jeff quadratic root 2

Percentage Accurate: 71.8% → 88.9%

Time: 23.6s

Alternatives: 8

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 71.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ t_1 := \mathsf{fma}\left(\sqrt{\frac{c}{a} \cdot -4}, 0.5, \frac{b}{a} \cdot -0.5\right)\\ t_2 := 2 \cdot \left(-c\right)\\ t_3 := \frac{t\_2}{b + t\_0}\\ \mathbf{if}\;b \leq -4 \cdot 10^{+99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq -2.6 \cdot 10^{-271}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + t\_0}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{+53}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{t\_2}{b + b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* (* 4.0 a) c))))
        (t_1 (fma (sqrt (* (/ c a) -4.0)) 0.5 (* (/ b a) -0.5)))
        (t_2 (* 2.0 (- c)))
        (t_3 (/ t_2 (+ b t_0))))
   (if (<= b -4e+99)
     (if (>= b 0.0) t_3 (/ (+ (- b) (- b)) (* 2.0 a)))
     (if (<= b -2.6e-271)
       (if (>= b 0.0) (- (sqrt (/ (- c) a))) (/ (+ (- b) t_0) (* 2.0 a)))
       (if (<= b 9.8e+53)
         (if (>= b 0.0) t_3 t_1)
         (if (>= b 0.0) (/ t_2 (+ b b)) t_1))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	double t_1 = fma(sqrt(((c / a) * -4.0)), 0.5, ((b / a) * -0.5));
	double t_2 = 2.0 * -c;
	double t_3 = t_2 / (b + t_0);
	double tmp_1;
	if (b <= -4e+99) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_3;
		} else {
			tmp_2 = (-b + -b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= -2.6e-271) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -sqrt((-c / a));
		} else {
			tmp_3 = (-b + t_0) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 9.8e+53) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = t_3;
		} else {
			tmp_4 = t_1;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = t_2 / (b + b);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	t_1 = fma(sqrt(Float64(Float64(c / a) * -4.0)), 0.5, Float64(Float64(b / a) * -0.5))
	t_2 = Float64(2.0 * Float64(-c))
	t_3 = Float64(t_2 / Float64(b + t_0))
	tmp_1 = 0.0
	if (b <= -4e+99)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_3;
		else
			tmp_2 = Float64(Float64(Float64(-b) + Float64(-b)) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= -2.6e-271)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(-sqrt(Float64(Float64(-c) / a)));
		else
			tmp_3 = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (b <= 9.8e+53)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = t_3;
		else
			tmp_4 = t_1;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(t_2 / Float64(b + b));
	else
		tmp_1 = t_1;
	end
	return tmp_1
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]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(c / a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] * 0.5 + N[(N[(b / a), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * (-c)), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(b + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4e+99], If[GreaterEqual[b, 0.0], t$95$3, N[(N[((-b) + (-b)), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -2.6e-271], If[GreaterEqual[b, 0.0], (-N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision]), N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 9.8e+53], If[GreaterEqual[b, 0.0], t$95$3, t$95$1], If[GreaterEqual[b, 0.0], N[(t$95$2 / N[(b + b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -3.9999999999999999e99

   1. Initial program 57.6%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 97.8

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

   5. Applied rewrites 97.8%

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

   if -3.9999999999999999e99 < b < -2.6e-271

   1. Initial program 88.2%

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

   3. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. sqrt-unprod N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 88.2

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-\frac{\mathsf{fma}\left(0.5, b, -\sqrt{\left(a \cdot c\right) \cdot -1}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Applied rewrites 88.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-\frac{\mathsf{fma}\left(0.5, b, -\sqrt{\left(a \cdot c\right) \cdot -1}\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   6. Taylor expanded in a around -inf

