            tmp_2 = 2.0d0 * (c / (t_1 - b))
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (b + t_0) / ((-2.0d0) * a)
    else
        tmp_1 = (c + c) / (t_0 - b)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(Math.abs(((a * c) * -4.0)));
	double t_1 = Math.sqrt(((c / a) * -4.0)) * a;
	double tmp_1;
	if (c <= -1.45e+135) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (t_1 + b) * (-0.5 / a);
		} else {
			tmp_2 = 2.0 * (c / (t_1 - b));
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (b + t_0) / (-2.0 * a);
	} else {
		tmp_1 = (c + c) / (t_0 - b);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(math.fabs(((a * c) * -4.0)))
	t_1 = math.sqrt(((c / a) * -4.0)) * a
	tmp_1 = 0
	if c <= -1.45e+135:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (t_1 + b) * (-0.5 / a)
		else:
			tmp_2 = 2.0 * (c / (t_1 - b))
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = (b + t_0) / (-2.0 * a)
	else:
		tmp_1 = (c + c) / (t_0 - b)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(abs(Float64(Float64(a * c) * -4.0)))
	t_1 = Float64(sqrt(Float64(Float64(c / a) * -4.0)) * a)
	tmp_1 = 0.0
	if (c <= -1.45e+135)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(t_1 + b) * Float64(-0.5 / a));
		else
			tmp_2 = Float64(2.0 * Float64(c / Float64(t_1 - b)));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(b + t_0) / Float64(-2.0 * a));
	else
		tmp_1 = Float64(Float64(c + c) / Float64(t_0 - b));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = sqrt(abs(((a * c) * -4.0)));
	t_1 = sqrt(((c / a) * -4.0)) * a;
	tmp_2 = 0.0;
	if (c <= -1.45e+135)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (t_1 + b) * (-0.5 / a);
		else
			tmp_3 = 2.0 * (c / (t_1 - b));
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = (b + t_0) / (-2.0 * a);
	else
		tmp_2 = (c + c) / (t_0 - b);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[Abs[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(c / a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[c, -1.45e+135], If[GreaterEqual[b, 0.0], N[(N[(t$95$1 + b), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(c / N[(t$95$1 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(b + t$95$0), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if c < -1.4499999999999999e135

   1. Initial program 71.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 48.3

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

   4. Applied rewrites 48.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 26.2

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

   7. Applied rewrites 26.2%

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

   9. Applied rewrites 26.2%

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

       1. lift-/.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. div-add N/A

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

       4. count-2 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 26.2

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

   11. Applied rewrites 26.2%

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

   if -1.4499999999999999e135 < c

   1. Initial program 71.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.6

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

   4. Applied rewrites 56.6%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 40.6

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

   7. Applied rewrites 40.6%

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

   9. Applied rewrites 40.6%

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

       1. rem-square-sqrt N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. sqr-abs-rev N/A

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

       5. mul-fabs N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. rem-square-sqrt N/A

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

       9. lower-fabs.f64 45.3

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

   11. Applied rewrites 45.3%

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

       1. rem-square-sqrt N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. sqr-abs-rev N/A

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

       5. mul-fabs N/A

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

       6. lift-sqrt.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. rem-square-sqrt N/A

