quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.4% → 82.9%

Time: 10.9s

Alternatives: 6

Speedup: 15.9×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

C

double code(double a, double b_2, double c) {
	return (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Initial Program: 52.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

C

double code(double a, double b_2, double c) {
	return (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Alternative 1: 82.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \frac{c}{b_2}\\ \mathbf{if}\;b_2 \leq -9.2 \cdot 10^{-52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq -4.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{\frac{1}{\frac{\frac{-1}{a}}{c}}}}{a}\\ \mathbf{elif}\;b_2 \leq -8 \cdot 10^{-223}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq 10^{+94}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (* -0.5 (/ c b_2))))
   (if (<= b_2 -9.2e-52)
     t_0
     (if (<= b_2 -4.5e-80)
       (/ (- (- b_2) (sqrt (/ 1.0 (/ (/ -1.0 a) c)))) a)
       (if (<= b_2 -8e-223)
         t_0
         (if (<= b_2 1e+94)
           (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
           (/ (* b_2 -2.0) a)))))))

C

double code(double a, double b_2, double c) {
	double t_0 = -0.5 * (c / b_2);
	double tmp;
	if (b_2 <= -9.2e-52) {
		tmp = t_0;
	} else if (b_2 <= -4.5e-80) {
		tmp = (-b_2 - sqrt((1.0 / ((-1.0 / a) / c)))) / a;
	} else if (b_2 <= -8e-223) {
		tmp = t_0;
	} else if (b_2 <= 1e+94) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-0.5d0) * (c / b_2)
    if (b_2 <= (-9.2d-52)) then
        tmp = t_0
    else if (b_2 <= (-4.5d-80)) then
        tmp = (-b_2 - sqrt((1.0d0 / (((-1.0d0) / a) / c)))) / a
    else if (b_2 <= (-8d-223)) then
        tmp = t_0
    else if (b_2 <= 1d+94) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double t_0 = -0.5 * (c / b_2);
	double tmp;
	if (b_2 <= -9.2e-52) {
		tmp = t_0;
	} else if (b_2 <= -4.5e-80) {
		tmp = (-b_2 - Math.sqrt((1.0 / ((-1.0 / a) / c)))) / a;
	} else if (b_2 <= -8e-223) {
		tmp = t_0;
	} else if (b_2 <= 1e+94) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	t_0 = -0.5 * (c / b_2)
	tmp = 0
	if b_2 <= -9.2e-52:
		tmp = t_0
	elif b_2 <= -4.5e-80:
		tmp = (-b_2 - math.sqrt((1.0 / ((-1.0 / a) / c)))) / a
	elif b_2 <= -8e-223:
		tmp = t_0
	elif b_2 <= 1e+94:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	t_0 = Float64(-0.5 * Float64(c / b_2))
	tmp = 0.0
	if (b_2 <= -9.2e-52)
		tmp = t_0;
	elseif (b_2 <= -4.5e-80)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(1.0 / Float64(Float64(-1.0 / a) / c)))) / a);
	elseif (b_2 <= -8e-223)
		tmp = t_0;
	elseif (b_2 <= 1e+94)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	t_0 = -0.5 * (c / b_2);
	tmp = 0.0;
	if (b_2 <= -9.2e-52)
		tmp = t_0;
	elseif (b_2 <= -4.5e-80)
		tmp = (-b_2 - sqrt((1.0 / ((-1.0 / a) / c)))) / a;
	elseif (b_2 <= -8e-223)
		tmp = t_0;
	elseif (b_2 <= 1e+94)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$2, -9.2e-52], t$95$0, If[LessEqual[b$95$2, -4.5e-80], N[(N[((-b$95$2) - N[Sqrt[N[(1.0 / N[(N[(-1.0 / a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, -8e-223], t$95$0, If[LessEqual[b$95$2, 1e+94], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -9.19999999999999977e-52 or -4.5000000000000003e-80 < b_2 < -7.9999999999999998e-223