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

      1. sqrt-prod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 88.2

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

   8. Applied rewrites 88.2%

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

   if -2.6e-271 < b < 9.80000000000000036e53

   1. Initial program 83.3%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 84.1

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

   5. Applied rewrites 84.1%

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

   if 9.80000000000000036e53 < b

   1. Initial program 61.0%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 95.9

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

   8. Applied rewrites 95.9%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot \left(-c\right)}{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq -2.6 \cdot 10^{-271}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{+53}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot \left(-c\right)}{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{c}{a} \cdot -4}, 0.5, \frac{b}{a} \cdot -0.5\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot \left(-c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{c}{a} \cdot -4}, 0.5, \frac{b}{a} \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ t_1 := 2 \cdot \left(-c\right)\\ t_2 := \frac{t\_1}{b + t\_0}\\ \mathbf{if}\;b \leq -4 \cdot 10^{+99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{+53}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + t\_0}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{t\_1}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{c}{a} \cdot -4}, 0.5, \frac{b}{a} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* (* 4.0 a) c))))
        (t_1 (* 2.0 (- c)))
        (t_2 (/ t_1 (+ b t_0))))
   (if (<= b -4e+99)
     (if (>= b 0.0) t_2 (/ (+ (- b) (- b)) (* 2.0 a)))
     (if (<= b 9.8e+53)
       (if (>= b 0.0) t_2 (/ (+ (- b) t_0) (* 2.0 a)))
       (if (>= b 0.0)
         (/ t_1 (+ b b))
         (fma (sqrt (* (/ c a) -4.0)) 0.5 (* (/ b a) -0.5)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	double t_1 = 2.0 * -c;
	double t_2 = t_1 / (b + t_0);
	double tmp_1;
	if (b <= -4e+99) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_2;
		} else {
			tmp_2 = (-b + -b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 9.8e+53) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_2;
		} else {
			tmp_3 = (-b + t_0) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = t_1 / (b + b);
	} else {
		tmp_1 = fma(sqrt(((c / a) * -4.0)), 0.5, ((b / a) * -0.5));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	t_1 = Float64(2.0 * Float64(-c))
	t_2 = Float64(t_1 / Float64(b + t_0))
	tmp_1 = 0.0
	if (b <= -4e+99)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_2;
		else
			tmp_2 = Float64(Float64(Float64(-b) + Float64(-b)) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 9.8e+53)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_2;
		else
			tmp_3 = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(t_1 / Float64(b + b));
	else
		tmp_1 = fma(sqrt(Float64(Float64(c / a) * -4.0)), 0.5, Float64(Float64(b / a) * -0.5));
	end
	return tmp_1
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]}, Block[{t$95$1 = N[(2.0 * (-c)), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(b + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4e+99], If[GreaterEqual[b, 0.0], t$95$2, N[(N[((-b) + (-b)), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 9.8e+53], If[GreaterEqual[b, 0.0], t$95$2, N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(t$95$1 / N[(b + b), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(c / a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] * 0.5 + N[(N[(b / a), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.9999999999999999e99

