jeff quadratic root 1

Percentage Accurate: 72.4% → 90.6%

Time: 4.9s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.4% 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: 90.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}\\ \mathbf{if}\;b \leq -1.5 \cdot 10^{+94}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{+72}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5 \cdot \left(b + t\_0\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;-\left(-\sqrt{-\frac{c}{a}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma b b (* (* -4.0 a) c)))))
   (if (<= b -1.5e+94)
     (if (>= b 0.0) (/ (- b) a) (- (/ c b)))
     (if (<= b 1.12e+72)
       (if (>= b 0.0) (/ (* -0.5 (+ b t_0)) a) (/ (+ c c) (- t_0 b)))
       (if (>= b 0.0) (* (/ (+ b b) a) -0.5) (- (- (sqrt (- (/ c a))))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma(b, b, ((-4.0 * a) * c)));
	double tmp_1;
	if (b <= -1.5e+94) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -b / a;
		} else {
			tmp_2 = -(c / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.12e+72) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-0.5 * (b + t_0)) / a;
		} else {
			tmp_3 = (c + c) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = ((b + b) / a) * -0.5;
	} else {
		tmp_1 = -(-sqrt(-(c / a)));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(b, b, Float64(Float64(-4.0 * a) * c)))
	tmp_1 = 0.0
	if (b <= -1.5e+94)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-b) / a);
		else
			tmp_2 = Float64(-Float64(c / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.12e+72)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-0.5 * Float64(b + t_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(Float64(b + b) / a) * -0.5);
	else
		tmp_1 = Float64(-Float64(-sqrt(Float64(-Float64(c / a)))));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(b * b + N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.5e+94], If[GreaterEqual[b, 0.0], N[((-b) / a), $MachinePrecision], (-N[(c / b), $MachinePrecision])], If[LessEqual[b, 1.12e+72], If[GreaterEqual[b, 0.0], N[(N[(-0.5 * N[(b + t$95$0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], (-(-N[Sqrt[(-N[(c / a), $MachinePrecision])], $MachinePrecision]))]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.5e94

   1. Initial program 52.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 52.7%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 52.7%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 2.2%

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

   11. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lift-/.f64 96.4

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

   13. Applied rewrites 96.4%

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

   14. Taylor expanded in a around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lift-neg.f64 96.4

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

   16. Applied rewrites 96.4%

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

   if -1.5e94 < b < 1.12000000000000001e72

   1. Initial program 86.2%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 86.2%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 86.2%

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

   if 1.12000000000000001e72 < b

   1. Initial program 60.3%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 94.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 94.9%

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

   11. Taylor expanded in a around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. sqrt-unprod N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 94.9

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

   13. Applied rewrites 94.9%

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

   14. Taylor expanded in c around -inf

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

       1. mul-1-neg N/A

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

       2. sqrt-prod N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-neg.f64 94.9

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

       7. lift-*.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. *-commutative N/A

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

       10. mul-1-neg N/A

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

       11. lower-neg.f64 N/A

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

       12. lift-/.f64 94.9

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

   16. Applied rewrites 94.9%

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

Alternative 2: 90.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;b \leq -1.5 \cdot 10^{+94}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{+72}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{t\_0 + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;-\left(-\sqrt{-\frac{c}{a}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 a) c (* b b)))))
   (if (<= b -1.5e+94)
     (if (>= b 0.0) (/ (- b) a) (- (/ c b)))
     (if (<= b 1.12e+72)
       (if (>= b 0.0) (* (/ (+ t_0 b) a) -0.5) (/ (+ c c) (- t_0 b)))
       (if (>= b 0.0) (* (/ (+ b b) a) -0.5) (- (- (sqrt (- (/ c a))))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-4.0 * a), c, (b * b)));
	double tmp_1;
	if (b <= -1.5e+94) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -b / a;
		} else {
			tmp_2 = -(c / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.12e+72) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = ((t_0 + b) / a) * -0.5;
		} else {
			tmp_3 = (c + c) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = ((b + b) / a) * -0.5;
	} else {
		tmp_1 = -(-sqrt(-(c / a)));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))
	tmp_1 = 0.0
	if (b <= -1.5e+94)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-b) / a);
		else
			tmp_2 = Float64(-Float64(c / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.12e+72)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(t_0 + b) / a) * -0.5);
		else
			tmp_3 = Float64(Float64(c + c) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(b + b) / a) * -0.5);
	else
		tmp_1 = Float64(-Float64(-sqrt(Float64(-Float64(c / a)))));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.5e+94], If[GreaterEqual[b, 0.0], N[((-b) / a), $MachinePrecision], (-N[(c / b), $MachinePrecision])], If[LessEqual[b, 1.12e+72], If[GreaterEqual[b, 0.0], N[(N[(N[(t$95$0 + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], (-(-N[Sqrt[(-N[(c / a), $MachinePrecision])], $MachinePrecision]))]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.5e94

   1. Initial program 52.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 52.7%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 52.7%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 2.2%

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

   11. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lift-/.f64 96.4

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

   13. Applied rewrites 96.4%

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

   14. Taylor expanded in a around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lift-neg.f64 96.4

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

   16. Applied rewrites 96.4%

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

   if -1.5e94 < b < 1.12000000000000001e72

   1. Initial program 86.2%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 86.2%

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

   if 1.12000000000000001e72 < b

   1. Initial program 60.3%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 94.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 94.9%

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

   11. Taylor expanded in a around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. sqrt-unprod N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 94.9

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

   13. Applied rewrites 94.9%

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

   14. Taylor expanded in c around -inf

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

       1. mul-1-neg N/A

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

       2. sqrt-prod N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-neg.f64 94.9

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

       7. lift-*.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. *-commutative N/A

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

       10. mul-1-neg N/A

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

       11. lower-neg.f64 N/A

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

       12. lift-/.f64 94.9

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

   16. Applied rewrites 94.9%

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

Alternative 3: 86.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-b}{a}\\ \mathbf{if}\;b \leq -1.5 \cdot 10^{+94}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{a \cdot c}{b}, b + b\right)}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-72}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\left(-4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_0 + \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- b) a)))
   (if (<= b -1.5e+94)
     (if (>= b 0.0) t_0 (- (/ c b)))
     (if (<= b -2e-310)
       (if (>= b 0.0)
         (* (/ (fma -2.0 (/ (* a c) b) (+ b b)) a) -0.5)
         (/ (+ c c) (- (sqrt (fma (* -4.0 a) c (* b b))) b)))
       (if (<= b 1.95e-72)
         (if (>= b 0.0)
           (/ (- (- b) (sqrt (* (* -4.0 a) c))) (* 2.0 a))
           (- (/ (fma a (/ (* c c) (* b b)) c) b)))
         (if (>= b 0.0) (+ t_0 (/ c b)) (/ (* 2.0 c) (+ (- b) b))))))))

C

double code(double a, double b, double c) {
	double t_0 = -b / a;
	double tmp_1;
	if (b <= -1.5e+94) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = -(c / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (fma(-2.0, ((a * c) / b), (b + b)) / a) * -0.5;
		} else {
			tmp_3 = (c + c) / (sqrt(fma((-4.0 * a), c, (b * b))) - b);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.95e-72) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (-b - sqrt(((-4.0 * a) * c))) / (2.0 * a);
		} else {
			tmp_4 = -(fma(a, ((c * c) / (b * b)), c) / b);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = t_0 + (c / b);
	} else {
		tmp_1 = (2.0 * c) / (-b + b);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-b) / a)
	tmp_1 = 0.0
	if (b <= -1.5e+94)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(-Float64(c / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= -2e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(fma(-2.0, Float64(Float64(a * c) / b), Float64(b + b)) / a) * -0.5);
		else
			tmp_3 = Float64(Float64(c + c) / Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) - b));
		end
		tmp_1 = tmp_3;
	elseif (b <= 1.95e-72)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(-4.0 * a) * c))) / Float64(2.0 * a));
		else
			tmp_4 = Float64(-Float64(fma(a, Float64(Float64(c * c) / Float64(b * b)), c) / b));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(t_0 + Float64(c / b));
	else
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + b));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) / a), $MachinePrecision]}, If[LessEqual[b, -1.5e+94], If[GreaterEqual[b, 0.0], t$95$0, (-N[(c / b), $MachinePrecision])], If[LessEqual[b, -2e-310], If[GreaterEqual[b, 0.0], N[(N[(N[(-2.0 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision] + N[(b + b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.95e-72], If[GreaterEqual[b, 0.0], N[(N[((-b) - N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], (-N[(N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / b), $MachinePrecision])], If[GreaterEqual[b, 0.0], N[(t$95$0 + N[(c / b), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + b), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.5e94