   1. Initial program 20.3%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 81.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]

   if -9.19999999999999977e-52 < b_2 < -4.5000000000000003e-80

   1. Initial program 83.5%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. flip3-- 33.3%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\frac{{\left(b_2 \cdot b_2\right)}^{3} - {\left(a \cdot c\right)}^{3}}{\left(b_2 \cdot b_2\right) \cdot \left(b_2 \cdot b_2\right) + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) + \left(b_2 \cdot b_2\right) \cdot \left(a \cdot c\right)\right)}}}}{a} \]

      2. clear-num 33.3%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\frac{1}{\frac{\left(b_2 \cdot b_2\right) \cdot \left(b_2 \cdot b_2\right) + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) + \left(b_2 \cdot b_2\right) \cdot \left(a \cdot c\right)\right)}{{\left(b_2 \cdot b_2\right)}^{3} - {\left(a \cdot c\right)}^{3}}}}}}{a} \]

   3. Applied egg-rr 33.3%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\frac{1}{\frac{{b_2}^{4} + \left(a \cdot c\right) \cdot \mathsf{fma}\left(a, c, {b_2}^{2}\right)}{{b_2}^{6} - {\left(a \cdot c\right)}^{3}}}}}}{a} \]

   4. Taylor expanded in b_2 around 0 83.7%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\frac{1}{\color{blue}{\frac{-1}{a \cdot c}}}}}{a} \]
   5. Step-by-step derivation

      1. associate-/r* 83.7%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\frac{1}{\color{blue}{\frac{\frac{-1}{a}}{c}}}}}{a} \]

   6. Simplified 83.7%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\frac{1}{\color{blue}{\frac{\frac{-1}{a}}{c}}}}}{a} \]

   if -7.9999999999999998e-223 < b_2 < 1e94

   1. Initial program 80.7%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   if 1e94 < b_2

   1. Initial program 59.2%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 97.1%

      \[\leadsto \frac{\color{blue}{-2 \cdot b_2}}{a} \]
   3. Step-by-step derivation

      1. *-commutative 97.1%

         \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]

   4. Simplified 97.1%

      \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -9.2 \cdot 10^{-52}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -4.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{\frac{1}{\frac{\frac{-1}{a}}{c}}}}{a}\\ \mathbf{elif}\;b_2 \leq -8 \cdot 10^{-223}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 10^{+94}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 2: 82.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \frac{c}{b_2}\\ \mathbf{if}\;b_2 \leq -2 \cdot 10^{-47}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq -1.8 \cdot 10^{-77}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{elif}\;b_2 \leq -8 \cdot 10^{-223}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq 10^{+94}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (* -0.5 (/ c b_2))))
   (if (<= b_2 -2e-47)
     t_0
     (if (<= b_2 -1.8e-77)
       (/ (- (- b_2) (sqrt (* c (- a)))) a)
       (if (<= b_2 -8e-223)
         t_0
         (if (<= b_2 1e+94)
           (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
           (/ (* b_2 -2.0) a)))))))