   1. Initial program 57.6%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 97.8

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

   5. Applied rewrites 97.8%

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

   if -3.9999999999999999e99 < b < 9.80000000000000036e53

   1. Initial program 86.0%

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

   if 9.80000000000000036e53 < b

   1. Initial program 61.0%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 95.9

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

   8. Applied rewrites 95.9%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot \left(-c\right)}{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{+53}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot \left(-c\right)}{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot \left(-c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{c}{a} \cdot -4}, 0.5, \frac{b}{a} \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ t_1 := 2 \cdot \left(-c\right)\\ t_2 := \frac{t\_1}{b + b}\\ t_3 := \mathsf{fma}\left(\sqrt{\frac{c}{a} \cdot -4}, 0.5, \frac{b}{a} \cdot -0.5\right)\\ \mathbf{if}\;b \leq -4 \cdot 10^{+99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) \cdot \mathsf{fma}\left(a, \frac{\frac{c}{b}}{b} \cdot -2, 2\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq -2.6 \cdot 10^{-271}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + t\_0}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{+53}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{t\_1}{b + t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* (* 4.0 a) c))))
        (t_1 (* 2.0 (- c)))
        (t_2 (/ t_1 (+ b b)))
        (t_3 (fma (sqrt (* (/ c a) -4.0)) 0.5 (* (/ b a) -0.5))))
   (if (<= b -4e+99)
     (if (>= b 0.0)
       t_2
       (/ (* (- b) (fma a (* (/ (/ c b) b) -2.0) 2.0)) (* 2.0 a)))
     (if (<= b -2.6e-271)
       (if (>= b 0.0) (- (sqrt (/ (- c) a))) (/ (+ (- b) t_0) (* 2.0 a)))
       (if (<= b 9.8e+53)
         (if (>= b 0.0) (/ t_1 (+ b t_0)) t_3)
         (if (>= b 0.0) t_2 t_3))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	double t_1 = 2.0 * -c;
	double t_2 = t_1 / (b + b);
	double t_3 = fma(sqrt(((c / a) * -4.0)), 0.5, ((b / a) * -0.5));
	double tmp_1;
	if (b <= -4e+99) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_2;
		} else {
			tmp_2 = (-b * fma(a, (((c / b) / b) * -2.0), 2.0)) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= -2.6e-271) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -sqrt((-c / a));
		} else {
			tmp_3 = (-b + t_0) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 9.8e+53) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = t_1 / (b + t_0);
		} else {
			tmp_4 = t_3;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = t_2;
	} else {
		tmp_1 = t_3;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	t_1 = Float64(2.0 * Float64(-c))
	t_2 = Float64(t_1 / Float64(b + b))
	t_3 = fma(sqrt(Float64(Float64(c / a) * -4.0)), 0.5, Float64(Float64(b / a) * -0.5))
	tmp_1 = 0.0
	if (b <= -4e+99)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_2;
		else
			tmp_2 = Float64(Float64(Float64(-b) * fma(a, Float64(Float64(Float64(c / b) / b) * -2.0), 2.0)) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= -2.6e-271)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(-sqrt(Float64(Float64(-c) / a)));
		else
			tmp_3 = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (b <= 9.8e+53)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(t_1 / Float64(b + t_0));
		else
			tmp_4 = t_3;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = t_2;
	else
		tmp_1 = t_3;
	end
	return tmp_1
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]}, Block[{t$95$1 = N[(2.0 * (-c)), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(b + b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(c / a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] * 0.5 + N[(N[(b / a), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4e+99], If[GreaterEqual[b, 0.0], t$95$2, N[(N[((-b) * N[(a * N[(N[(N[(c / b), $MachinePrecision] / b), $MachinePrecision] * -2.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -2.6e-271], If[GreaterEqual[b, 0.0], (-N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision]), N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 9.8e+53], If[GreaterEqual[b, 0.0], N[(t$95$1 / N[(b + t$95$0), $MachinePrecision]), $MachinePrecision], t$95$3], If[GreaterEqual[b, 0.0], t$95$2, t$95$3]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -3.9999999999999999e99

   1. Initial program 57.6%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

   8. Applied rewrites 97.8%

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

   if -3.9999999999999999e99 < b < -2.6e-271

   1. Initial program 88.2%

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

   3. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. sqrt-unprod N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 88.2

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-\frac{\mathsf{fma}\left(0.5, b, -\sqrt{\left(a \cdot c\right) \cdot -1}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Applied rewrites 88.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-\frac{\mathsf{fma}\left(0.5, b, -\sqrt{\left(a \cdot c\right) \cdot -1}\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   6. Taylor expanded in a around -inf