   1. Initial program 52.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 52.7%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 52.7%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 2.2%

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

   11. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lift-/.f64 96.4

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

   13. Applied rewrites 96.4%

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

   14. Taylor expanded in a around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lift-neg.f64 96.4

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

   16. Applied rewrites 96.4%

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

   if -1.5e94 < b < -1.999999999999994e-310

   1. Initial program 86.7%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 86.7%

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

   5. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. count-2-rev N/A

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

      5. lower-+.f64 86.7

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

   7. Applied rewrites 86.7%

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

   if -1.999999999999994e-310 < b < 1.95e-72

   1. Initial program 80.6%

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

   2. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 80.6

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

   4. Applied rewrites 80.6%

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

   5. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lift-*.f64 70.0

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

   7. Applied rewrites 70.0%

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

   if 1.95e-72 < b

   1. Initial program 70.7%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 85.9%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 85.9%

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

   8. Taylor expanded in c around 0

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

      1. lower-+.f64 N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 86.3

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

   10. Applied rewrites 86.3%

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

Alternative 4: 82.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left(-4 \cdot a\right) \cdot c}\\ t_1 := \frac{-b}{a}\\ \mathbf{if}\;b \leq -1.32 \cdot 10^{-43}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-72}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{t\_0 + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_1 + \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* (* -4.0 a) c))) (t_1 (/ (- b) a)))
   (if (<= b -1.32e-43)
     (if (>= b 0.0) t_1 (- (/ c b)))
     (if (<= b 1.95e-72)
       (if (>= b 0.0) (* (/ (+ t_0 b) a) -0.5) (/ (+ c c) (- t_0 b)))
       (if (>= b 0.0) (+ t_1 (/ c b)) (/ (* 2.0 c) (+ (- b) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((-4.0 * a) * c));
	double t_1 = -b / a;
	double tmp_1;
	if (b <= -1.32e-43) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = -(c / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.95e-72) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = ((t_0 + b) / a) * -0.5;
		} else {
			tmp_3 = (c + c) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = t_1 + (c / b);
	} else {
		tmp_1 = (2.0 * c) / (-b + b);
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = sqrt((((-4.0d0) * a) * c))
    t_1 = -b / a
    if (b <= (-1.32d-43)) then
        if (b >= 0.0d0) then
            tmp_2 = t_1
        else
            tmp_2 = -(c / b)
        end if
        tmp_1 = tmp_2
    else if (b <= 1.95d-72) then
        if (b >= 0.0d0) then
            tmp_3 = ((t_0 + b) / a) * (-0.5d0)
        else
            tmp_3 = (c + c) / (t_0 - b)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = t_1 + (c / b)
    else
        tmp_1 = (2.0d0 * c) / (-b + b)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((-4.0 * a) * c));
	double t_1 = -b / a;
	double tmp_1;
	if (b <= -1.32e-43) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = -(c / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.95e-72) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = ((t_0 + b) / a) * -0.5;
		} else {
			tmp_3 = (c + c) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = t_1 + (c / b);
	} else {
		tmp_1 = (2.0 * c) / (-b + b);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((-4.0 * a) * c))
	t_1 = -b / a
	tmp_1 = 0
	if b <= -1.32e-43:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_1
		else:
			tmp_2 = -(c / b)
		tmp_1 = tmp_2
	elif b <= 1.95e-72:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = ((t_0 + b) / a) * -0.5
		else:
			tmp_3 = (c + c) / (t_0 - b)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = t_1 + (c / b)
	else:
		tmp_1 = (2.0 * c) / (-b + b)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(-4.0 * a) * c))
	t_1 = Float64(Float64(-b) / a)
	tmp_1 = 0.0
	if (b <= -1.32e-43)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(-Float64(c / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.95e-72)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(t_0 + b) / a) * -0.5);
		else
			tmp_3 = Float64(Float64(c + c) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(t_1 + Float64(c / b));
	else
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + b));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((-4.0 * a) * c));
	t_1 = -b / a;
	tmp_2 = 0.0;
	if (b <= -1.32e-43)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = -(c / b);
		end
		tmp_2 = tmp_3;
	elseif (b <= 1.95e-72)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = ((t_0 + b) / a) * -0.5;
		else
			tmp_4 = (c + c) / (t_0 - b);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = t_1 + (c / b);
	else
		tmp_2 = (2.0 * c) / (-b + b);
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-b) / a), $MachinePrecision]}, If[LessEqual[b, -1.32e-43], If[GreaterEqual[b, 0.0], t$95$1, (-N[(c / b), $MachinePrecision])], If[LessEqual[b, 1.95e-72], If[GreaterEqual[b, 0.0], N[(N[(N[(t$95$0 + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(t$95$1 + N[(c / b), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + b), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.32000000000000002e-43

   1. Initial program 65.7%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 65.8%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 65.8%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 2.4%

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

   11. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lift-/.f64 89.3

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

   13. Applied rewrites 89.3%

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

   14. Taylor expanded in a around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lift-neg.f64 89.3

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

   16. Applied rewrites 89.3%

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

   if -1.32000000000000002e-43 < b < 1.95e-72

   1. Initial program 81.7%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 81.7%

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

   5. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lift-*.f64 76.6

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

   7. Applied rewrites 76.6%

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

   8. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lift-*.f64 68.7

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

   10. Applied rewrites 68.7%

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

   if 1.95e-72 < b

   1. Initial program 70.7%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 85.9%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 85.9%

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

   8. Taylor expanded in c around 0

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

      1. lower-+.f64 N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 86.3

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

   10. Applied rewrites 86.3%

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

Alternative 5: 79.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-b}{a}\\ t_1 := \sqrt{\left(-4 \cdot a\right) \cdot c}\\ \mathbf{if}\;b \leq -1.15 \cdot 10^{-165}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{t\_1}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-72}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{t\_1}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{b - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_0 + \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- b) a)) (t_1 (sqrt (* (* -4.0 a) c))))
   (if (<= b -1.15e-165)
     (if (>= b 0.0) t_0 (- (/ c b)))
     (if (<= b -2e-310)
       (if (>= b 0.0) (* (/ (+ b b) a) -0.5) (/ (+ c c) t_1))
       (if (<= b 1.95e-72)
         (if (>= b 0.0) (* (/ t_1 a) -0.5) (/ (+ c c) (- b b)))
         (if (>= b 0.0) (+ t_0 (/ c b)) (/ (* 2.0 c) (+ (- b) b))))))))