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

       9. lower-fabs.f64 50.0

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

   13. Applied rewrites 50.0%

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

Alternative 13: 50.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.4%

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

2. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 56.6

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

4. Applied rewrites 56.6%

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

5. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 40.6

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

7. Applied rewrites 40.6%

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

9. Applied rewrites 40.6%

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

    1. rem-square-sqrt N/A

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

    2. lift-sqrt.f64 N/A

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

    3. lift-sqrt.f64 N/A

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

    4. sqr-abs-rev N/A

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

    5. mul-fabs N/A

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

    6. lift-sqrt.f64 N/A

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

    7. lift-sqrt.f64 N/A

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

    8. rem-square-sqrt N/A

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

    9. lower-fabs.f64 45.3

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

11. Applied rewrites 45.3%

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

    1. rem-square-sqrt N/A

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

    2. lift-sqrt.f64 N/A

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

    3. lift-sqrt.f64 N/A

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

    4. sqr-abs-rev N/A

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

    5. mul-fabs N/A

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

    6. lift-sqrt.f64 N/A

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

    7. lift-sqrt.f64 N/A

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

    8. rem-square-sqrt N/A

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

    9. lower-fabs.f64 50.0

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

13. Applied rewrites 50.0%

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

Alternative 14: 45.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot c\right) \cdot -4\\ \mathbf{if}\;b \leq -4.3 \cdot 10^{-268}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left|t\_0\right|} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot c\right) \cdot a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{t\_0} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* a c) -4.0)))
   (if (<= b -4.3e-268)
     (if (>= b 0.0)
       (/ (sqrt (- (* 4.0 (* a c)))) (* -2.0 a))
       (/ (+ c c) (- (sqrt (fabs t_0)) b)))
     (if (>= b 0.0)
       (/ (+ b (sqrt (* (* -4.0 c) a))) (* -2.0 a))
       (* c (/ 2.0 (- (sqrt t_0) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = (a * c) * -4.0;
	double tmp_1;
	if (b <= -4.3e-268) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = sqrt(-(4.0 * (a * c))) / (-2.0 * a);
		} else {
			tmp_2 = (c + c) / (sqrt(fabs(t_0)) - b);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (b + sqrt(((-4.0 * c) * a))) / (-2.0 * a);
	} else {
		tmp_1 = c * (2.0 / (sqrt(t_0) - b));
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = (a * c) * (-4.0d0)
    if (b <= (-4.3d-268)) then
        if (b >= 0.0d0) then
            tmp_2 = sqrt(-(4.0d0 * (a * c))) / ((-2.0d0) * a)
        else
            tmp_2 = (c + c) / (sqrt(abs(t_0)) - b)
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (b + sqrt((((-4.0d0) * c) * a))) / ((-2.0d0) * a)
    else
        tmp_1 = c * (2.0d0 / (sqrt(t_0) - b))
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (a * c) * -4.0;
	double tmp_1;
	if (b <= -4.3e-268) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = Math.sqrt(-(4.0 * (a * c))) / (-2.0 * a);
		} else {
			tmp_2 = (c + c) / (Math.sqrt(Math.abs(t_0)) - b);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (b + Math.sqrt(((-4.0 * c) * a))) / (-2.0 * a);
	} else {
		tmp_1 = c * (2.0 / (Math.sqrt(t_0) - b));
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (a * c) * -4.0
	tmp_1 = 0
	if b <= -4.3e-268:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = math.sqrt(-(4.0 * (a * c))) / (-2.0 * a)
		else:
			tmp_2 = (c + c) / (math.sqrt(math.fabs(t_0)) - b)
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = (b + math.sqrt(((-4.0 * c) * a))) / (-2.0 * a)
	else:
		tmp_1 = c * (2.0 / (math.sqrt(t_0) - b))
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(a * c) * -4.0)
	tmp_1 = 0.0
	if (b <= -4.3e-268)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(sqrt(Float64(-Float64(4.0 * Float64(a * c)))) / Float64(-2.0 * a));
		else
			tmp_2 = Float64(Float64(c + c) / Float64(sqrt(abs(t_0)) - b));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(b + sqrt(Float64(Float64(-4.0 * c) * a))) / Float64(-2.0 * a));
	else
		tmp_1 = Float64(c * Float64(2.0 / Float64(sqrt(t_0) - b)));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = (a * c) * -4.0;
	tmp_2 = 0.0;
	if (b <= -4.3e-268)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = sqrt(-(4.0 * (a * c))) / (-2.0 * a);
		else
			tmp_3 = (c + c) / (sqrt(abs(t_0)) - b);
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = (b + sqrt(((-4.0 * c) * a))) / (-2.0 * a);
	else
		tmp_2 = c * (2.0 / (sqrt(t_0) - b));
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]}, If[LessEqual[b, -4.3e-268], If[GreaterEqual[b, 0.0], N[(N[Sqrt[(-N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[Abs[t$95$0], $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(b + N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[(c * N[(2.0 / N[(N[Sqrt[t$95$0], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.3e-268

   1. Initial program 71.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.6

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

   4. Applied rewrites 56.6%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 40.6

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

   7. Applied rewrites 40.6%

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

   9. Applied rewrites 40.6%

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

   10. Taylor expanded in b around 0

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

       1. lower-sqrt.f64 N/A

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

       2. lower-neg.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 33.4

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

   12. Applied rewrites 33.4%

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

       1. rem-square-sqrt N/A

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

       2. lift-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\sqrt{\left(-4 \cdot c\right) \cdot a} \cdot \sqrt{\left(-4 \cdot c\right) \cdot a}} - b}\\ \end{array} \]

       3. lift-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\sqrt{\left(-4 \cdot c\right) \cdot a} \cdot \sqrt{\left(-4 \cdot c\right) \cdot a}} - b}\\ \end{array} \]

       4. sqr-abs-rev N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{c} + c}{\sqrt{\left|\sqrt{\left(-4 \cdot c\right) \cdot a}\right| \cdot \left|\sqrt{\left(-4 \cdot c\right) \cdot a}\right|} - b}\\ \end{array} \]

       5. mul-fabs N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{c} + c}{\sqrt{\left|\sqrt{\left(-4 \cdot c\right) \cdot a} \cdot \sqrt{\left(-4 \cdot c\right) \cdot a}\right|} - b}\\ \end{array} \]

       6. lift-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left|\sqrt{\left(-4 \cdot c\right) \cdot a} \cdot \sqrt{\left(-4 \cdot c\right) \cdot a}\right|} - b}\\ \end{array} \]

       7. lift-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left|\sqrt{\left(-4 \cdot c\right) \cdot a} \cdot \sqrt{\left(-4 \cdot c\right) \cdot a}\right|} - b}\\ \end{array} \]

       8. rem-square-sqrt N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left|\left(-4 \cdot c\right) \cdot a\right|} - b}\\ \end{array} \]

       9. lower-fabs.f64 38.1

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{c} + c}{\sqrt{\left|\left(-4 \cdot c\right) \cdot a\right|} - b}\\ \end{array} \]

   14. Applied rewrites 38.1%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{c} + c}{\sqrt{\left|\left(a \cdot c\right) \cdot -4\right|} - b}\\ \end{array} \]

   if -4.3e-268 < b

   1. Initial program 71.4%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      2. lower-*.f64 56.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   4. Applied rewrites 56.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]