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

      1. sqrt-prod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 88.2

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

   8. Applied rewrites 88.2%

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

   if -2.6e-271 < b < 9.80000000000000036e53

   1. Initial program 83.3%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 84.1

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

   5. Applied rewrites 84.1%

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

   if 9.80000000000000036e53 < b

   1. Initial program 61.0%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 95.9

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

   8. Applied rewrites 95.9%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot \left(-c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) \cdot \mathsf{fma}\left(a, \frac{\frac{c}{b}}{b} \cdot -2, 2\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq -2.6 \cdot 10^{-271}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{+53}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot \left(-c\right)}{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{c}{a} \cdot -4}, 0.5, \frac{b}{a} \cdot -0.5\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot \left(-c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{c}{a} \cdot -4}, 0.5, \frac{b}{a} \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2 \cdot \left(-c\right)}{b + b}\\ t_1 := \mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)\\ \mathbf{if}\;b \leq -4 \cdot 10^{+99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) \cdot \mathsf{fma}\left(a, \frac{\frac{c}{b}}{b} \cdot -2, 2\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-271}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, b, -\sqrt{\left(a \cdot c\right) \cdot -1}\right)}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{+53}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-2\right) \cdot \frac{c}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{{t\_1}^{1.5} - {b}^{3}}{a \cdot \left(\mathsf{fma}\left(-4, a \cdot c, 2 \cdot \left(b \cdot b\right)\right) + b \cdot \sqrt{t\_1}\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{c}{a} \cdot -4}, 0.5, \frac{b}{a} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* 2.0 (- c)) (+ b b))) (t_1 (fma -4.0 (* a c) (* b b))))
   (if (<= b -4e+99)
     (if (>= b 0.0)
       t_0
       (/ (* (- b) (fma a (* (/ (/ c b) b) -2.0) 2.0)) (* 2.0 a)))
     (if (<= b 1.25e-271)
       (if (>= b 0.0)
         (/ (fma 0.5 b (- (sqrt (* (* a c) -1.0)))) (- a))
         (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
       (if (<= b 9.8e+53)
         (if (>= b 0.0)
           (* (- 2.0) (/ c (+ b (sqrt (- (* b b) (* 4.0 (* a c)))))))
           (*
            0.5
            (/
             (- (pow t_1 1.5) (pow b 3.0))
             (* a (+ (fma -4.0 (* a c) (* 2.0 (* b b))) (* b (sqrt t_1)))))))
         (if (>= b 0.0)
           t_0
           (fma (sqrt (* (/ c a) -4.0)) 0.5 (* (/ b a) -0.5))))))))

C

double code(double a, double b, double c) {
	double t_0 = (2.0 * -c) / (b + b);
	double t_1 = fma(-4.0, (a * c), (b * b));
	double tmp_1;
	if (b <= -4e+99) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = (-b * fma(a, (((c / b) / b) * -2.0), 2.0)) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.25e-271) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = fma(0.5, b, -sqrt(((a * c) * -1.0))) / -a;
		} else {
			tmp_3 = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 9.8e+53) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = -2.0 * (c / (b + sqrt(((b * b) - (4.0 * (a * c))))));
		} else {
			tmp_4 = 0.5 * ((pow(t_1, 1.5) - pow(b, 3.0)) / (a * (fma(-4.0, (a * c), (2.0 * (b * b))) + (b * sqrt(t_1)))));
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = fma(sqrt(((c / a) * -4.0)), 0.5, ((b / a) * -0.5));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(2.0 * Float64(-c)) / Float64(b + b))
	t_1 = fma(-4.0, Float64(a * c), Float64(b * b))
	tmp_1 = 0.0
	if (b <= -4e+99)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(Float64(Float64(-b) * fma(a, Float64(Float64(Float64(c / b) / b) * -2.0), 2.0)) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.25e-271)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(fma(0.5, b, Float64(-sqrt(Float64(Float64(a * c) * -1.0)))) / Float64(-a));
		else
			tmp_3 = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (b <= 9.8e+53)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(-2.0) * Float64(c / Float64(b + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))))));
		else
			tmp_4 = Float64(0.5 * Float64(Float64((t_1 ^ 1.5) - (b ^ 3.0)) / Float64(a * Float64(fma(-4.0, Float64(a * c), Float64(2.0 * Float64(b * b))) + Float64(b * sqrt(t_1))))));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = t_0;
	else
		tmp_1 = fma(sqrt(Float64(Float64(c / a) * -4.0)), 0.5, Float64(Float64(b / a) * -0.5));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(2.0 * (-c)), $MachinePrecision] / N[(b + b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4e+99], If[GreaterEqual[b, 0.0], t$95$0, N[(N[((-b) * N[(a * N[(N[(N[(c / b), $MachinePrecision] / b), $MachinePrecision] * -2.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.25e-271], If[GreaterEqual[b, 0.0], N[(N[(0.5 * b + (-N[Sqrt[N[(N[(a * c), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / (-a)), $MachinePrecision], N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 9.8e+53], If[GreaterEqual[b, 0.0], N[((-2.0) * N[(c / N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(N[Power[t$95$1, 1.5], $MachinePrecision] - N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] / N[(a * N[(N[(-4.0 * N[(a * c), $MachinePrecision] + N[(2.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$0, N[(N[Sqrt[N[(N[(c / a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] * 0.5 + N[(N[(b / a), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -3.9999999999999999e99

   1. Initial program 57.6%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

   8. Applied rewrites 97.8%

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

   if -3.9999999999999999e99 < b < 1.2500000000000001e-271

   1. Initial program 86.9%

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

   3. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. sqrt-unprod N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 86.9