C

double code(double a, double b, double c) {
	double t_0 = -b / a;
	double t_1 = sqrt(((-4.0 * a) * c));
	double tmp_1;
	if (b <= -1.15e-165) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = -(c / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = ((b + b) / a) * -0.5;
		} else {
			tmp_3 = (c + c) / t_1;
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.95e-72) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (t_1 / a) * -0.5;
		} else {
			tmp_4 = (c + c) / (b - b);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = t_0 + (c / b);
	} else {
		tmp_1 = (2.0 * c) / (-b + b);
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    real(8) :: tmp_4
    t_0 = -b / a
    t_1 = sqrt((((-4.0d0) * a) * c))
    if (b <= (-1.15d-165)) then
        if (b >= 0.0d0) then
            tmp_2 = t_0
        else
            tmp_2 = -(c / b)
        end if
        tmp_1 = tmp_2
    else if (b <= (-2d-310)) then
        if (b >= 0.0d0) then
            tmp_3 = ((b + b) / a) * (-0.5d0)
        else
            tmp_3 = (c + c) / t_1
        end if
        tmp_1 = tmp_3
    else if (b <= 1.95d-72) then
        if (b >= 0.0d0) then
            tmp_4 = (t_1 / a) * (-0.5d0)
        else
            tmp_4 = (c + c) / (b - b)
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = t_0 + (c / b)
    else
        tmp_1 = (2.0d0 * c) / (-b + b)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -b / a;
	double t_1 = Math.sqrt(((-4.0 * a) * c));
	double tmp_1;
	if (b <= -1.15e-165) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = -(c / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= -2e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = ((b + b) / a) * -0.5;
		} else {
			tmp_3 = (c + c) / t_1;
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.95e-72) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (t_1 / a) * -0.5;
		} else {
			tmp_4 = (c + c) / (b - b);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = t_0 + (c / b);
	} else {
		tmp_1 = (2.0 * c) / (-b + b);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = -b / a
	t_1 = math.sqrt(((-4.0 * a) * c))
	tmp_1 = 0
	if b <= -1.15e-165:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_0
		else:
			tmp_2 = -(c / b)
		tmp_1 = tmp_2
	elif b <= -2e-310:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = ((b + b) / a) * -0.5
		else:
			tmp_3 = (c + c) / t_1
		tmp_1 = tmp_3
	elif b <= 1.95e-72:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = (t_1 / a) * -0.5
		else:
			tmp_4 = (c + c) / (b - b)
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = t_0 + (c / b)
	else:
		tmp_1 = (2.0 * c) / (-b + b)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-b) / a)
	t_1 = sqrt(Float64(Float64(-4.0 * a) * c))
	tmp_1 = 0.0
	if (b <= -1.15e-165)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(-Float64(c / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= -2e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(b + b) / a) * -0.5);
		else
			tmp_3 = Float64(Float64(c + c) / t_1);
		end
		tmp_1 = tmp_3;
	elseif (b <= 1.95e-72)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(t_1 / a) * -0.5);
		else
			tmp_4 = Float64(Float64(c + c) / Float64(b - b));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(t_0 + Float64(c / b));
	else
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + b));
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = -b / a;
	t_1 = sqrt(((-4.0 * a) * c));
	tmp_2 = 0.0;
	if (b <= -1.15e-165)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_0;
		else
			tmp_3 = -(c / b);
		end
		tmp_2 = tmp_3;
	elseif (b <= -2e-310)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = ((b + b) / a) * -0.5;
		else
			tmp_4 = (c + c) / t_1;
		end
		tmp_2 = tmp_4;
	elseif (b <= 1.95e-72)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = (t_1 / a) * -0.5;
		else
			tmp_5 = (c + c) / (b - b);
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = t_0 + (c / b);
	else
		tmp_2 = (2.0 * c) / (-b + b);
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) / a), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.15e-165], If[GreaterEqual[b, 0.0], t$95$0, (-N[(c / b), $MachinePrecision])], If[LessEqual[b, -2e-310], If[GreaterEqual[b, 0.0], N[(N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / t$95$1), $MachinePrecision]], If[LessEqual[b, 1.95e-72], If[GreaterEqual[b, 0.0], N[(N[(t$95$1 / a), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[(b - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(t$95$0 + N[(c / b), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + b), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.15e-165

   1. Initial program 70.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 70.6%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 70.6%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 2.6%

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

   11. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lift-/.f64 80.0

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

   13. Applied rewrites 80.0%

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

   14. Taylor expanded in a around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lift-neg.f64 80.0

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

   16. Applied rewrites 80.0%

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

   if -1.15e-165 < b < -1.999999999999994e-310

   1. Initial program 74.6%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 74.6%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 74.6%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 3.3%

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

   11. Taylor expanded in a around inf

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

       1. sqrt-unprod N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 72.9

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

   13. Applied rewrites 72.9%

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

   if -1.999999999999994e-310 < b < 1.95e-72

   1. Initial program 80.6%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 80.6%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 20.3%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 20.3%

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

   11. Taylor expanded in a around inf

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

       1. sqrt-unprod N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 67.2

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

   13. Applied rewrites 67.2%

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

   if 1.95e-72 < b

   1. Initial program 70.7%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 85.9%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 85.9%

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

   8. Taylor expanded in c around 0

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

      1. lower-+.f64 N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 86.3

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

   10. Applied rewrites 86.3%

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

Alternative 6: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-b}{a}\\ \mathbf{if}\;b \leq -1.15 \cdot 10^{-165}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-266}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(-4 \cdot a\right) \cdot c}}\\ \end{array}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{-146}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{\frac{c}{a} \cdot -4} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-2 \cdot b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_0 + \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- b) a)))
   (if (<= b -1.15e-165)
     (if (>= b 0.0) t_0 (- (/ c b)))
     (if (<= b 7e-266)
       (if (>= b 0.0)
         (* (/ (+ b b) a) -0.5)
         (/ (+ c c) (sqrt (* (* -4.0 a) c))))
       (if (<= b 5.4e-146)
         (if (>= b 0.0)
           (* (sqrt (* (/ c a) -4.0)) -0.5)
           (/ (* 2.0 c) (* -2.0 b)))
         (if (>= b 0.0) (+ t_0 (/ c b)) (/ (* 2.0 c) (+ (- b) b))))))))

C

double code(double a, double b, double c) {
	double t_0 = -b / a;
	double tmp_1;
	if (b <= -1.15e-165) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = -(c / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 7e-266) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = ((b + b) / a) * -0.5;
		} else {
			tmp_3 = (c + c) / sqrt(((-4.0 * a) * c));
		}
		tmp_1 = tmp_3;
	} else if (b <= 5.4e-146) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = sqrt(((c / a) * -4.0)) * -0.5;
		} else {
			tmp_4 = (2.0 * c) / (-2.0 * b);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = t_0 + (c / b);
	} else {
		tmp_1 = (2.0 * c) / (-b + b);
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    real(8) :: tmp_4
    t_0 = -b / a
    if (b <= (-1.15d-165)) then
        if (b >= 0.0d0) then
            tmp_2 = t_0
        else
            tmp_2 = -(c / b)
        end if
        tmp_1 = tmp_2
    else if (b <= 7d-266) then
        if (b >= 0.0d0) then
            tmp_3 = ((b + b) / a) * (-0.5d0)
        else
            tmp_3 = (c + c) / sqrt((((-4.0d0) * a) * c))
        end if
        tmp_1 = tmp_3
    else if (b <= 5.4d-146) then
        if (b >= 0.0d0) then
            tmp_4 = sqrt(((c / a) * (-4.0d0))) * (-0.5d0)
        else
            tmp_4 = (2.0d0 * c) / ((-2.0d0) * b)
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = t_0 + (c / b)
    else
        tmp_1 = (2.0d0 * c) / (-b + b)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -b / a;
	double tmp_1;
	if (b <= -1.15e-165) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = -(c / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 7e-266) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = ((b + b) / a) * -0.5;
		} else {
			tmp_3 = (c + c) / Math.sqrt(((-4.0 * a) * c));
		}
		tmp_1 = tmp_3;
	} else if (b <= 5.4e-146) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = Math.sqrt(((c / a) * -4.0)) * -0.5;
		} else {
			tmp_4 = (2.0 * c) / (-2.0 * b);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = t_0 + (c / b);
	} else {
		tmp_1 = (2.0 * c) / (-b + b);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = -b / a
	tmp_1 = 0
	if b <= -1.15e-165:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_0
		else:
			tmp_2 = -(c / b)
		tmp_1 = tmp_2
	elif b <= 7e-266:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = ((b + b) / a) * -0.5
		else:
			tmp_3 = (c + c) / math.sqrt(((-4.0 * a) * c))
		tmp_1 = tmp_3
	elif b <= 5.4e-146:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = math.sqrt(((c / a) * -4.0)) * -0.5
		else:
			tmp_4 = (2.0 * c) / (-2.0 * b)
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = t_0 + (c / b)
	else:
		tmp_1 = (2.0 * c) / (-b + b)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-b) / a)
	tmp_1 = 0.0
	if (b <= -1.15e-165)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(-Float64(c / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 7e-266)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(b + b) / a) * -0.5);
		else
			tmp_3 = Float64(Float64(c + c) / sqrt(Float64(Float64(-4.0 * a) * c)));
		end
		tmp_1 = tmp_3;
	elseif (b <= 5.4e-146)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(sqrt(Float64(Float64(c / a) * -4.0)) * -0.5);
		else
			tmp_4 = Float64(Float64(2.0 * c) / Float64(-2.0 * b));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(t_0 + Float64(c / b));
	else
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + b));
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = -b / a;
	tmp_2 = 0.0;
	if (b <= -1.15e-165)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_0;
		else
			tmp_3 = -(c / b);
		end
		tmp_2 = tmp_3;
	elseif (b <= 7e-266)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = ((b + b) / a) * -0.5;
		else
			tmp_4 = (c + c) / sqrt(((-4.0 * a) * c));
		end
		tmp_2 = tmp_4;
	elseif (b <= 5.4e-146)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = sqrt(((c / a) * -4.0)) * -0.5;
		else
			tmp_5 = (2.0 * c) / (-2.0 * b);
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = t_0 + (c / b);
	else
		tmp_2 = (2.0 * c) / (-b + b);
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) / a), $MachinePrecision]}, If[LessEqual[b, -1.15e-165], If[GreaterEqual[b, 0.0], t$95$0, (-N[(c / b), $MachinePrecision])], If[LessEqual[b, 7e-266], If[GreaterEqual[b, 0.0], N[(N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(c + c), $MachinePrecision] / N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 5.4e-146], If[GreaterEqual[b, 0.0], N[(N[Sqrt[N[(N[(c / a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[(-2.0 * b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(t$95$0 + N[(c / b), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + b), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.15e-165