C

double code(double a, double b_2, double c) {
	double t_0 = -0.5 * (c / b_2);
	double tmp;
	if (b_2 <= -2e-47) {
		tmp = t_0;
	} else if (b_2 <= -1.8e-77) {
		tmp = (-b_2 - sqrt((c * -a))) / a;
	} else if (b_2 <= -8e-223) {
		tmp = t_0;
	} else if (b_2 <= 1e+94) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-0.5d0) * (c / b_2)
    if (b_2 <= (-2d-47)) then
        tmp = t_0
    else if (b_2 <= (-1.8d-77)) then
        tmp = (-b_2 - sqrt((c * -a))) / a
    else if (b_2 <= (-8d-223)) then
        tmp = t_0
    else if (b_2 <= 1d+94) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double t_0 = -0.5 * (c / b_2);
	double tmp;
	if (b_2 <= -2e-47) {
		tmp = t_0;
	} else if (b_2 <= -1.8e-77) {
		tmp = (-b_2 - Math.sqrt((c * -a))) / a;
	} else if (b_2 <= -8e-223) {
		tmp = t_0;
	} else if (b_2 <= 1e+94) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	t_0 = -0.5 * (c / b_2)
	tmp = 0
	if b_2 <= -2e-47:
		tmp = t_0
	elif b_2 <= -1.8e-77:
		tmp = (-b_2 - math.sqrt((c * -a))) / a
	elif b_2 <= -8e-223:
		tmp = t_0
	elif b_2 <= 1e+94:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	t_0 = Float64(-0.5 * Float64(c / b_2))
	tmp = 0.0
	if (b_2 <= -2e-47)
		tmp = t_0;
	elseif (b_2 <= -1.8e-77)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(c * Float64(-a)))) / a);
	elseif (b_2 <= -8e-223)
		tmp = t_0;
	elseif (b_2 <= 1e+94)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	t_0 = -0.5 * (c / b_2);
	tmp = 0.0;
	if (b_2 <= -2e-47)
		tmp = t_0;
	elseif (b_2 <= -1.8e-77)
		tmp = (-b_2 - sqrt((c * -a))) / a;
	elseif (b_2 <= -8e-223)
		tmp = t_0;
	elseif (b_2 <= 1e+94)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$2, -2e-47], t$95$0, If[LessEqual[b$95$2, -1.8e-77], N[(N[((-b$95$2) - N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, -8e-223], t$95$0, If[LessEqual[b$95$2, 1e+94], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.9999999999999999e-47 or -1.8e-77 < b_2 < -7.9999999999999998e-223

   1. Initial program 20.3%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 81.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]

   if -1.9999999999999999e-47 < b_2 < -1.8e-77

   1. Initial program 83.5%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around 0 83.5%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]
   3. Step-by-step derivation

      1. associate-*r* 83.5%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}}}{a} \]

      2. neg-mul-1 83.5%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(-a\right)} \cdot c}}{a} \]

   4. Simplified 83.5%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(-a\right) \cdot c}}}{a} \]

   if -7.9999999999999998e-223 < b_2 < 1e94

   1. Initial program 80.7%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   if 1e94 < b_2

   1. Initial program 59.2%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 97.1%

      \[\leadsto \frac{\color{blue}{-2 \cdot b_2}}{a} \]
   3. Step-by-step derivation

      1. *-commutative 97.1%

         \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]

   4. Simplified 97.1%

      \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{-47}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -1.8 \cdot 10^{-77}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{elif}\;b_2 \leq -8 \cdot 10^{-223}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 10^{+94}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 3: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \frac{c}{b_2}\\ t_1 := \frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{if}\;b_2 \leq -8.6 \cdot 10^{-50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq -1.66 \cdot 10^{-80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b_2 \leq -8 \cdot 10^{-223}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq 7 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (* -0.5 (/ c b_2))) (t_1 (/ (- (- b_2) (sqrt (* c (- a)))) a)))
   (if (<= b_2 -8.6e-50)
     t_0
     (if (<= b_2 -1.66e-80)
       t_1
       (if (<= b_2 -8e-223)
         t_0
         (if (<= b_2 7e-19) t_1 (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5))))))))

C

double code(double a, double b_2, double c) {
	double t_0 = -0.5 * (c / b_2);
	double t_1 = (-b_2 - sqrt((c * -a))) / a;
	double tmp;
	if (b_2 <= -8.6e-50) {
		tmp = t_0;
	} else if (b_2 <= -1.66e-80) {
		tmp = t_1;
	} else if (b_2 <= -8e-223) {
		tmp = t_0;
	} else if (b_2 <= 7e-19) {
		tmp = t_1;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-0.5d0) * (c / b_2)
    t_1 = (-b_2 - sqrt((c * -a))) / a
    if (b_2 <= (-8.6d-50)) then
        tmp = t_0
    else if (b_2 <= (-1.66d-80)) then
        tmp = t_1
    else if (b_2 <= (-8d-223)) then
        tmp = t_0
    else if (b_2 <= 7d-19) then
        tmp = t_1
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((c / b_2) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double t_0 = -0.5 * (c / b_2);
	double t_1 = (-b_2 - Math.sqrt((c * -a))) / a;
	double tmp;
	if (b_2 <= -8.6e-50) {
		tmp = t_0;
	} else if (b_2 <= -1.66e-80) {
		tmp = t_1;
	} else if (b_2 <= -8e-223) {
		tmp = t_0;
	} else if (b_2 <= 7e-19) {
		tmp = t_1;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	t_0 = -0.5 * (c / b_2)
	t_1 = (-b_2 - math.sqrt((c * -a))) / a
	tmp = 0
	if b_2 <= -8.6e-50:
		tmp = t_0
	elif b_2 <= -1.66e-80:
		tmp = t_1
	elif b_2 <= -8e-223:
		tmp = t_0
	elif b_2 <= 7e-19:
		tmp = t_1
	else:
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5)
	return tmp