      2. lower-*.f64 40.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]

   7. Applied rewrites 40.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 40.6%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot c\right) \cdot a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ } \end{array}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot c\right) \cdot a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       2. lift-+.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot c\right) \cdot a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       3. count-2-rev N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot c\right) \cdot a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       4. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot c\right) \cdot a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       5. associate-/l* N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot c\right) \cdot a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       6. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot c\right) \cdot a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       7. lower-/.f64 40.5

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot c\right) \cdot a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{c \cdot \frac{2}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}}\\ \end{array} \]

       8. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot c\right) \cdot a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       9. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot c\right) \cdot a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       10. associate-*l* N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot c\right) \cdot a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{-4 \cdot \left(c \cdot a\right)} - b}\\ \end{array} \]

       11. lift-*.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot c\right) \cdot a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{-4 \cdot \left(c \cdot a\right)} - b}\\ \end{array} \]

       12. *-commutative N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot c\right) \cdot a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{\left(c \cdot a\right) \cdot -4} - b}\\ \end{array} \]

       13. lower-*.f64 40.6

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot c\right) \cdot a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{\left(c \cdot a\right) \cdot -4} - b}\\ \end{array} \]

       14. lift-*.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot c\right) \cdot a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{\left(c \cdot a\right) \cdot -4} - b}\\ \end{array} \]

       15. *-commutative N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot c\right) \cdot a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{\left(a \cdot c\right) \cdot -4} - b}\\ \end{array} \]

       16. lower-*.f64 40.6

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot c\right) \cdot a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{\left(a \cdot c\right) \cdot -4} - b}\\ \end{array} \]

   11. Applied rewrites 40.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot c\right) \cdot a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\sqrt{\left(a \cdot c\right) \cdot -4} - b}\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 40.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left(-4 \cdot c\right) \cdot a}\\ \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + t\_0}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0 - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* (* -4.0 c) a))))
   (if (>= b 0.0) (/ (+ b t_0) (* -2.0 a)) (/ (+ c c) (- t_0 b)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((-4.0 * c) * a));
	double tmp;
	if (b >= 0.0) {
		tmp = (b + t_0) / (-2.0 * a);
	} else {
		tmp = (c + c) / (t_0 - b);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((((-4.0d0) * c) * a))
    if (b >= 0.0d0) then
        tmp = (b + t_0) / ((-2.0d0) * a)
    else
        tmp = (c + c) / (t_0 - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((-4.0 * c) * a));
	double tmp;
	if (b >= 0.0) {
		tmp = (b + t_0) / (-2.0 * a);
	} else {
		tmp = (c + c) / (t_0 - b);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((-4.0 * c) * a))
	tmp = 0
	if b >= 0.0:
		tmp = (b + t_0) / (-2.0 * a)
	else:
		tmp = (c + c) / (t_0 - b)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(-4.0 * c) * a))
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(b + t_0) / Float64(-2.0 * a));
	else
		tmp = Float64(Float64(c + c) / Float64(t_0 - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((-4.0 * c) * a));
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (b + t_0) / (-2.0 * a);
	else
		tmp = (c + c) / (t_0 - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision]}, If[GreaterEqual[b, 0.0], N[(N[(b + t$95$0), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 71.4%

   \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

2. Taylor expanded in a around inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. lower-*.f64 56.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

4. Applied rewrites 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

5. Taylor expanded in a around inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]

   2. lower-*.f64 40.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]

7. Applied rewrites 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]
8. Step-by-step derivation

9. Applied rewrites 40.6%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot c\right) \cdot a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ } \end{array}} \]
10. Add Preprocessing

Alternative 16: 40.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(\sqrt{\left(a \cdot c\right) \cdot -4} + b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0)
   (* (/ -0.5 a) (+ (sqrt (* (* a c) -4.0)) b))
   (/ (+ c c) (- (sqrt (* (* -4.0 c) a)) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (-0.5 / a) * (sqrt(((a * c) * -4.0)) + b);
	} else {
		tmp = (c + c) / (sqrt(((-4.0 * c) * a)) - b);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b >= 0.0d0) then
        tmp = ((-0.5d0) / a) * (sqrt(((a * c) * (-4.0d0))) + b)
    else
        tmp = (c + c) / (sqrt((((-4.0d0) * c) * a)) - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (-0.5 / a) * (Math.sqrt(((a * c) * -4.0)) + b);
	} else {
		tmp = (c + c) / (Math.sqrt(((-4.0 * c) * a)) - b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = (-0.5 / a) * (math.sqrt(((a * c) * -4.0)) + b)
	else:
		tmp = (c + c) / (math.sqrt(((-4.0 * c) * a)) - b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(-0.5 / a) * Float64(sqrt(Float64(Float64(a * c) * -4.0)) + b));
	else
		tmp = Float64(Float64(c + c) / Float64(sqrt(Float64(Float64(-4.0 * c) * a)) - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (-0.5 / a) * (sqrt(((a * c) * -4.0)) + b);
	else
		tmp = (c + c) / (sqrt(((-4.0 * c) * a)) - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(N[(-0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 71.4%