   1. Initial program 70.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 70.6%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 70.6%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 2.6%

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

   11. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lift-/.f64 80.0

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

   13. Applied rewrites 80.0%

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

   14. Taylor expanded in a around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lift-neg.f64 80.0

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

   16. Applied rewrites 80.0%

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

   if -1.15e-165 < b < 7.00000000000000058e-266

   1. Initial program 75.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 75.1%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 58.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 3.5%

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

   11. Taylor expanded in a around inf

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

       1. sqrt-unprod N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 57.6

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

   13. Applied rewrites 57.6%

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

   if 7.00000000000000058e-266 < b < 5.3999999999999999e-146

   1. Initial program 75.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 15.1%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 15.1%

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

   8. Taylor expanded in b around -inf

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

      1. lower-*.f64 15.1

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

   10. Applied rewrites 15.1%

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

   11. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. sqrt-unprod N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 35.6

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

   13. Applied rewrites 35.6%

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

   if 5.3999999999999999e-146 < b

   1. Initial program 73.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 79.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 79.4%

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

   8. Taylor expanded in c around 0

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

      1. lower-+.f64 N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 79.9

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

   10. Applied rewrites 79.9%

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

Alternative 7: 71.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\sqrt{-\frac{c}{a}}\\ t_1 := \frac{-b}{a}\\ \mathbf{if}\;b \leq -3.4 \cdot 10^{-184}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-266}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{-146}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{b - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_1 + \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (sqrt (- (/ c a))))) (t_1 (/ (- b) a)))
   (if (<= b -3.4e-184)
     (if (>= b 0.0) t_1 (- (/ c b)))
     (if (<= b 7e-266)
       (if (>= b 0.0) (* (/ (+ b b) a) -0.5) t_0)
       (if (<= b 5.4e-146)
         (if (>= b 0.0) t_0 (/ (+ c c) (- b b)))
         (if (>= b 0.0) (+ t_1 (/ c b)) (/ (* 2.0 c) (+ (- b) b))))))))

C

double code(double a, double b, double c) {
	double t_0 = -sqrt(-(c / a));
	double t_1 = -b / a;
	double tmp_1;
	if (b <= -3.4e-184) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = -(c / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 7e-266) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = ((b + b) / a) * -0.5;
		} else {
			tmp_3 = t_0;
		}
		tmp_1 = tmp_3;
	} else if (b <= 5.4e-146) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = t_0;
		} else {
			tmp_4 = (c + c) / (b - b);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = t_1 + (c / b);
	} else {
		tmp_1 = (2.0 * c) / (-b + b);
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    real(8) :: tmp_4
    t_0 = -sqrt(-(c / a))
    t_1 = -b / a
    if (b <= (-3.4d-184)) then
        if (b >= 0.0d0) then
            tmp_2 = t_1
        else
            tmp_2 = -(c / b)
        end if
        tmp_1 = tmp_2
    else if (b <= 7d-266) then
        if (b >= 0.0d0) then
            tmp_3 = ((b + b) / a) * (-0.5d0)
        else
            tmp_3 = t_0
        end if
        tmp_1 = tmp_3
    else if (b <= 5.4d-146) then
        if (b >= 0.0d0) then
            tmp_4 = t_0
        else
            tmp_4 = (c + c) / (b - b)
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = t_1 + (c / b)
    else
        tmp_1 = (2.0d0 * c) / (-b + b)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -Math.sqrt(-(c / a));
	double t_1 = -b / a;
	double tmp_1;
	if (b <= -3.4e-184) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = -(c / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 7e-266) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = ((b + b) / a) * -0.5;
		} else {
			tmp_3 = t_0;
		}
		tmp_1 = tmp_3;
	} else if (b <= 5.4e-146) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = t_0;
		} else {
			tmp_4 = (c + c) / (b - b);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = t_1 + (c / b);
	} else {
		tmp_1 = (2.0 * c) / (-b + b);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = -math.sqrt(-(c / a))
	t_1 = -b / a
	tmp_1 = 0
	if b <= -3.4e-184:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_1
		else:
			tmp_2 = -(c / b)
		tmp_1 = tmp_2
	elif b <= 7e-266:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = ((b + b) / a) * -0.5
		else:
			tmp_3 = t_0
		tmp_1 = tmp_3
	elif b <= 5.4e-146:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = t_0
		else:
			tmp_4 = (c + c) / (b - b)
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = t_1 + (c / b)
	else:
		tmp_1 = (2.0 * c) / (-b + b)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(-sqrt(Float64(-Float64(c / a))))
	t_1 = Float64(Float64(-b) / a)
	tmp_1 = 0.0
	if (b <= -3.4e-184)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(-Float64(c / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 7e-266)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(b + b) / a) * -0.5);
		else
			tmp_3 = t_0;
		end
		tmp_1 = tmp_3;
	elseif (b <= 5.4e-146)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = t_0;
		else
			tmp_4 = Float64(Float64(c + c) / Float64(b - b));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(t_1 + Float64(c / b));
	else
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + b));
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = -sqrt(-(c / a));
	t_1 = -b / a;
	tmp_2 = 0.0;
	if (b <= -3.4e-184)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = -(c / b);
		end
		tmp_2 = tmp_3;
	elseif (b <= 7e-266)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = ((b + b) / a) * -0.5;
		else
			tmp_4 = t_0;
		end
		tmp_2 = tmp_4;
	elseif (b <= 5.4e-146)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = t_0;
		else
			tmp_5 = (c + c) / (b - b);
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = t_1 + (c / b);
	else
		tmp_2 = (2.0 * c) / (-b + b);
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = (-N[Sqrt[(-N[(c / a), $MachinePrecision])], $MachinePrecision])}, Block[{t$95$1 = N[((-b) / a), $MachinePrecision]}, If[LessEqual[b, -3.4e-184], If[GreaterEqual[b, 0.0], t$95$1, (-N[(c / b), $MachinePrecision])], If[LessEqual[b, 7e-266], If[GreaterEqual[b, 0.0], N[(N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], t$95$0], If[LessEqual[b, 5.4e-146], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(c + c), $MachinePrecision] / N[(b - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(t$95$1 + N[(c / b), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + b), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -3.40000000000000004e-184

   1. Initial program 70.6%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 70.7%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 70.7%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 2.6%

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

   11. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lift-/.f64 78.6

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

   13. Applied rewrites 78.6%

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

   14. Taylor expanded in a around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lift-neg.f64 78.6

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

   16. Applied rewrites 78.6%

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

   if -3.40000000000000004e-184 < b < 7.00000000000000058e-266

   1. Initial program 75.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 75.1%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 57.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 3.6%