Julia

function code(a, b_2, c)
	t_0 = Float64(-0.5 * Float64(c / b_2))
	t_1 = Float64(Float64(Float64(-b_2) - sqrt(Float64(c * Float64(-a)))) / a)
	tmp = 0.0
	if (b_2 <= -8.6e-50)
		tmp = t_0;
	elseif (b_2 <= -1.66e-80)
		tmp = t_1;
	elseif (b_2 <= -8e-223)
		tmp = t_0;
	elseif (b_2 <= 7e-19)
		tmp = t_1;
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c / b_2) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	t_0 = -0.5 * (c / b_2);
	t_1 = (-b_2 - sqrt((c * -a))) / a;
	tmp = 0.0;
	if (b_2 <= -8.6e-50)
		tmp = t_0;
	elseif (b_2 <= -1.66e-80)
		tmp = t_1;
	elseif (b_2 <= -8e-223)
		tmp = t_0;
	elseif (b_2 <= 7e-19)
		tmp = t_1;
	else
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[((-b$95$2) - N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[b$95$2, -8.6e-50], t$95$0, If[LessEqual[b$95$2, -1.66e-80], t$95$1, If[LessEqual[b$95$2, -8e-223], t$95$0, If[LessEqual[b$95$2, 7e-19], t$95$1, N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -8.59999999999999995e-50 or -1.66000000000000003e-80 < b_2 < -7.9999999999999998e-223

   1. Initial program 20.3%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 81.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]

   if -8.59999999999999995e-50 < b_2 < -1.66000000000000003e-80 or -7.9999999999999998e-223 < b_2 < 7.00000000000000031e-19

   1. Initial program 73.9%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around 0 66.7%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]
   3. Step-by-step derivation

      1. associate-*r* 66.7%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}}}{a} \]

      2. neg-mul-1 66.7%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(-a\right)} \cdot c}}{a} \]

   4. Simplified 66.7%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(-a\right) \cdot c}}}{a} \]

   if 7.00000000000000031e-19 < b_2

   1. Initial program 70.3%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 88.4%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -8.6 \cdot 10^{-50}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -1.66 \cdot 10^{-80}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{elif}\;b_2 \leq -8 \cdot 10^{-223}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 7 \cdot 10^{-19}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 4: 68.6% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e-310)
   (* -0.5 (/ c b_2))
   (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-4d-310)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((c / b_2) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4e-310:
		tmp = -0.5 * (c / b_2)
	else:
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4e-310)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c / b_2) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4e-310)
		tmp = -0.5 * (c / b_2);
	else
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4e-310], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.999999999999988e-310

   1. Initial program 27.9%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 72.5%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]

   if -3.999999999999988e-310 < b_2

   1. Initial program 70.3%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 63.6%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 5: 68.4% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -1.7 \cdot 10^{-305}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.7e-305) (* -0.5 (/ c b_2)) (/ (* b_2 -2.0) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.7e-305) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.7d-305)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.7e-305) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.7e-305:
		tmp = -0.5 * (c / b_2)
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.7e-305)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.7e-305)
		tmp = -0.5 * (c / b_2);
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.7e-305], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.7e-305