   \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

2. Taylor expanded in a around inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. lower-*.f64 56.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

4. Applied rewrites 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

5. Taylor expanded in a around inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]

   2. lower-*.f64 40.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]

7. Applied rewrites 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]
8. Step-by-step derivation

9. Applied rewrites 40.6%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot c\right) \cdot a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ } \end{array}} \]
10. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{b + \sqrt{\left(-4 \cdot c\right) \cdot a}}{-2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    2. mult-flip N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\left(b + \sqrt{\left(-4 \cdot c\right) \cdot a}\right) \cdot \frac{1}{-2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    3. lift-*.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(b + \sqrt{\left(-4 \cdot c\right) \cdot a}\right) \cdot \frac{1}{\color{blue}{-2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    4. associate-/r* N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(b + \sqrt{\left(-4 \cdot c\right) \cdot a}\right) \cdot \color{blue}{\frac{\frac{1}{-2}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    5. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(b + \sqrt{\left(-4 \cdot c\right) \cdot a}\right) \cdot \frac{\color{blue}{\frac{-1}{2}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    6. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(b + \sqrt{\left(-4 \cdot c\right) \cdot a}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    7. *-commutative N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{\frac{-1}{2}}{a} \cdot \left(b + \sqrt{\left(-4 \cdot c\right) \cdot a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    8. lower-*.f64 40.6

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-0.5}{a} \cdot \left(b + \sqrt{\left(-4 \cdot c\right) \cdot a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    9. lift-+.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{-1}{2}}{a} \cdot \color{blue}{\left(b + \sqrt{\left(-4 \cdot c\right) \cdot a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    10. +-commutative N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{-1}{2}}{a} \cdot \color{blue}{\left(\sqrt{\left(-4 \cdot c\right) \cdot a} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    11. lower-+.f64 40.6

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \color{blue}{\left(\sqrt{\left(-4 \cdot c\right) \cdot a} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

11. Applied rewrites 40.6%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-0.5}{a} \cdot \left(\sqrt{\left(a \cdot c\right) \cdot -4} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]
12. Add Preprocessing

Alternative 17: 37.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \mathbf{if}\;c \leq -4.9 \cdot 10^{-306}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot c} \cdot \sqrt{a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{a \cdot -4} \cdot \sqrt{c}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (+ c c) (- (sqrt (* (* -4.0 c) a)) b))))
   (if (<= c -4.9e-306)
     (if (>= b 0.0) (/ (* (sqrt (* -4.0 c)) (sqrt a)) (* -2.0 a)) t_0)
     (if (>= b 0.0) (/ (* (sqrt (* a -4.0)) (sqrt c)) (* -2.0 a)) t_0))))

C

double code(double a, double b, double c) {
	double t_0 = (c + c) / (sqrt(((-4.0 * c) * a)) - b);
	double tmp_1;
	if (c <= -4.9e-306) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (sqrt((-4.0 * c)) * sqrt(a)) / (-2.0 * a);
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (sqrt((a * -4.0)) * sqrt(c)) / (-2.0 * a);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = (c + c) / (sqrt((((-4.0d0) * c) * a)) - b)
    if (c <= (-4.9d-306)) then
        if (b >= 0.0d0) then
            tmp_2 = (sqrt(((-4.0d0) * c)) * sqrt(a)) / ((-2.0d0) * a)
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (sqrt((a * (-4.0d0))) * sqrt(c)) / ((-2.0d0) * a)
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (c + c) / (Math.sqrt(((-4.0 * c) * a)) - b);
	double tmp_1;
	if (c <= -4.9e-306) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (Math.sqrt((-4.0 * c)) * Math.sqrt(a)) / (-2.0 * a);
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (Math.sqrt((a * -4.0)) * Math.sqrt(c)) / (-2.0 * a);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (c + c) / (math.sqrt(((-4.0 * c) * a)) - b)
	tmp_1 = 0
	if c <= -4.9e-306:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (math.sqrt((-4.0 * c)) * math.sqrt(a)) / (-2.0 * a)
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = (math.sqrt((a * -4.0)) * math.sqrt(c)) / (-2.0 * a)
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c + c) / Float64(sqrt(Float64(Float64(-4.0 * c) * a)) - b))
	tmp_1 = 0.0
	if (c <= -4.9e-306)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(sqrt(Float64(-4.0 * c)) * sqrt(a)) / Float64(-2.0 * a));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(sqrt(Float64(a * -4.0)) * sqrt(c)) / Float64(-2.0 * a));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = (c + c) / (sqrt(((-4.0 * c) * a)) - b);
	tmp_2 = 0.0;
	if (c <= -4.9e-306)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (sqrt((-4.0 * c)) * sqrt(a)) / (-2.0 * a);
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = (sqrt((a * -4.0)) * sqrt(c)) / (-2.0 * a);
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -4.9e-306], If[GreaterEqual[b, 0.0], N[(N[(N[Sqrt[N[(-4.0 * c), $MachinePrecision]], $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(N[Sqrt[N[(a * -4.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[c], $MachinePrecision]), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -4.90000000000000025e-306

   1. Initial program 71.4%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      2. lower-*.f64 56.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   4. Applied rewrites 56.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]

