quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.1% → 85.9%

Time: 13.3s

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: 52.1% 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: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -6.8 \cdot 10^{-59}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 5 \cdot 10^{+82}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 + \left(b_2 + -0.5 \cdot \frac{a}{\frac{b_2}{c}}\right)}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -6.8e-59)
   (* -0.5 (/ c b_2))
   (if (<= b_2 5e+82)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (/ (+ b_2 (+ b_2 (* -0.5 (/ a (/ b_2 c))))) (- a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6.8e-59) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 5e+82) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (b_2 + (b_2 + (-0.5 * (a / (b_2 / c))))) / -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 <= (-6.8d-59)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 5d+82) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = (b_2 + (b_2 + ((-0.5d0) * (a / (b_2 / c))))) / -a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6.8e-59) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 5e+82) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (b_2 + (b_2 + (-0.5 * (a / (b_2 / c))))) / -a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -6.8e-59:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 5e+82:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (b_2 + (b_2 + (-0.5 * (a / (b_2 / c))))) / -a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -6.8e-59)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 5e+82)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(b_2 + Float64(b_2 + Float64(-0.5 * Float64(a / Float64(b_2 / c))))) / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -6.8e-59)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 5e+82)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (b_2 + (b_2 + (-0.5 * (a / (b_2 / c))))) / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -6.8e-59], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 5e+82], 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 + N[(b$95$2 + N[(-0.5 * N[(a / N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-a)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -6.80000000000000035e-59

   1. Initial program 16.6%

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

      1. sub-neg 16.6%

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

      2. sqr-neg 16.6%

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

      3. distribute-neg-out 16.6%

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

      4. neg-mul-1 16.6%

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

      5. *-commutative 16.6%

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

      6. associate-/l* 16.6%

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

      7. remove-double-neg 16.6%

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

      8. neg-mul-1 16.6%

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

      9. *-commutative 16.6%

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

      10. associate-/l* 16.6%

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

      11. metadata-eval 16.6%

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

      12. /-rgt-identity 16.6%

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

   3. Simplified 16.6%

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

   4. Taylor expanded in b_2 around -inf 84.7%

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

   if -6.80000000000000035e-59 < b_2 < 5.00000000000000015e82

   1. Initial program 77.8%

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

   if 5.00000000000000015e82 < b_2

   1. Initial program 62.2%

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

      1. sub-neg 62.2%

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

      2. sqr-neg 62.2%

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

      3. distribute-neg-out 62.2%

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

      4. neg-mul-1 62.2%

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

      5. *-commutative 62.2%

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

      6. associate-/l* 62.2%

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

      7. remove-double-neg 62.2%

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

      8. neg-mul-1 62.2%

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

      9. *-commutative 62.2%

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

      10. associate-/l* 62.2%

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

      11. metadata-eval 62.2%

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

      12. /-rgt-identity 62.2%

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

   3. Simplified 62.2%

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

   4. Taylor expanded in a around 0 93.8%

      \[\leadsto \frac{b_2 + \color{blue}{\left(b_2 + -0.5 \cdot \frac{a \cdot c}{b_2}\right)}}{-a} \]
   5. Step-by-step derivation

      1. associate-/l* 99.1%

         \[\leadsto \frac{b_2 + \left(b_2 + -0.5 \cdot \color{blue}{\frac{a}{\frac{b_2}{c}}}\right)}{-a} \]

   6. Simplified 99.1%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -6.8 \cdot 10^{-59}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 5 \cdot 10^{+82}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 + \left(b_2 + -0.5 \cdot \frac{a}{\frac{b_2}{c}}\right)}{-a}\\ \end{array} \]

Alternative 2: 81.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -1.55 \cdot 10^{-55}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 9.4 \cdot 10^{-57}:\\ \;\;\;\;\frac{b_2 + \sqrt{a \cdot \left(-c\right)}}{-a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.55e-55)
   (* -0.5 (/ c b_2))
   (if (<= b_2 9.4e-57)
     (/ (+ b_2 (sqrt (* a (- c)))) (- a))
     (+ (* -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 <= -1.55e-55) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 9.4e-57) {
		tmp = (b_2 + sqrt((a * -c))) / -a;
	} 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 <= (-1.55d-55)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 9.4d-57) then
        tmp = (b_2 + sqrt((a * -c))) / -a
    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 <= -1.55e-55) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 9.4e-57) {
		tmp = (b_2 + Math.sqrt((a * -c))) / -a;
	} 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 <= -1.55e-55:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 9.4e-57:
		tmp = (b_2 + math.sqrt((a * -c))) / -a
	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 <= -1.55e-55)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 9.4e-57)
		tmp = Float64(Float64(b_2 + sqrt(Float64(a * Float64(-c)))) / Float64(-a));
	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 <= -1.55e-55)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 9.4e-57)
		tmp = (b_2 + sqrt((a * -c))) / -a;
	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, -1.55e-55], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 9.4e-57], N[(N[(b$95$2 + N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-a)), $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 3 regimes