   1. Initial program 26.7%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 73.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]

   if -1.7e-305 < b_2

   1. Initial program 70.7%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 62.6%

      \[\leadsto \frac{\color{blue}{-2 \cdot b_2}}{a} \]
   3. Step-by-step derivation

      1. *-commutative 62.6%

         \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]

   4. Simplified 62.6%

      \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.7 \cdot 10^{-305}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 6: 35.6% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b_2} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* -0.5 (/ c b_2)))

C

double code(double a, double b_2, double c) {
	return -0.5 * (c / b_2);
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-0.5d0) * (c / b_2)
end function

Java

public static double code(double a, double b_2, double c) {
	return -0.5 * (c / b_2);
}

Python

def code(a, b_2, c):
	return -0.5 * (c / b_2)

Julia

function code(a, b_2, c)
	return Float64(-0.5 * Float64(c / b_2))
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = -0.5 * (c / b_2);
end

Wolfram

code[a_, b$95$2_, c_] := N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.2%

   \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

2. Taylor expanded in b_2 around -inf 35.3%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]

3. Final simplification 35.3%

   \[\leadsto -0.5 \cdot \frac{c}{b_2} \]

Developer target: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_1 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{\left|b_2\right| - t_0} \cdot \sqrt{\left|b_2\right| + t_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b_2, t_0\right)\\ \end{array}\\ \mathbf{if}\;b_2 < 0:\\ \;\;\;\;\frac{c}{t_1 - b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 + t_1}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_1
         (if (== (copysign a c) a)
           (* (sqrt (- (fabs b_2) t_0)) (sqrt (+ (fabs b_2) t_0)))
           (hypot b_2 t_0))))
   (if (< b_2 0.0) (/ c (- t_1 b_2)) (/ (+ b_2 t_1) (- a)))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((fabs(b_2) - t_0)) * sqrt((fabs(b_2) + t_0));
	} else {
		tmp = hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = c / (t_1 - b_2);
	} else {
		tmp_1 = (b_2 + t_1) / -a;
	}
	return tmp_1;
}

Java

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((Math.abs(b_2) - t_0)) * Math.sqrt((Math.abs(b_2) + t_0));
	} else {
		tmp = Math.hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = c / (t_1 - b_2);
	} else {
		tmp_1 = (b_2 + t_1) / -a;
	}
	return tmp_1;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((math.fabs(b_2) - t_0)) * math.sqrt((math.fabs(b_2) + t_0))
	else:
		tmp = math.hypot(b_2, t_0)
	t_1 = tmp
	tmp_1 = 0
	if b_2 < 0.0:
		tmp_1 = c / (t_1 - b_2)
	else:
		tmp_1 = (b_2 + t_1) / -a
	return tmp_1

Julia

function code(a, b_2, c)
	t_0 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(abs(b_2) - t_0)) * sqrt(Float64(abs(b_2) + t_0)));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp
	tmp_1 = 0.0
	if (b_2 < 0.0)
		tmp_1 = Float64(c / Float64(t_1 - b_2));
	else
		tmp_1 = Float64(Float64(b_2 + t_1) / Float64(-a));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b_2, c)
	t_0 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((abs(b_2) - t_0)) * sqrt((abs(b_2) + t_0));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp;
	tmp_2 = 0.0;
	if (b_2 < 0.0)
		tmp_2 = c / (t_1 - b_2);
	else
		tmp_2 = (b_2 + t_1) / -a;
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(N[Abs[b$95$2], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[Abs[b$95$2], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[b$95$2 ^ 2 + t$95$0 ^ 2], $MachinePrecision]]}, If[Less[b$95$2, 0.0], N[(c / N[(t$95$1 - b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$2 + t$95$1), $MachinePrecision] / (-a)), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2023335 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  :herbie-expected 10

  :herbie-target
  (if (< b_2 0.0) (/ c (- (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot b_2 (* (sqrt (fabs a)) (sqrt (fabs c))))) b_2)) (/ (+ b_2 (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot b_2 (* (sqrt (fabs a)) (sqrt (fabs c)))))) (- a)))

  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))