      2. lower-*.f64 40.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]

   7. Applied rewrites 40.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 40.6%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot c\right) \cdot a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ } \end{array}} \]

   10. Taylor expanded in b around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       2. lower-neg.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       3. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       4. lower-*.f64 33.4

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

   12. Applied rewrites 33.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right)}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]
   13. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       2. lift-neg.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       3. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       4. distribute-lft-neg-in N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       5. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       6. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       7. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       8. associate-*l* N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       9. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       10. *-commutative N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot -4\right) \cdot a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       11. sqrt-prod N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{c \cdot -4} \cdot \color{blue}{\sqrt{a}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       12. lower-unsound-*.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{c \cdot -4} \cdot \color{blue}{\sqrt{a}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       13. lower-unsound-sqrt.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{c \cdot -4} \cdot \sqrt{\color{blue}{a}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       14. lift-*.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{c \cdot -4} \cdot \sqrt{a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       15. *-commutative N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot c} \cdot \sqrt{a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       16. lift-*.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot c} \cdot \sqrt{a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       17. lower-unsound-sqrt.f64 28.8

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot c} \cdot \sqrt{a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

   14. Applied rewrites 28.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot c} \cdot \color{blue}{\sqrt{a}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

   if -4.90000000000000025e-306 < c

   1. Initial program 71.4%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      2. lower-*.f64 56.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   4. Applied rewrites 56.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]

      2. lower-*.f64 40.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]

   7. Applied rewrites 40.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 40.6%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot c\right) \cdot a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ } \end{array}} \]

   10. Taylor expanded in b around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       2. lower-neg.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       3. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       4. lower-*.f64 33.4

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

   12. Applied rewrites 33.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right)}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]
   13. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       2. lift-neg.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       3. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       4. distribute-lft-neg-in N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       5. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       6. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       7. associate-*r* N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       8. sqrt-prod N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot a} \cdot \color{blue}{\sqrt{c}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       9. lower-unsound-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot a} \cdot \color{blue}{\sqrt{c}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       10. lower-unsound-sqrt.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot a} \cdot \sqrt{\color{blue}{c}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       11. *-commutative N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{a \cdot -4} \cdot \sqrt{c}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       12. lower-*.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{a \cdot -4} \cdot \sqrt{c}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       13. lower-unsound-sqrt.f64 28.6

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{a \cdot -4} \cdot \sqrt{c}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

   14. Applied rewrites 28.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{a \cdot -4} \cdot \color{blue}{\sqrt{c}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 35.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ t_1 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t\_1}\\ \end{array} \leq -1 \cdot 10^{-120}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot c} \cdot \sqrt{a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left|\left(a \cdot c\right) \cdot -4\right|}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (+ c c) (- (sqrt (* (* -4.0 c) a)) b)))
        (t_1 (sqrt (- (* b b) (* (* 4.0 a) c)))))
   (if (<=
        (if (>= b 0.0) (/ (- (- b) t_1) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) t_1)))
        -1e-120)
     (if (>= b 0.0) (/ (* (sqrt (* -4.0 c)) (sqrt a)) (* -2.0 a)) t_0)
     (if (>= b 0.0) (/ (sqrt (fabs (* (* a c) -4.0))) (* -2.0 a)) t_0))))

C

double code(double a, double b, double c) {
	double t_0 = (c + c) / (sqrt(((-4.0 * c) * a)) - b);
	double t_1 = sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - t_1) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + t_1);
	}
	double tmp_2;
	if (tmp <= -1e-120) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (sqrt((-4.0 * c)) * sqrt(a)) / (-2.0 * a);
		} else {
			tmp_3 = t_0;
		}
		tmp_2 = tmp_3;
	} else if (b >= 0.0) {
		tmp_2 = sqrt(fabs(((a * c) * -4.0))) / (-2.0 * a);
	} else {
		tmp_2 = t_0;
	}
	return tmp_2;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = (c + c) / (sqrt((((-4.0d0) * c) * a)) - b)
    t_1 = sqrt(((b * b) - ((4.0d0 * a) * c)))
    if (b >= 0.0d0) then
        tmp = (-b - t_1) / (2.0d0 * a)
    else
        tmp = (2.0d0 * c) / (-b + t_1)
    end if
    if (tmp <= (-1d-120)) then
        if (b >= 0.0d0) then
            tmp_3 = (sqrt(((-4.0d0) * c)) * sqrt(a)) / ((-2.0d0) * a)
        else
            tmp_3 = t_0
        end if
        tmp_2 = tmp_3
    else if (b >= 0.0d0) then
        tmp_2 = sqrt(abs(((a * c) * (-4.0d0)))) / ((-2.0d0) * a)
    else
        tmp_2 = t_0
    end if
    code = tmp_2
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (c + c) / (Math.sqrt(((-4.0 * c) * a)) - b);
	double t_1 = Math.sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - t_1) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + t_1);
	}
	double tmp_2;
	if (tmp <= -1e-120) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (Math.sqrt((-4.0 * c)) * Math.sqrt(a)) / (-2.0 * a);
		} else {
			tmp_3 = t_0;
		}
		tmp_2 = tmp_3;
	} else if (b >= 0.0) {
		tmp_2 = Math.sqrt(Math.abs(((a * c) * -4.0))) / (-2.0 * a);
	} else {
		tmp_2 = t_0;
	}
	return tmp_2;
}