2. if b_2 < -1.54999999999999998e-55

   1. Initial program 16.6%

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

      1. sub-neg 16.6%

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

      2. sqr-neg 16.6%

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

      3. distribute-neg-out 16.6%

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

      4. neg-mul-1 16.6%

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

      5. *-commutative 16.6%

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

      6. associate-/l* 16.6%

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

      7. remove-double-neg 16.6%

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

      8. neg-mul-1 16.6%

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

      9. *-commutative 16.6%

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

      10. associate-/l* 16.6%

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

      11. metadata-eval 16.6%

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

      12. /-rgt-identity 16.6%

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

   3. Simplified 16.6%

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

   4. Taylor expanded in b_2 around -inf 84.7%

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

   if -1.54999999999999998e-55 < b_2 < 9.3999999999999996e-57

   1. Initial program 74.8%

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

      1. sub-neg 74.8%

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

      2. sqr-neg 74.8%

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

      3. distribute-neg-out 74.8%

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

      4. neg-mul-1 74.8%

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

      5. *-commutative 74.8%

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

      6. associate-/l* 74.8%

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

      7. remove-double-neg 74.8%

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

      8. neg-mul-1 74.8%

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

      9. *-commutative 74.8%

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

      10. associate-/l* 74.8%

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

      11. metadata-eval 74.8%

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

      12. /-rgt-identity 74.8%

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

   3. Simplified 74.8%

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

   4. Taylor expanded in a around inf 70.3%

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

      1. mul-1-neg 70.3%

         \[\leadsto \frac{b_2 + \sqrt{\color{blue}{-a \cdot c}}}{-a} \]

   6. Simplified 70.3%

      \[\leadsto \frac{b_2 + \sqrt{\color{blue}{-a \cdot c}}}{-a} \]

   if 9.3999999999999996e-57 < b_2

   1. Initial program 69.2%

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

      1. sub-neg 69.2%

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

      2. sqr-neg 69.2%

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

      3. distribute-neg-out 69.2%

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

      4. neg-mul-1 69.2%

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

      5. *-commutative 69.2%

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

      6. associate-/l* 69.2%

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

      7. remove-double-neg 69.2%

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

      8. neg-mul-1 69.2%

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

      9. *-commutative 69.2%

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

      10. associate-/l* 69.2%

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

      11. metadata-eval 69.2%

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

      12. /-rgt-identity 69.2%

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

   3. Simplified 69.2%

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

   4. Taylor expanded in b_2 around inf 90.7%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.55 \cdot 10^{-55}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 9.4 \cdot 10^{-57}:\\ \;\;\;\;\frac{b_2 + \sqrt{a \cdot \left(-c\right)}}{-a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 3: 67.9% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-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 <= -5e-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 <= (-5d-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 <= -5e-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 <= -5e-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 <= -5e-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 <= -5e-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, -5e-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 < -4.999999999999985e-310

   1. Initial program 30.7%

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

      1. sub-neg 30.7%

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

      2. sqr-neg 30.7%

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

      3. distribute-neg-out 30.7%

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

      4. neg-mul-1 30.7%

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

      5. *-commutative 30.7%

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

      6. associate-/l* 30.7%

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

      7. remove-double-neg 30.7%

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

      8. neg-mul-1 30.7%

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

      9. *-commutative 30.7%

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

      10. associate-/l* 30.7%

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

      11. metadata-eval 30.7%

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

      12. /-rgt-identity 30.7%

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

   3. Simplified 30.7%

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

   4. Taylor expanded in b_2 around -inf 66.5%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 73.7%

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

      1. sub-neg 73.7%

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

      2. sqr-neg 73.7%

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

      3. distribute-neg-out 73.7%

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

      4. neg-mul-1 73.7%

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

      5. *-commutative 73.7%

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

      6. associate-/l* 73.7%

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

      7. remove-double-neg 73.7%

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

      8. neg-mul-1 73.7%

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

      9. *-commutative 73.7%

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

      10. associate-/l* 73.7%

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

      11. metadata-eval 73.7%

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

      12. /-rgt-identity 73.7%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in b_2 around inf 71.1%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 4: 47.9% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-285}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-285) (* -0.5 (/ c b_2)) (/ (- b_2) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-285) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = -b_2 / 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 <= (-1d-285)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = -b_2 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-285) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e-285:
		tmp = -0.5 * (c / b_2)
	else:
		tmp = -b_2 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-285)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(Float64(-b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e-285)
		tmp = -0.5 * (c / b_2);
	else
		tmp = -b_2 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e-285], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[((-b$95$2) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.00000000000000007e-285