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-\frac{\mathsf{fma}\left(0.5, b, -\sqrt{\left(a \cdot c\right) \cdot -1}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Applied rewrites 86.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-\frac{\mathsf{fma}\left(0.5, b, -\sqrt{\left(a \cdot c\right) \cdot -1}\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   if 1.2500000000000001e-271 < b < 9.80000000000000036e53

   1. Initial program 84.6%

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

   4. Applied rewrites 84.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left({b}^{3}, -1, {\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right)}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in b around -inf

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

   7. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{-1 \cdot b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{{\left(\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)\right)}^{1.5} + -1 \cdot {b}^{3}}{a \cdot \left(\mathsf{fma}\left(-4, a \cdot c, 2 \cdot \left(b \cdot b\right)\right) - -1 \cdot \left(b \cdot \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}\right)\right)}\\ } \end{array}} \]

   if 9.80000000000000036e53 < b

   1. Initial program 61.0%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 95.9

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

   8. Applied rewrites 95.9%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot \left(-c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) \cdot \mathsf{fma}\left(a, \frac{\frac{c}{b}}{b} \cdot -2, 2\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-271}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, b, -\sqrt{\left(a \cdot c\right) \cdot -1}\right)}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{+53}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-2\right) \cdot \frac{c}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{{\left(\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)\right)}^{1.5} - {b}^{3}}{a \cdot \left(\mathsf{fma}\left(-4, a \cdot c, 2 \cdot \left(b \cdot b\right)\right) + b \cdot \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot \left(-c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{c}{a} \cdot -4}, 0.5, \frac{b}{a} \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)\\ \mathbf{if}\;b \leq 1.25 \cdot 10^{-271}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, b, -\sqrt{\left(a \cdot c\right) \cdot -1}\right)}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{+53}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-2\right) \cdot \frac{c}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{{t\_0}^{1.5} - {b}^{3}}{a \cdot \left(\mathsf{fma}\left(-4, a \cdot c, 2 \cdot \left(b \cdot b\right)\right) + b \cdot \sqrt{t\_0}\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot \left(-c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{c}{a} \cdot -4}, 0.5, \frac{b}{a} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* a c) (* b b))))
   (if (<= b 1.25e-271)
     (if (>= b 0.0)
       (/ (fma 0.5 b (- (sqrt (* (* a c) -1.0)))) (- a))
       (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
     (if (<= b 9.8e+53)
       (if (>= b 0.0)
         (* (- 2.0) (/ c (+ b (sqrt (- (* b b) (* 4.0 (* a c)))))))
         (*
          0.5
          (/
           (- (pow t_0 1.5) (pow b 3.0))
           (* a (+ (fma -4.0 (* a c) (* 2.0 (* b b))) (* b (sqrt t_0)))))))
       (if (>= b 0.0)
         (/ (* 2.0 (- c)) (+ b b))
         (fma (sqrt (* (/ c a) -4.0)) 0.5 (* (/ b a) -0.5)))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-4.0, (a * c), (b * b));
	double tmp_1;
	if (b <= 1.25e-271) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = fma(0.5, b, -sqrt(((a * c) * -1.0))) / -a;
		} else {
			tmp_2 = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 9.8e+53) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -2.0 * (c / (b + sqrt(((b * b) - (4.0 * (a * c))))));
		} else {
			tmp_3 = 0.5 * ((pow(t_0, 1.5) - pow(b, 3.0)) / (a * (fma(-4.0, (a * c), (2.0 * (b * b))) + (b * sqrt(t_0)))));
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * -c) / (b + b);
	} else {
		tmp_1 = fma(sqrt(((c / a) * -4.0)), 0.5, ((b / a) * -0.5));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = fma(-4.0, Float64(a * c), Float64(b * b))
	tmp_1 = 0.0
	if (b <= 1.25e-271)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(fma(0.5, b, Float64(-sqrt(Float64(Float64(a * c) * -1.0)))) / Float64(-a));
		else
			tmp_2 = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 9.8e+53)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-2.0) * Float64(c / Float64(b + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))))));
		else
			tmp_3 = Float64(0.5 * Float64(Float64((t_0 ^ 1.5) - (b ^ 3.0)) / Float64(a * Float64(fma(-4.0, Float64(a * c), Float64(2.0 * Float64(b * b))) + Float64(b * sqrt(t_0))))));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * Float64(-c)) / Float64(b + b));
	else
		tmp_1 = fma(sqrt(Float64(Float64(c / a) * -4.0)), 0.5, Float64(Float64(b / a) * -0.5));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-4.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.25e-271], If[GreaterEqual[b, 0.0], N[(N[(0.5 * b + (-N[Sqrt[N[(N[(a * c), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / (-a)), $MachinePrecision], N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 9.8e+53], If[GreaterEqual[b, 0.0], N[((-2.0) * N[(c / N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(N[Power[t$95$0, 1.5], $MachinePrecision] - N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] / N[(a * N[(N[(-4.0 * N[(a * c), $MachinePrecision] + N[(2.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * (-c)), $MachinePrecision] / N[(b + b), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(c / a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] * 0.5 + N[(N[(b / a), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.2500000000000001e-271