Python

def code(a, b, c):
	t_0 = (c + c) / (math.sqrt(((-4.0 * c) * a)) - b)
	t_1 = math.sqrt(((b * b) - ((4.0 * a) * c)))
	tmp = 0
	if b >= 0.0:
		tmp = (-b - t_1) / (2.0 * a)
	else:
		tmp = (2.0 * c) / (-b + t_1)
	tmp_2 = 0
	if tmp <= -1e-120:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (math.sqrt((-4.0 * c)) * math.sqrt(a)) / (-2.0 * a)
		else:
			tmp_3 = t_0
		tmp_2 = tmp_3
	elif b >= 0.0:
		tmp_2 = math.sqrt(math.fabs(((a * c) * -4.0))) / (-2.0 * a)
	else:
		tmp_2 = t_0
	return tmp_2

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c + c) / Float64(sqrt(Float64(Float64(-4.0 * c) * a)) - b))
	t_1 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(Float64(-b) - t_1) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_1));
	end
	tmp_2 = 0.0
	if (tmp <= -1e-120)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(sqrt(Float64(-4.0 * c)) * sqrt(a)) / Float64(-2.0 * a));
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = Float64(sqrt(abs(Float64(Float64(a * c) * -4.0))) / Float64(-2.0 * a));
	else
		tmp_2 = t_0;
	end
	return tmp_2
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = (c + c) / (sqrt(((-4.0 * c) * a)) - b);
	t_1 = sqrt(((b * b) - ((4.0 * a) * c)));
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (-b - t_1) / (2.0 * a);
	else
		tmp = (2.0 * c) / (-b + t_1);
	end
	tmp_3 = 0.0;
	if (tmp <= -1e-120)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (sqrt((-4.0 * c)) * sqrt(a)) / (-2.0 * a);
		else
			tmp_4 = t_0;
		end
		tmp_3 = tmp_4;
	elseif (b >= 0.0)
		tmp_3 = sqrt(abs(((a * c) * -4.0))) / (-2.0 * a);
	else
		tmp_3 = t_0;
	end
	tmp_5 = tmp_3;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$1), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$1), $MachinePrecision]), $MachinePrecision]], -1e-120], If[GreaterEqual[b, 0.0], N[(N[(N[Sqrt[N[(-4.0 * c), $MachinePrecision]], $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[Sqrt[N[Abs[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (if (>=.f64 b #s(literal 0 binary64)) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) (/.f64 (*.f64 #s(literal 2 binary64) c) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))))) < -9.99999999999999979e-121

   1. Initial program 71.4%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      2. lower-*.f64 56.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   4. Applied rewrites 56.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]

      2. lower-*.f64 40.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]

   7. Applied rewrites 40.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 40.6%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot c\right) \cdot a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ } \end{array}} \]

   10. Taylor expanded in b around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       2. lower-neg.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       3. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       4. lower-*.f64 33.4

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

   12. Applied rewrites 33.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right)}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]
   13. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       2. lift-neg.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       3. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       4. distribute-lft-neg-in N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       5. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       6. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       7. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       8. associate-*l* N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       9. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       10. *-commutative N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot -4\right) \cdot a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       11. sqrt-prod N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{c \cdot -4} \cdot \color{blue}{\sqrt{a}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       12. lower-unsound-*.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{c \cdot -4} \cdot \color{blue}{\sqrt{a}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       13. lower-unsound-sqrt.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{c \cdot -4} \cdot \sqrt{\color{blue}{a}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       14. lift-*.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{c \cdot -4} \cdot \sqrt{a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       15. *-commutative N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot c} \cdot \sqrt{a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       16. lift-*.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot c} \cdot \sqrt{a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       17. lower-unsound-sqrt.f64 28.8

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot c} \cdot \sqrt{a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

   14. Applied rewrites 28.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot c} \cdot \color{blue}{\sqrt{a}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

   if -9.99999999999999979e-121 < (if (>=.f64 b #s(literal 0 binary64)) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) (/.f64 (*.f64 #s(literal 2 binary64) c) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c))))))

   1. Initial program 71.4%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

      2. lower-*.f64 56.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   4. Applied rewrites 56.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]

      2. lower-*.f64 40.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]

   7. Applied rewrites 40.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 40.6%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot c\right) \cdot a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ } \end{array}} \]

   10. Taylor expanded in b around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       2. lower-neg.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       3. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       4. lower-*.f64 33.4

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

   12. Applied rewrites 33.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right)}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]
   13. Step-by-step derivation