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

   11. Taylor expanded in a around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. sqrt-unprod N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 28.6

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

   13. Applied rewrites 28.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{c}{a} \cdot -1}\\ \end{array} \]
   14. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. *-commutative N/A

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

       4. mul-1-neg N/A

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

       5. lower-neg.f64 N/A

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

       6. lift-/.f64 28.6

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

   15. Applied rewrites 28.6%

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

   if 7.00000000000000058e-266 < b < 5.3999999999999999e-146

   1. Initial program 75.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 75.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 15.1%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 15.1%

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

   11. Taylor expanded in a around -inf

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

       1. sqrt-unprod N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 32.3

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

   13. Applied rewrites 32.3%

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

   14. Taylor expanded in c around -inf

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

       1. mul-1-neg N/A

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

       2. sqrt-prod N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-neg.f64 35.6

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

       7. lift-*.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. *-commutative N/A

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

       10. mul-1-neg N/A

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

       11. lower-neg.f64 N/A

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

       12. lift-/.f64 35.6

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

   16. Applied rewrites 35.6%

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

   if 5.3999999999999999e-146 < b

   1. Initial program 73.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 79.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 79.4%

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

   8. Taylor expanded in c around 0

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

      1. lower-+.f64 N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 79.9

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

   10. Applied rewrites 79.9%

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

Alternative 8: 71.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-b}{a}\\ \mathbf{if}\;b \leq -3.4 \cdot 10^{-184}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-266}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-\frac{c}{a}}\\ \end{array}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{-146}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\sqrt{\frac{c}{a} \cdot -4} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-2 \cdot b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_0 + \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- b) a)))
   (if (<= b -3.4e-184)
     (if (>= b 0.0) t_0 (- (/ c b)))
     (if (<= b 7e-266)
       (if (>= b 0.0) (* (/ (+ b b) a) -0.5) (- (sqrt (- (/ c a)))))
       (if (<= b 5.4e-146)
         (if (>= b 0.0)
           (* (sqrt (* (/ c a) -4.0)) -0.5)
           (/ (* 2.0 c) (* -2.0 b)))
         (if (>= b 0.0) (+ t_0 (/ c b)) (/ (* 2.0 c) (+ (- b) b))))))))

C

double code(double a, double b, double c) {
	double t_0 = -b / a;
	double tmp_1;
	if (b <= -3.4e-184) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = -(c / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 7e-266) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = ((b + b) / a) * -0.5;
		} else {
			tmp_3 = -sqrt(-(c / a));
		}
		tmp_1 = tmp_3;
	} else if (b <= 5.4e-146) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = sqrt(((c / a) * -4.0)) * -0.5;
		} else {
			tmp_4 = (2.0 * c) / (-2.0 * b);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = t_0 + (c / b);
	} else {
		tmp_1 = (2.0 * c) / (-b + b);
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    real(8) :: tmp_4
    t_0 = -b / a
    if (b <= (-3.4d-184)) then
        if (b >= 0.0d0) then
            tmp_2 = t_0
        else
            tmp_2 = -(c / b)
        end if
        tmp_1 = tmp_2
    else if (b <= 7d-266) then
        if (b >= 0.0d0) then
            tmp_3 = ((b + b) / a) * (-0.5d0)
        else
            tmp_3 = -sqrt(-(c / a))
        end if
        tmp_1 = tmp_3
    else if (b <= 5.4d-146) then
        if (b >= 0.0d0) then
            tmp_4 = sqrt(((c / a) * (-4.0d0))) * (-0.5d0)
        else
            tmp_4 = (2.0d0 * c) / ((-2.0d0) * b)
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = t_0 + (c / b)
    else
        tmp_1 = (2.0d0 * c) / (-b + b)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -b / a;
	double tmp_1;
	if (b <= -3.4e-184) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = -(c / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 7e-266) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = ((b + b) / a) * -0.5;
		} else {
			tmp_3 = -Math.sqrt(-(c / a));
		}
		tmp_1 = tmp_3;
	} else if (b <= 5.4e-146) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = Math.sqrt(((c / a) * -4.0)) * -0.5;
		} else {
			tmp_4 = (2.0 * c) / (-2.0 * b);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = t_0 + (c / b);
	} else {
		tmp_1 = (2.0 * c) / (-b + b);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = -b / a
	tmp_1 = 0
	if b <= -3.4e-184:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_0
		else:
			tmp_2 = -(c / b)
		tmp_1 = tmp_2
	elif b <= 7e-266:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = ((b + b) / a) * -0.5
		else:
			tmp_3 = -math.sqrt(-(c / a))
		tmp_1 = tmp_3
	elif b <= 5.4e-146:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = math.sqrt(((c / a) * -4.0)) * -0.5
		else:
			tmp_4 = (2.0 * c) / (-2.0 * b)
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = t_0 + (c / b)
	else:
		tmp_1 = (2.0 * c) / (-b + b)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-b) / a)
	tmp_1 = 0.0
	if (b <= -3.4e-184)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(-Float64(c / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 7e-266)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(b + b) / a) * -0.5);
		else
			tmp_3 = Float64(-sqrt(Float64(-Float64(c / a))));
		end
		tmp_1 = tmp_3;
	elseif (b <= 5.4e-146)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(sqrt(Float64(Float64(c / a) * -4.0)) * -0.5);
		else
			tmp_4 = Float64(Float64(2.0 * c) / Float64(-2.0 * b));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(t_0 + Float64(c / b));
	else
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + b));
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = -b / a;
	tmp_2 = 0.0;
	if (b <= -3.4e-184)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_0;
		else
			tmp_3 = -(c / b);
		end
		tmp_2 = tmp_3;
	elseif (b <= 7e-266)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = ((b + b) / a) * -0.5;
		else
			tmp_4 = -sqrt(-(c / a));
		end
		tmp_2 = tmp_4;
	elseif (b <= 5.4e-146)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = sqrt(((c / a) * -4.0)) * -0.5;
		else
			tmp_5 = (2.0 * c) / (-2.0 * b);
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = t_0 + (c / b);
	else
		tmp_2 = (2.0 * c) / (-b + b);
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) / a), $MachinePrecision]}, If[LessEqual[b, -3.4e-184], If[GreaterEqual[b, 0.0], t$95$0, (-N[(c / b), $MachinePrecision])], If[LessEqual[b, 7e-266], If[GreaterEqual[b, 0.0], N[(N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], (-N[Sqrt[(-N[(c / a), $MachinePrecision])], $MachinePrecision])], If[LessEqual[b, 5.4e-146], If[GreaterEqual[b, 0.0], N[(N[Sqrt[N[(N[(c / a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[(-2.0 * b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(t$95$0 + N[(c / b), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + b), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -3.40000000000000004e-184

   1. Initial program 70.6%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 70.7%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 70.7%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 2.6%

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

   11. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lift-/.f64 78.6

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

   13. Applied rewrites 78.6%

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

   14. Taylor expanded in a around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lift-neg.f64 78.6

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

   16. Applied rewrites 78.6%

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

   if -3.40000000000000004e-184 < b < 7.00000000000000058e-266

   1. Initial program 75.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 75.1%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 57.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 3.6%

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

   11. Taylor expanded in a around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. sqrt-unprod N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 28.6

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

   13. Applied rewrites 28.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{c}{a} \cdot -1}\\ \end{array} \]
   14. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. *-commutative N/A

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

       4. mul-1-neg N/A

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

       5. lower-neg.f64 N/A

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

       6. lift-/.f64 28.6

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

   15. Applied rewrites 28.6%

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

   if 7.00000000000000058e-266 < b < 5.3999999999999999e-146

   1. Initial program 75.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 15.1%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 15.1%

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

   8. Taylor expanded in b around -inf

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

      1. lower-*.f64 15.1

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

   10. Applied rewrites 15.1%

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

   11. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. sqrt-unprod N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 35.6

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

   13. Applied rewrites 35.6%

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

   if 5.3999999999999999e-146 < b

   1. Initial program 73.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 79.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 79.4%