   1. Initial program 77.1%

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

   3. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. sqrt-unprod N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 77.1

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-\frac{\mathsf{fma}\left(0.5, b, -\sqrt{\left(a \cdot c\right) \cdot -1}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Applied rewrites 77.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-\frac{\mathsf{fma}\left(0.5, b, -\sqrt{\left(a \cdot c\right) \cdot -1}\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   if 1.2500000000000001e-271 < b < 9.80000000000000036e53

   1. Initial program 84.6%

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

   4. Applied rewrites 84.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left({b}^{3}, -1, {\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right)}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in b around -inf

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

   7. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{-1 \cdot b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{{\left(\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)\right)}^{1.5} + -1 \cdot {b}^{3}}{a \cdot \left(\mathsf{fma}\left(-4, a \cdot c, 2 \cdot \left(b \cdot b\right)\right) - -1 \cdot \left(b \cdot \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}\right)\right)}\\ } \end{array}} \]

   if 9.80000000000000036e53 < b

   1. Initial program 61.0%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 95.9

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

   8. Applied rewrites 95.9%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{-271}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, b, -\sqrt{\left(a \cdot c\right) \cdot -1}\right)}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{+53}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-2\right) \cdot \frac{c}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{{\left(\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)\right)}^{1.5} - {b}^{3}}{a \cdot \left(\mathsf{fma}\left(-4, a \cdot c, 2 \cdot \left(b \cdot b\right)\right) + b \cdot \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot \left(-c\right)}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{c}{a} \cdot -4}, 0.5, \frac{b}{a} \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)\\ \mathbf{if}\;b \leq 1.25 \cdot 10^{-271}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, b, -\sqrt{\left(a \cdot c\right) \cdot -1}\right)}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\left(-2\right) \cdot \frac{c}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{{t\_0}^{1.5} - {b}^{3}}{a \cdot \left(\mathsf{fma}\left(-4, a \cdot c, 2 \cdot \left(b \cdot b\right)\right) + b \cdot \sqrt{t\_0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* a c) (* b b))))
   (if (<= b 1.25e-271)
     (if (>= b 0.0)
       (/ (fma 0.5 b (- (sqrt (* (* a c) -1.0)))) (- a))
       (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
     (if (>= b 0.0)
       (* (- 2.0) (/ c (+ b (sqrt (- (* b b) (* 4.0 (* a c)))))))
       (*
        0.5
        (/
         (- (pow t_0 1.5) (pow b 3.0))
         (* a (+ (fma -4.0 (* a c) (* 2.0 (* b b))) (* b (sqrt t_0))))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-4.0, (a * c), (b * b));
	double tmp_1;
	if (b <= 1.25e-271) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = fma(0.5, b, -sqrt(((a * c) * -1.0))) / -a;
		} else {
			tmp_2 = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = -2.0 * (c / (b + sqrt(((b * b) - (4.0 * (a * c))))));
	} else {
		tmp_1 = 0.5 * ((pow(t_0, 1.5) - pow(b, 3.0)) / (a * (fma(-4.0, (a * c), (2.0 * (b * b))) + (b * sqrt(t_0)))));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = fma(-4.0, Float64(a * c), Float64(b * b))
	tmp_1 = 0.0
	if (b <= 1.25e-271)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(fma(0.5, b, Float64(-sqrt(Float64(Float64(a * c) * -1.0)))) / Float64(-a));
		else
			tmp_2 = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-2.0) * Float64(c / Float64(b + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))))));
	else
		tmp_1 = Float64(0.5 * Float64(Float64((t_0 ^ 1.5) - (b ^ 3.0)) / Float64(a * Float64(fma(-4.0, Float64(a * c), Float64(2.0 * Float64(b * b))) + Float64(b * sqrt(t_0))))));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-4.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.25e-271], If[GreaterEqual[b, 0.0], N[(N[(0.5 * b + (-N[Sqrt[N[(N[(a * c), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / (-a)), $MachinePrecision], N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[((-2.0) * N[(c / N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(N[Power[t$95$0, 1.5], $MachinePrecision] - N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] / N[(a * N[(N[(-4.0 * N[(a * c), $MachinePrecision] + N[(2.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.2500000000000001e-271