       1. lift-neg.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       2. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       3. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{neg}\left(\left(a \cdot c\right) \cdot 4\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       4. distribute-rgt-neg-out N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot \left(\mathsf{neg}\left(4\right)\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       5. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       6. rem-square-sqrt N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\sqrt{\left(a \cdot c\right) \cdot -4} \cdot \sqrt{\left(a \cdot c\right) \cdot -4}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       7. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\left(a \cdot c\right) \cdot -4}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       8. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} \cdot \sqrt{\left(a \cdot c\right) \cdot -4}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       9. distribute-lft-neg-in N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)} \cdot \sqrt{\left(a \cdot c\right) \cdot -4}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       10. lift-*.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)} \cdot \sqrt{\left(a \cdot c\right) \cdot -4}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       11. lift-neg.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\left(a \cdot c\right) \cdot -4}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       12. lift-sqrt.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\left(a \cdot c\right) \cdot -4}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       13. *-commutative N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       14. metadata-eval N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       15. distribute-lft-neg-in N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       16. lift-*.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       17. lift-neg.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       18. lift-sqrt.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       19. sqr-abs-rev N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left|\sqrt{-4 \cdot \left(a \cdot c\right)}\right| \cdot \left|\sqrt{-4 \cdot \left(a \cdot c\right)}\right|}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

       20. mul-fabs N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left|\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}\right|}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

   14. Applied rewrites 34.1%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left|\left(a \cdot c\right) \cdot -4\right|}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 34.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left|\left(a \cdot c\right) \cdot -4\right|}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0)
   (/ (sqrt (fabs (* (* a c) -4.0))) (* -2.0 a))
   (/ (+ c c) (- (sqrt (* (* -4.0 c) a)) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = sqrt(fabs(((a * c) * -4.0))) / (-2.0 * a);
	} else {
		tmp = (c + c) / (sqrt(((-4.0 * c) * a)) - b);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b >= 0.0d0) then
        tmp = sqrt(abs(((a * c) * (-4.0d0)))) / ((-2.0d0) * a)
    else
        tmp = (c + c) / (sqrt((((-4.0d0) * c) * a)) - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = Math.sqrt(Math.abs(((a * c) * -4.0))) / (-2.0 * a);
	} else {
		tmp = (c + c) / (Math.sqrt(((-4.0 * c) * a)) - b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = math.sqrt(math.fabs(((a * c) * -4.0))) / (-2.0 * a)
	else:
		tmp = (c + c) / (math.sqrt(((-4.0 * c) * a)) - b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(sqrt(abs(Float64(Float64(a * c) * -4.0))) / Float64(-2.0 * a));
	else
		tmp = Float64(Float64(c + c) / Float64(sqrt(Float64(Float64(-4.0 * c) * a)) - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = sqrt(abs(((a * c) * -4.0))) / (-2.0 * a);
	else
		tmp = (c + c) / (sqrt(((-4.0 * c) * a)) - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(N[Sqrt[N[Abs[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 71.4%

   \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

2. Taylor expanded in a around inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. lower-*.f64 56.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

4. Applied rewrites 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

5. Taylor expanded in a around inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]

   2. lower-*.f64 40.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]

7. Applied rewrites 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]
8. Step-by-step derivation

9. Applied rewrites 40.6%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot c\right) \cdot a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ } \end{array}} \]

10. Taylor expanded in b around 0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]
11. Step-by-step derivation

    1. lower-sqrt.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    2. lower-neg.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    3. lower-*.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    4. lower-*.f64 33.4

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

12. Applied rewrites 33.4%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right)}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]
13. Step-by-step derivation

    1. lift-neg.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    2. lift-*.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    3. *-commutative N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{neg}\left(\left(a \cdot c\right) \cdot 4\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    4. distribute-rgt-neg-out N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot \left(\mathsf{neg}\left(4\right)\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    5. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -4}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    6. rem-square-sqrt N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\sqrt{\left(a \cdot c\right) \cdot -4} \cdot \sqrt{\left(a \cdot c\right) \cdot -4}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    7. *-commutative N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\left(a \cdot c\right) \cdot -4}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    8. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} \cdot \sqrt{\left(a \cdot c\right) \cdot -4}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    9. distribute-lft-neg-in N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)} \cdot \sqrt{\left(a \cdot c\right) \cdot -4}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    10. lift-*.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)} \cdot \sqrt{\left(a \cdot c\right) \cdot -4}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    11. lift-neg.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\left(a \cdot c\right) \cdot -4}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    12. lift-sqrt.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\left(a \cdot c\right) \cdot -4}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    13. *-commutative N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    14. metadata-eval N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    15. distribute-lft-neg-in N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    16. lift-*.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    17. lift-neg.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    18. lift-sqrt.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    19. sqr-abs-rev N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left|\sqrt{-4 \cdot \left(a \cdot c\right)}\right| \cdot \left|\sqrt{-4 \cdot \left(a \cdot c\right)}\right|}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    20. mul-fabs N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left|\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}\right|}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

14. Applied rewrites 34.1%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\left|\left(a \cdot c\right) \cdot -4\right|}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]
15. Add Preprocessing