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

   8. Taylor expanded in c around 0

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

      1. lower-+.f64 N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 79.9

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

   10. Applied rewrites 79.9%

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

Alternative 9: 71.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b + b}{a} \cdot -0.5\\ t_1 := -\sqrt{-\frac{c}{a}}\\ \mathbf{if}\;b \leq -3.4 \cdot 10^{-184}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-266}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{-146}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{b - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (/ (+ b b) a) -0.5)) (t_1 (- (sqrt (- (/ c a))))))
   (if (<= b -3.4e-184)
     (if (>= b 0.0) (/ (- b) a) (- (/ c b)))
     (if (<= b 7e-266)
       (if (>= b 0.0) t_0 t_1)
       (if (<= b 5.4e-146)
         (if (>= b 0.0) t_1 (/ (+ c c) (- b b)))
         (if (>= b 0.0) t_0 (- t_1)))))))

C

double code(double a, double b, double c) {
	double t_0 = ((b + b) / a) * -0.5;
	double t_1 = -sqrt(-(c / a));
	double tmp_1;
	if (b <= -3.4e-184) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -b / a;
		} else {
			tmp_2 = -(c / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 7e-266) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_0;
		} else {
			tmp_3 = t_1;
		}
		tmp_1 = tmp_3;
	} else if (b <= 5.4e-146) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = t_1;
		} else {
			tmp_4 = (c + c) / (b - b);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = -t_1;
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    real(8) :: tmp_4
    t_0 = ((b + b) / a) * (-0.5d0)
    t_1 = -sqrt(-(c / a))
    if (b <= (-3.4d-184)) then
        if (b >= 0.0d0) then
            tmp_2 = -b / a
        else
            tmp_2 = -(c / b)
        end if
        tmp_1 = tmp_2
    else if (b <= 7d-266) then
        if (b >= 0.0d0) then
            tmp_3 = t_0
        else
            tmp_3 = t_1
        end if
        tmp_1 = tmp_3
    else if (b <= 5.4d-146) then
        if (b >= 0.0d0) then
            tmp_4 = t_1
        else
            tmp_4 = (c + c) / (b - b)
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = t_0
    else
        tmp_1 = -t_1
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = ((b + b) / a) * -0.5;
	double t_1 = -Math.sqrt(-(c / a));
	double tmp_1;
	if (b <= -3.4e-184) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -b / a;
		} else {
			tmp_2 = -(c / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 7e-266) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_0;
		} else {
			tmp_3 = t_1;
		}
		tmp_1 = tmp_3;
	} else if (b <= 5.4e-146) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = t_1;
		} else {
			tmp_4 = (c + c) / (b - b);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = -t_1;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = ((b + b) / a) * -0.5
	t_1 = -math.sqrt(-(c / a))
	tmp_1 = 0
	if b <= -3.4e-184:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -b / a
		else:
			tmp_2 = -(c / b)
		tmp_1 = tmp_2
	elif b <= 7e-266:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = t_0
		else:
			tmp_3 = t_1
		tmp_1 = tmp_3
	elif b <= 5.4e-146:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = t_1
		else:
			tmp_4 = (c + c) / (b - b)
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = t_0
	else:
		tmp_1 = -t_1
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(b + b) / a) * -0.5)
	t_1 = Float64(-sqrt(Float64(-Float64(c / a))))
	tmp_1 = 0.0
	if (b <= -3.4e-184)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-b) / a);
		else
			tmp_2 = Float64(-Float64(c / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 7e-266)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_0;
		else
			tmp_3 = t_1;
		end
		tmp_1 = tmp_3;
	elseif (b <= 5.4e-146)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = t_1;
		else
			tmp_4 = Float64(Float64(c + c) / Float64(b - b));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = t_0;
	else
		tmp_1 = Float64(-t_1);
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = ((b + b) / a) * -0.5;
	t_1 = -sqrt(-(c / a));
	tmp_2 = 0.0;
	if (b <= -3.4e-184)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -b / a;
		else
			tmp_3 = -(c / b);
		end
		tmp_2 = tmp_3;
	elseif (b <= 7e-266)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = t_0;
		else
			tmp_4 = t_1;
		end
		tmp_2 = tmp_4;
	elseif (b <= 5.4e-146)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = t_1;
		else
			tmp_5 = (c + c) / (b - b);
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = t_0;
	else
		tmp_2 = -t_1;
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[(-N[(c / a), $MachinePrecision])], $MachinePrecision])}, If[LessEqual[b, -3.4e-184], If[GreaterEqual[b, 0.0], N[((-b) / a), $MachinePrecision], (-N[(c / b), $MachinePrecision])], If[LessEqual[b, 7e-266], If[GreaterEqual[b, 0.0], t$95$0, t$95$1], If[LessEqual[b, 5.4e-146], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(c + c), $MachinePrecision] / N[(b - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$0, (-t$95$1)]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -3.40000000000000004e-184

   1. Initial program 70.6%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 70.7%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 70.7%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 2.6%

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

   11. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lift-/.f64 78.6

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

   13. Applied rewrites 78.6%

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

   14. Taylor expanded in a around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lift-neg.f64 78.6

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

   16. Applied rewrites 78.6%

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

   if -3.40000000000000004e-184 < b < 7.00000000000000058e-266

   1. Initial program 75.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 75.1%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 57.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 3.6%

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

   11. Taylor expanded in a around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. sqrt-unprod N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 28.6

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

   13. Applied rewrites 28.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{c}{a} \cdot -1}\\ \end{array} \]
   14. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. *-commutative N/A

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

       4. mul-1-neg N/A

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

       5. lower-neg.f64 N/A

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

       6. lift-/.f64 28.6

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

   15. Applied rewrites 28.6%

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

   if 7.00000000000000058e-266 < b < 5.3999999999999999e-146

   1. Initial program 75.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 75.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 15.1%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 15.1%

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

   11. Taylor expanded in a around -inf

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

       1. sqrt-unprod N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 32.3

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

   13. Applied rewrites 32.3%

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

   14. Taylor expanded in c around -inf

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

       1. mul-1-neg N/A

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

       2. sqrt-prod N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-neg.f64 35.6

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

       7. lift-*.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. *-commutative N/A

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

       10. mul-1-neg N/A

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

       11. lower-neg.f64 N/A

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

       12. lift-/.f64 35.6

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

   16. Applied rewrites 35.6%

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

   if 5.3999999999999999e-146 < b

   1. Initial program 73.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 73.1%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 79.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 79.4%

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

   11. Taylor expanded in a around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. sqrt-unprod N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 79.4

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

   13. Applied rewrites 79.4%

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

   14. Taylor expanded in c around -inf

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

       1. mul-1-neg N/A

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

       2. sqrt-prod N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-neg.f64 79.4

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

       7. lift-*.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. *-commutative N/A

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

       10. mul-1-neg N/A

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

       11. lower-neg.f64 N/A

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

       12. lift-/.f64 79.4

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

   16. Applied rewrites 79.4%

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

Alternative 10: 71.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\sqrt{-\frac{c}{a}}\\ t_1 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;-t\_0\\ \end{array}\\ \mathbf{if}\;b \leq -4.2 \cdot 10^{-165}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-266}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{-146}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{b - b}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (sqrt (- (/ c a)))))
        (t_1 (if (>= b 0.0) (* (/ (+ b b) a) -0.5) (- t_0))))
   (if (<= b -4.2e-165)
     (if (>= b 0.0) (/ (- b) a) (- (/ c b)))
     (if (<= b 7e-266)
       t_1
       (if (<= b 5.4e-146) (if (>= b 0.0) t_0 (/ (+ c c) (- b b))) t_1)))))