   1. Initial program 77.1%

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

   3. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. sqrt-unprod N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 77.1

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-\frac{\mathsf{fma}\left(0.5, b, -\sqrt{\left(a \cdot c\right) \cdot -1}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Applied rewrites 77.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-\frac{\mathsf{fma}\left(0.5, b, -\sqrt{\left(a \cdot c\right) \cdot -1}\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   if 1.2500000000000001e-271 < b

   1. Initial program 71.8%

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

   4. Applied rewrites 71.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left({b}^{3}, -1, {\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right)}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in b around -inf

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

   7. Applied rewrites 71.7%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{-1 \cdot b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{{\left(\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)\right)}^{1.5} + -1 \cdot {b}^{3}}{a \cdot \left(\mathsf{fma}\left(-4, a \cdot c, 2 \cdot \left(b \cdot b\right)\right) - -1 \cdot \left(b \cdot \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}\right)\right)}\\ } \end{array}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{-271}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, b, -\sqrt{\left(a \cdot c\right) \cdot -1}\right)}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\left(-2\right) \cdot \frac{c}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{{\left(\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)\right)}^{1.5} - {b}^{3}}{a \cdot \left(\mathsf{fma}\left(-4, a \cdot c, 2 \cdot \left(b \cdot b\right)\right) + b \cdot \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)\\ \mathbf{if}\;b \leq -1.66 \cdot 10^{-292}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\left(-2\right) \cdot \frac{c}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{{t\_0}^{1.5} - {b}^{3}}{a \cdot \left(\mathsf{fma}\left(-4, a \cdot c, 2 \cdot \left(b \cdot b\right)\right) + b \cdot \sqrt{t\_0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* a c) (* b b))))
   (if (<= b -1.66e-292)
     (if (>= b 0.0)
       (- (sqrt (/ (- c) a)))
       (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
     (if (>= b 0.0)
       (* (- 2.0) (/ c (+ b (sqrt (- (* b b) (* 4.0 (* a c)))))))
       (*
        0.5
        (/
         (- (pow t_0 1.5) (pow b 3.0))
         (* a (+ (fma -4.0 (* a c) (* 2.0 (* b b))) (* b (sqrt t_0))))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-4.0, (a * c), (b * b));
	double tmp_1;
	if (b <= -1.66e-292) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -sqrt((-c / a));
		} else {
			tmp_2 = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = -2.0 * (c / (b + sqrt(((b * b) - (4.0 * (a * c))))));
	} else {
		tmp_1 = 0.5 * ((pow(t_0, 1.5) - pow(b, 3.0)) / (a * (fma(-4.0, (a * c), (2.0 * (b * b))) + (b * sqrt(t_0)))));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = fma(-4.0, Float64(a * c), Float64(b * b))
	tmp_1 = 0.0
	if (b <= -1.66e-292)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(-sqrt(Float64(Float64(-c) / a)));
		else
			tmp_2 = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(-2.0) * Float64(c / Float64(b + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))))));
	else
		tmp_1 = Float64(0.5 * Float64(Float64((t_0 ^ 1.5) - (b ^ 3.0)) / Float64(a * Float64(fma(-4.0, Float64(a * c), Float64(2.0 * Float64(b * b))) + Float64(b * sqrt(t_0))))));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-4.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.66e-292], If[GreaterEqual[b, 0.0], (-N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision]), N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[((-2.0) * N[(c / N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(N[Power[t$95$0, 1.5], $MachinePrecision] - N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] / N[(a * N[(N[(-4.0 * N[(a * c), $MachinePrecision] + N[(2.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.66e-292