Alternative 20: 33.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \sqrt{\left(a \cdot c\right) \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0)
   (* (/ -0.5 a) (sqrt (* (* a c) -4.0)))
   (/ (+ c c) (- (sqrt (* (* -4.0 c) a)) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (-0.5 / a) * sqrt(((a * c) * -4.0));
	} else {
		tmp = (c + c) / (sqrt(((-4.0 * c) * a)) - b);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b >= 0.0d0) then
        tmp = ((-0.5d0) / a) * sqrt(((a * c) * (-4.0d0)))
    else
        tmp = (c + c) / (sqrt((((-4.0d0) * c) * a)) - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (-0.5 / a) * Math.sqrt(((a * c) * -4.0));
	} else {
		tmp = (c + c) / (Math.sqrt(((-4.0 * c) * a)) - b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = (-0.5 / a) * math.sqrt(((a * c) * -4.0))
	else:
		tmp = (c + c) / (math.sqrt(((-4.0 * c) * a)) - b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(-0.5 / a) * sqrt(Float64(Float64(a * c) * -4.0)));
	else
		tmp = Float64(Float64(c + c) / Float64(sqrt(Float64(Float64(-4.0 * c) * a)) - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (-0.5 / a) * sqrt(((a * c) * -4.0));
	else
		tmp = (c + c) / (sqrt(((-4.0 * c) * a)) - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(N[(-0.5 / a), $MachinePrecision] * N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 71.4%

   \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

2. Taylor expanded in a around inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. lower-*.f64 56.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

4. Applied rewrites 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

5. Taylor expanded in a around inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]

   2. lower-*.f64 40.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]

7. Applied rewrites 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]
8. Step-by-step derivation

9. Applied rewrites 40.6%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot c\right) \cdot a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ } \end{array}} \]

10. Taylor expanded in b around 0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]
11. Step-by-step derivation

    1. lower-sqrt.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    2. lower-neg.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    3. lower-*.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    4. lower-*.f64 33.4

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

12. Applied rewrites 33.4%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right)}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]
13. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    2. mult-flip N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \frac{1}{-2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    3. *-commutative N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{1}{-2 \cdot a} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    4. lift-*.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{1}{\color{blue}{-2 \cdot a}} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    5. associate-/r* N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{\frac{1}{-2}}{a}} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    6. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\frac{-1}{2}}}{a} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    7. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{\frac{-1}{2}}{a}} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    8. lower-*.f64 33.3

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-0.5}{a} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    9. lift-neg.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{-1}{2}}{a} \cdot \sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    10. lift-*.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{-1}{2}}{a} \cdot \sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    11. distribute-lft-neg-in N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{-1}{2}}{a} \cdot \sqrt{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    12. metadata-eval N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{-1}{2}}{a} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    13. *-commutative N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{-1}{2}}{a} \cdot \sqrt{\left(a \cdot c\right) \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    14. lower-*.f64 33.3

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \sqrt{\left(a \cdot c\right) \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

14. Applied rewrites 33.3%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-0.5}{a} \cdot \sqrt{\left(a \cdot c\right) \cdot -4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]
15. Add Preprocessing

Alternative 21: 26.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{t\_0}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* 4.0 (* a c))))))
   (if (>= b 0.0) (/ t_0 (* -2.0 a)) (* 2.0 (/ c t_0)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(-(4.0 * (a * c)));
	double tmp;
	if (b >= 0.0) {
		tmp = t_0 / (-2.0 * a);
	} else {
		tmp = 2.0 * (c / t_0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(-(4.0d0 * (a * c)))
    if (b >= 0.0d0) then
        tmp = t_0 / ((-2.0d0) * a)
    else
        tmp = 2.0d0 * (c / t_0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(-(4.0 * (a * c)));
	double tmp;
	if (b >= 0.0) {
		tmp = t_0 / (-2.0 * a);
	} else {
		tmp = 2.0 * (c / t_0);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(-(4.0 * (a * c)))
	tmp = 0
	if b >= 0.0:
		tmp = t_0 / (-2.0 * a)
	else:
		tmp = 2.0 * (c / t_0)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(-Float64(4.0 * Float64(a * c))))
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(t_0 / Float64(-2.0 * a));
	else
		tmp = Float64(2.0 * Float64(c / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(-(4.0 * (a * c)));
	tmp = 0.0;
	if (b >= 0.0)
		tmp = t_0 / (-2.0 * a);
	else
		tmp = 2.0 * (c / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[(-N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]}, If[GreaterEqual[b, 0.0], N[(t$95$0 / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(c / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 71.4%

   \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

2. Taylor expanded in a around inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. lower-*.f64 56.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

4. Applied rewrites 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

5. Taylor expanded in a around inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]

   2. lower-*.f64 40.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]

7. Applied rewrites 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]
8. Step-by-step derivation

9. Applied rewrites 40.6%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + \sqrt{\left(-4 \cdot c\right) \cdot a}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ } \end{array}} \]

10. Taylor expanded in b around 0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]
11. Step-by-step derivation

    1. lower-sqrt.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    2. lower-neg.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    3. lower-*.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

    4. lower-*.f64 33.4

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

12. Applied rewrites 33.4%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right)}}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot c\right) \cdot a} - b}\\ \end{array} \]

13. Taylor expanded in b around 0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}\\ \end{array} \]
14. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{2 \cdot \frac{c}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}\\ \end{array} \]

    2. lower-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \color{blue}{\frac{c}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}\\ \end{array} \]

    3. lower-sqrt.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\color{blue}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}\\ \end{array} \]

    4. lower-neg.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \end{array} \]

    5. lower-*.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-\color{blue}{4 \cdot \left(a \cdot c\right)}}}\\ \end{array} \]

    6. lower-*.f64 26.6

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \end{array} \]

15. Applied rewrites 26.6%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \end{array} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2025154 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  :precision binary64
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))