   1. Initial program 28.4%

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

      1. sub-neg 28.4%

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

      2. sqr-neg 28.4%

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

      3. distribute-neg-out 28.4%

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

      4. neg-mul-1 28.4%

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

      5. *-commutative 28.4%

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

      6. associate-/l* 28.4%

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

      7. remove-double-neg 28.4%

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

      8. neg-mul-1 28.4%

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

      9. *-commutative 28.4%

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

      10. associate-/l* 28.4%

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

      11. metadata-eval 28.4%

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

      12. /-rgt-identity 28.4%

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

   3. Simplified 28.4%

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

   4. Taylor expanded in b_2 around -inf 70.0%

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

   if -1.00000000000000007e-285 < b_2

   1. Initial program 73.6%

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

      1. sub-neg 73.6%

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

      2. sqr-neg 73.6%

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

      3. distribute-neg-out 73.6%

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

      4. neg-mul-1 73.6%

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

      5. *-commutative 73.6%

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

      6. associate-/l* 73.6%

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

      7. remove-double-neg 73.6%

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

      8. neg-mul-1 73.6%

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

      9. *-commutative 73.6%

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

      10. associate-/l* 73.6%

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

      11. metadata-eval 73.6%

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

      12. /-rgt-identity 73.6%

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

   3. Simplified 73.7%

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

   4. Taylor expanded in a around inf 44.8%

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

      1. mul-1-neg 44.8%

         \[\leadsto \frac{b_2 + \sqrt{\color{blue}{-a \cdot c}}}{-a} \]

   6. Simplified 44.8%

      \[\leadsto \frac{b_2 + \sqrt{\color{blue}{-a \cdot c}}}{-a} \]

   7. Taylor expanded in b_2 around inf 31.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{b_2}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-285}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 5: 67.7% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-285}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-285) (* -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 <= -1e-285) {
		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 <= (-1d-285)) 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 <= -1e-285) {
		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 <= -1e-285:
		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 <= -1e-285)
		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 <= -1e-285)
		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, -1e-285], 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 < -1.00000000000000007e-285

   1. Initial program 28.4%

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

      1. sub-neg 28.4%

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

      2. sqr-neg 28.4%

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

      3. distribute-neg-out 28.4%

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

      4. neg-mul-1 28.4%

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

      5. *-commutative 28.4%

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

      6. associate-/l* 28.4%

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

      7. remove-double-neg 28.4%

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

      8. neg-mul-1 28.4%

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

      9. *-commutative 28.4%

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

      10. associate-/l* 28.4%

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

      11. metadata-eval 28.4%

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

      12. /-rgt-identity 28.4%

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

   3. Simplified 28.4%

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

   4. Taylor expanded in b_2 around -inf 70.0%

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

   if -1.00000000000000007e-285 < b_2

   1. Initial program 73.6%

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

      1. sub-neg 73.6%

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

      2. sqr-neg 73.6%

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

      3. distribute-neg-out 73.6%

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

      4. neg-mul-1 73.6%

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

      5. *-commutative 73.6%

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

      6. associate-/l* 73.6%

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

      7. remove-double-neg 73.6%

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

      8. neg-mul-1 73.6%

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

      9. *-commutative 73.6%

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

      10. associate-/l* 73.6%

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

      11. metadata-eval 73.6%

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

      12. /-rgt-identity 73.6%

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

   3. Simplified 73.7%

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

   4. Taylor expanded in b_2 around inf 66.4%

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

      1. associate-*r/ 66.4%

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

      2. *-commutative 66.4%

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

   6. Simplified 66.4%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-285}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 6: 15.4% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ \frac{-b_2}{a} \end{array} \]

FPCore

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

C

double code(double a, double b_2, double c) {
	return -b_2 / 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 / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.0%

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

   1. sub-neg 51.0%

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

   2. sqr-neg 51.0%

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

   3. distribute-neg-out 51.0%

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

   4. neg-mul-1 51.0%

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

   5. *-commutative 51.0%

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

   6. associate-/l* 51.0%

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

   7. remove-double-neg 51.0%

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

   8. neg-mul-1 51.0%

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

   9. *-commutative 51.0%

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

   10. associate-/l* 51.0%

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

   11. metadata-eval 51.0%

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

   12. /-rgt-identity 51.0%

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

3. Simplified 51.0%

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

4. Taylor expanded in a around inf 34.4%

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

   1. mul-1-neg 34.4%

      \[\leadsto \frac{b_2 + \sqrt{\color{blue}{-a \cdot c}}}{-a} \]

6. Simplified 34.4%

   \[\leadsto \frac{b_2 + \sqrt{\color{blue}{-a \cdot c}}}{-a} \]

7. Taylor expanded in b_2 around inf 17.3%

   \[\leadsto \color{blue}{-1 \cdot \frac{b_2}{a}} \]

8. Final simplification 17.3%

   \[\leadsto \frac{-b_2}{a} \]

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 2023301 
(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))