   1. Initial program 76.8%

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

   3. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. sqrt-unprod N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 76.8

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-\frac{\mathsf{fma}\left(0.5, b, -\sqrt{\left(a \cdot c\right) \cdot -1}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Applied rewrites 76.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-\frac{\mathsf{fma}\left(0.5, b, -\sqrt{\left(a \cdot c\right) \cdot -1}\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   6. Taylor expanded in a around -inf

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

      1. sqrt-prod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 76.8

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

   8. Applied rewrites 76.8%

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

   if -1.66e-292 < b

   1. Initial program 72.3%

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

   4. Applied rewrites 72.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left({b}^{3}, -1, {\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right)}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in b around -inf

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

   7. Applied rewrites 72.3%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{-1 \cdot b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{{\left(\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)\right)}^{1.5} + -1 \cdot {b}^{3}}{a \cdot \left(\mathsf{fma}\left(-4, a \cdot c, 2 \cdot \left(b \cdot b\right)\right) - -1 \cdot \left(b \cdot \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}\right)\right)}\\ } \end{array}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.66 \cdot 10^{-292}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\left(-2\right) \cdot \frac{c}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{{\left(\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)\right)}^{1.5} - {b}^{3}}{a \cdot \left(\mathsf{fma}\left(-4, a \cdot c, 2 \cdot \left(b \cdot b\right)\right) + b \cdot \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)\\ \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-2\right) \cdot \frac{c}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{{t\_0}^{1.5} - {b}^{3}}{a \cdot \left(\mathsf{fma}\left(-4, a \cdot c, 2 \cdot \left(b \cdot b\right)\right) + b \cdot \sqrt{t\_0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* a c) (* b b))))
   (if (>= b 0.0)
     (* (- 2.0) (/ c (+ b (sqrt (- (* b b) (* 4.0 (* a c)))))))
     (*
      0.5
      (/
       (- (pow t_0 1.5) (pow b 3.0))
       (* a (+ (fma -4.0 (* a c) (* 2.0 (* b b))) (* b (sqrt t_0)))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-4.0, (a * c), (b * b));
	double tmp;
	if (b >= 0.0) {
		tmp = -2.0 * (c / (b + sqrt(((b * b) - (4.0 * (a * c))))));
	} else {
		tmp = 0.5 * ((pow(t_0, 1.5) - pow(b, 3.0)) / (a * (fma(-4.0, (a * c), (2.0 * (b * b))) + (b * sqrt(t_0)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(-4.0, Float64(a * c), Float64(b * b))
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(-2.0) * Float64(c / Float64(b + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))))));
	else
		tmp = Float64(0.5 * Float64(Float64((t_0 ^ 1.5) - (b ^ 3.0)) / Float64(a * Float64(fma(-4.0, Float64(a * c), Float64(2.0 * Float64(b * b))) + Float64(b * sqrt(t_0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-4.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[GreaterEqual[b, 0.0], N[((-2.0) * N[(c / N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(N[Power[t$95$0, 1.5], $MachinePrecision] - N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] / N[(a * N[(N[(-4.0 * N[(a * c), $MachinePrecision] + N[(2.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 74.5%

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

4. Applied rewrites 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left({b}^{3}, -1, {\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right)}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}}{2 \cdot a}\\ \end{array} \]

5. Taylor expanded in b around -inf

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

7. Applied rewrites 57.0%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{-1 \cdot b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{{\left(\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)\right)}^{1.5} + -1 \cdot {b}^{3}}{a \cdot \left(\mathsf{fma}\left(-4, a \cdot c, 2 \cdot \left(b \cdot b\right)\right) - -1 \cdot \left(b \cdot \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}\right)\right)}\\ } \end{array}} \]

8. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(-2\right) \cdot \frac{c}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{{\left(\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)\right)}^{1.5} - {b}^{3}}{a \cdot \left(\mathsf{fma}\left(-4, a \cdot c, 2 \cdot \left(b \cdot b\right)\right) + b \cdot \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}\right)}\\ \end{array} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))