C

double code(double a, double b, double c) {
	double t_0 = -sqrt(-(c / a));
	double tmp;
	if (b >= 0.0) {
		tmp = ((b + b) / a) * -0.5;
	} else {
		tmp = -t_0;
	}
	double t_1 = tmp;
	double tmp_2;
	if (b <= -4.2e-165) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -b / a;
		} else {
			tmp_3 = -(c / b);
		}
		tmp_2 = tmp_3;
	} else if (b <= 7e-266) {
		tmp_2 = t_1;
	} else if (b <= 5.4e-146) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = t_0;
		} else {
			tmp_4 = (c + c) / (b - b);
		}
		tmp_2 = tmp_4;
	} else {
		tmp_2 = t_1;
	}
	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
    real(8) :: tmp_4
    t_0 = -sqrt(-(c / a))
    if (b >= 0.0d0) then
        tmp = ((b + b) / a) * (-0.5d0)
    else
        tmp = -t_0
    end if
    t_1 = tmp
    if (b <= (-4.2d-165)) then
        if (b >= 0.0d0) then
            tmp_3 = -b / a
        else
            tmp_3 = -(c / b)
        end if
        tmp_2 = tmp_3
    else if (b <= 7d-266) then
        tmp_2 = t_1
    else if (b <= 5.4d-146) then
        if (b >= 0.0d0) then
            tmp_4 = t_0
        else
            tmp_4 = (c + c) / (b - b)
        end if
        tmp_2 = tmp_4
    else
        tmp_2 = t_1
    end if
    code = tmp_2
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -Math.sqrt(-(c / a));
	double tmp;
	if (b >= 0.0) {
		tmp = ((b + b) / a) * -0.5;
	} else {
		tmp = -t_0;
	}
	double t_1 = tmp;
	double tmp_2;
	if (b <= -4.2e-165) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -b / a;
		} else {
			tmp_3 = -(c / b);
		}
		tmp_2 = tmp_3;
	} else if (b <= 7e-266) {
		tmp_2 = t_1;
	} else if (b <= 5.4e-146) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = t_0;
		} else {
			tmp_4 = (c + c) / (b - b);
		}
		tmp_2 = tmp_4;
	} else {
		tmp_2 = t_1;
	}
	return tmp_2;
}

Python

def code(a, b, c):
	t_0 = -math.sqrt(-(c / a))
	tmp = 0
	if b >= 0.0:
		tmp = ((b + b) / a) * -0.5
	else:
		tmp = -t_0
	t_1 = tmp
	tmp_2 = 0
	if b <= -4.2e-165:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = -b / a
		else:
			tmp_3 = -(c / b)
		tmp_2 = tmp_3
	elif b <= 7e-266:
		tmp_2 = t_1
	elif b <= 5.4e-146:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = t_0
		else:
			tmp_4 = (c + c) / (b - b)
		tmp_2 = tmp_4
	else:
		tmp_2 = t_1
	return tmp_2

Julia

function code(a, b, c)
	t_0 = Float64(-sqrt(Float64(-Float64(c / a))))
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(Float64(b + b) / a) * -0.5);
	else
		tmp = Float64(-t_0);
	end
	t_1 = tmp
	tmp_2 = 0.0
	if (b <= -4.2e-165)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-b) / a);
		else
			tmp_3 = Float64(-Float64(c / b));
		end
		tmp_2 = tmp_3;
	elseif (b <= 7e-266)
		tmp_2 = t_1;
	elseif (b <= 5.4e-146)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = t_0;
		else
			tmp_4 = Float64(Float64(c + c) / Float64(b - b));
		end
		tmp_2 = tmp_4;
	else
		tmp_2 = t_1;
	end
	return tmp_2
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = -sqrt(-(c / a));
	tmp = 0.0;
	if (b >= 0.0)
		tmp = ((b + b) / a) * -0.5;
	else
		tmp = -t_0;
	end
	t_1 = tmp;
	tmp_3 = 0.0;
	if (b <= -4.2e-165)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = -b / a;
		else
			tmp_4 = -(c / b);
		end
		tmp_3 = tmp_4;
	elseif (b <= 7e-266)
		tmp_3 = t_1;
	elseif (b <= 5.4e-146)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = t_0;
		else
			tmp_5 = (c + c) / (b - b);
		end
		tmp_3 = tmp_5;
	else
		tmp_3 = t_1;
	end
	tmp_6 = tmp_3;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = (-N[Sqrt[(-N[(c / a), $MachinePrecision])], $MachinePrecision])}, Block[{t$95$1 = If[GreaterEqual[b, 0.0], N[(N[(N[(b + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], (-t$95$0)]}, If[LessEqual[b, -4.2e-165], If[GreaterEqual[b, 0.0], N[((-b) / a), $MachinePrecision], (-N[(c / b), $MachinePrecision])], If[LessEqual[b, 7e-266], t$95$1, If[LessEqual[b, 5.4e-146], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(c + c), $MachinePrecision] / N[(b - b), $MachinePrecision]), $MachinePrecision]], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.1999999999999999e-165

   1. Initial program 70.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 70.6%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 70.6%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 2.6%

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

   11. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lift-/.f64 80.0

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

   13. Applied rewrites 80.0%

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

   14. Taylor expanded in a around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lift-neg.f64 80.0

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

   16. Applied rewrites 80.0%

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

   if -4.1999999999999999e-165 < b < 7.00000000000000058e-266 or 5.3999999999999999e-146 < b

   1. Initial program 73.5%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 73.5%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 75.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 64.0%

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

   11. Taylor expanded in a around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. sqrt-unprod N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 69.3

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

   13. Applied rewrites 69.3%

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

   14. Taylor expanded in c around -inf

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

       1. mul-1-neg N/A

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

       2. sqrt-prod N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-neg.f64 69.0

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

       7. lift-*.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. *-commutative N/A

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

       10. mul-1-neg N/A

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

       11. lower-neg.f64 N/A

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

       12. lift-/.f64 69.0

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

   16. Applied rewrites 69.0%

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

   if 7.00000000000000058e-266 < b < 5.3999999999999999e-146

   1. Initial program 75.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 75.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 15.1%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 15.1%

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

   11. Taylor expanded in a around -inf

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

       1. sqrt-unprod N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 32.3

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

   13. Applied rewrites 32.3%

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

   14. Taylor expanded in c around -inf

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

       1. mul-1-neg N/A

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

       2. sqrt-prod N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-neg.f64 35.6

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

       7. lift-*.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. *-commutative N/A

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

       10. mul-1-neg N/A

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

       11. lower-neg.f64 N/A

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

       12. lift-/.f64 35.6

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

   16. Applied rewrites 35.6%

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

Alternative 11: 69.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c + c}{b - b}\\ t_1 := \frac{-b}{a}\\ \mathbf{if}\;b \leq 7 \cdot 10^{-266}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{-146}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-\sqrt{-\frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (+ c c) (- b b))) (t_1 (/ (- b) a)))
   (if (<= b 7e-266)
     (if (>= b 0.0) t_1 (- (/ c b)))
     (if (<= b 5.4e-146)
       (if (>= b 0.0) (- (sqrt (- (/ c a)))) t_0)
       (if (>= b 0.0) t_1 t_0)))))

C

double code(double a, double b, double c) {
	double t_0 = (c + c) / (b - b);
	double t_1 = -b / a;
	double tmp_1;
	if (b <= 7e-266) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = -(c / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 5.4e-146) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -sqrt(-(c / a));
		} else {
			tmp_3 = t_0;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = t_1;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = (c + c) / (b - b)
    t_1 = -b / a
    if (b <= 7d-266) then
        if (b >= 0.0d0) then
            tmp_2 = t_1
        else
            tmp_2 = -(c / b)
        end if
        tmp_1 = tmp_2
    else if (b <= 5.4d-146) then
        if (b >= 0.0d0) then
            tmp_3 = -sqrt(-(c / a))
        else
            tmp_3 = t_0
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = t_1
    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) / (b - b);
	double t_1 = -b / a;
	double tmp_1;
	if (b <= 7e-266) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = -(c / b);
		}
		tmp_1 = tmp_2;
	} else if (b <= 5.4e-146) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -Math.sqrt(-(c / a));
		} else {
			tmp_3 = t_0;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = t_1;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (c + c) / (b - b)
	t_1 = -b / a
	tmp_1 = 0
	if b <= 7e-266:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_1
		else:
			tmp_2 = -(c / b)
		tmp_1 = tmp_2
	elif b <= 5.4e-146:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = -math.sqrt(-(c / a))
		else:
			tmp_3 = t_0
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = t_1
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c + c) / Float64(b - b))
	t_1 = Float64(Float64(-b) / a)
	tmp_1 = 0.0
	if (b <= 7e-266)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(-Float64(c / b));
		end
		tmp_1 = tmp_2;
	elseif (b <= 5.4e-146)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(-sqrt(Float64(-Float64(c / a))));
		else
			tmp_3 = t_0;
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = t_1;
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = (c + c) / (b - b);
	t_1 = -b / a;
	tmp_2 = 0.0;
	if (b <= 7e-266)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = -(c / b);
		end
		tmp_2 = tmp_3;
	elseif (b <= 5.4e-146)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = -sqrt(-(c / a));
		else
			tmp_4 = t_0;
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = t_1;
	else
		tmp_2 = t_0;
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c + c), $MachinePrecision] / N[(b - b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-b) / a), $MachinePrecision]}, If[LessEqual[b, 7e-266], If[GreaterEqual[b, 0.0], t$95$1, (-N[(c / b), $MachinePrecision])], If[LessEqual[b, 5.4e-146], If[GreaterEqual[b, 0.0], (-N[Sqrt[(-N[(c / a), $MachinePrecision])], $MachinePrecision]), t$95$0], If[GreaterEqual[b, 0.0], t$95$1, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 7.00000000000000058e-266

   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 0

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

   4. Applied rewrites 71.5%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 68.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 2.8%

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

   11. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lift-/.f64 65.7

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

   13. Applied rewrites 65.7%

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

   14. Taylor expanded in a around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lift-neg.f64 65.7

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

   16. Applied rewrites 65.7%

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

   if 7.00000000000000058e-266 < b < 5.3999999999999999e-146

   1. Initial program 75.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 75.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 15.1%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 15.1%

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

   11. Taylor expanded in a around -inf

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

       1. sqrt-unprod N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 32.3

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

   13. Applied rewrites 32.3%

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

   14. Taylor expanded in c around -inf

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

       1. mul-1-neg N/A

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

       2. sqrt-prod N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-neg.f64 35.6

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

       7. lift-*.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. *-commutative N/A

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

       10. mul-1-neg N/A

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

       11. lower-neg.f64 N/A

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

       12. lift-/.f64 35.6

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

   16. Applied rewrites 35.6%

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

   if 5.3999999999999999e-146 < b

   1. Initial program 73.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 73.1%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 79.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 79.4%

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

   11. Taylor expanded in a around 0

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

       1. mul-1-neg N/A

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

       2. distribute-frac-neg N/A

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

       3. lift-neg.f64 N/A

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

       4. lift-/.f64 79.4

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

   13. Applied rewrites 79.4%

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

Alternative 12: 69.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\\ \mathbf{if}\;b \leq 7 \cdot 10^{-266}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{-146}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-\sqrt{-\frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{b - b}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (if (>= b 0.0) (/ (- b) a) (- (/ c b)))))
   (if (<= b 7e-266)
     t_0
     (if (<= b 5.4e-146)
       (if (>= b 0.0) (- (sqrt (- (/ c a)))) (/ (+ c c) (- b b)))
       t_0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -b / a;
	} else {
		tmp = -(c / b);
	}
	double t_0 = tmp;
	double tmp_1;
	if (b <= 7e-266) {
		tmp_1 = t_0;
	} else if (b <= 5.4e-146) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -sqrt(-(c / a));
		} else {
			tmp_2 = (c + c) / (b - b);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    if (b >= 0.0d0) then
        tmp = -b / a
    else
        tmp = -(c / b)
    end if
    t_0 = tmp
    if (b <= 7d-266) then
        tmp_1 = t_0
    else if (b <= 5.4d-146) then
        if (b >= 0.0d0) then
            tmp_2 = -sqrt(-(c / a))
        else
            tmp_2 = (c + c) / (b - b)
        end if
        tmp_1 = tmp_2
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -b / a;
	} else {
		tmp = -(c / b);
	}
	double t_0 = tmp;
	double tmp_1;
	if (b <= 7e-266) {
		tmp_1 = t_0;
	} else if (b <= 5.4e-146) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -Math.sqrt(-(c / a));
		} else {
			tmp_2 = (c + c) / (b - b);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = -b / a
	else:
		tmp = -(c / b)
	t_0 = tmp
	tmp_1 = 0
	if b <= 7e-266:
		tmp_1 = t_0
	elif b <= 5.4e-146:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -math.sqrt(-(c / a))
		else:
			tmp_2 = (c + c) / (b - b)
		tmp_1 = tmp_2
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(-Float64(c / b));
	end
	t_0 = tmp
	tmp_1 = 0.0
	if (b <= 7e-266)
		tmp_1 = t_0;
	elseif (b <= 5.4e-146)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-sqrt(Float64(-Float64(c / a))));
		else
			tmp_2 = Float64(Float64(c + c) / Float64(b - b));
		end
		tmp_1 = tmp_2;
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = -b / a;
	else
		tmp = -(c / b);
	end
	t_0 = tmp;
	tmp_2 = 0.0;
	if (b <= 7e-266)
		tmp_2 = t_0;
	elseif (b <= 5.4e-146)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -sqrt(-(c / a));
		else
			tmp_3 = (c + c) / (b - b);
		end
		tmp_2 = tmp_3;
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = If[GreaterEqual[b, 0.0], N[((-b) / a), $MachinePrecision], (-N[(c / b), $MachinePrecision])]}, If[LessEqual[b, 7e-266], t$95$0, If[LessEqual[b, 5.4e-146], If[GreaterEqual[b, 0.0], (-N[Sqrt[(-N[(c / a), $MachinePrecision])], $MachinePrecision]), N[(N[(c + c), $MachinePrecision] / N[(b - b), $MachinePrecision]), $MachinePrecision]], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < 7.00000000000000058e-266 or 5.3999999999999999e-146 < b

   1. Initial program 72.2%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 73.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 36.9%

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

   11. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lift-/.f64 71.8

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

   13. Applied rewrites 71.8%

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

   14. Taylor expanded in a around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lift-neg.f64 71.8

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

   16. Applied rewrites 71.8%

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

   if 7.00000000000000058e-266 < b < 5.3999999999999999e-146

   1. Initial program 75.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 75.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 15.1%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 15.1%

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

   11. Taylor expanded in a around -inf

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

       1. sqrt-unprod N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 32.3

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

   13. Applied rewrites 32.3%

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

   14. Taylor expanded in c around -inf

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

       1. mul-1-neg N/A

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

       2. sqrt-prod N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-neg.f64 35.6

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

       7. lift-*.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. *-commutative N/A

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

       10. mul-1-neg N/A

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

       11. lower-neg.f64 N/A

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

       12. lift-/.f64 35.6

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

   16. Applied rewrites 35.6%

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

Alternative 13: 67.8% accurate, 3.1× speedup

Math

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

FPCore

(FPCore (a b c) :precision binary64 (if (>= b 0.0) (/ (- b) a) (- (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = -b / a;
	} else {
		tmp = -(c / 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 = -b / a
    else
        tmp = -(c / 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 = -b / a;
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = -b / a
	else:
		tmp = -(c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(-Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = -b / a;
	else
		tmp = -(c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[((-b) / a), $MachinePrecision], (-N[(c / b), $MachinePrecision])]

Derivation

1. Initial program 72.4%

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

2. Taylor expanded in a around 0

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

4. Applied rewrites 72.5%

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

5. Taylor expanded in a around 0

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

7. Applied rewrites 69.0%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 35.3%

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

11. Taylor expanded in b around -inf

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

    1. mul-1-neg N/A

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

    2. lower-neg.f64 N/A

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

    3. lift-/.f64 67.7

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

13. Applied rewrites 67.7%

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

14. Taylor expanded in a around 0

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

    1. associate-*r/ N/A

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

    2. lower-/.f64 N/A

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

    3. mul-1-neg N/A

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

    4. lift-neg.f64 67.8

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

16. Applied rewrites 67.8%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{\color{blue}{a}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2025117 
(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)))))))