quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.2% → 84.3%

Time: 7.5s

Alternatives: 7

Speedup: 1.7×

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: 52.2% 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: 84.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{+146}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 510000:\\ \;\;\;\;\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.125}{b\_2}, a \cdot \frac{c}{b\_2}, -0.5\right) \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e+146)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 510000.0)
     (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
     (/ (* (fma (/ -0.125 b_2) (* a (/ c b_2)) -0.5) c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e+146) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 510000.0) {
		tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = (fma((-0.125 / b_2), (a * (c / b_2)), -0.5) * c) / b_2;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e+146)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 510000.0)
		tmp = Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	else
		tmp = Float64(Float64(fma(Float64(-0.125 / b_2), Float64(a * Float64(c / b_2)), -0.5) * c) / b_2);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e+146], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 510000.0], 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], N[(N[(N[(N[(-0.125 / b$95$2), $MachinePrecision] * N[(a * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -9.99999999999999934e145

   1. Initial program 53.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   4. Step-by-step derivation

      1. lower-*.f64 95.0

         \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]

   5. Applied rewrites 95.0%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]

   if -9.99999999999999934e145 < b_2 < 5.1e5

   1. Initial program 82.8%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   if 5.1e5 < b_2

   1. Initial program 13.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}}{b\_2}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}} + \frac{-1}{2} \cdot c}}{b\_2} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{-1}{8} \cdot \left(a \cdot {c}^{2}\right)}{{b\_2}^{2}}} + \frac{-1}{2} \cdot c}{b\_2} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{\frac{-1}{8} \cdot \left(a \cdot {c}^{2}\right)}{\color{blue}{b\_2 \cdot b\_2}} + \frac{-1}{2} \cdot c}{b\_2} \]

      5. times-frac N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{-1}{8}}{b\_2} \cdot \frac{a \cdot {c}^{2}}{b\_2}} + \frac{-1}{2} \cdot c}{b\_2} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\frac{-1}{8}}{b\_2}, \frac{a \cdot {c}^{2}}{b\_2}, \frac{-1}{2} \cdot c\right)}}{b\_2} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{8}}{b\_2}}, \frac{a \cdot {c}^{2}}{b\_2}, \frac{-1}{2} \cdot c\right)}{b\_2} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{8}}{b\_2}, \color{blue}{\frac{a \cdot {c}^{2}}{b\_2}}, \frac{-1}{2} \cdot c\right)}{b\_2} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{8}}{b\_2}, \frac{\color{blue}{{c}^{2} \cdot a}}{b\_2}, \frac{-1}{2} \cdot c\right)}{b\_2} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{8}}{b\_2}, \frac{\color{blue}{{c}^{2} \cdot a}}{b\_2}, \frac{-1}{2} \cdot c\right)}{b\_2} \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{8}}{b\_2}, \frac{\color{blue}{\left(c \cdot c\right)} \cdot a}{b\_2}, \frac{-1}{2} \cdot c\right)}{b\_2} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{8}}{b\_2}, \frac{\color{blue}{\left(c \cdot c\right)} \cdot a}{b\_2}, \frac{-1}{2} \cdot c\right)}{b\_2} \]

      13. lower-*.f64 73.5

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-0.125}{b\_2}, \frac{\left(c \cdot c\right) \cdot a}{b\_2}, \color{blue}{-0.5 \cdot c}\right)}{b\_2} \]

   5. Applied rewrites 73.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-0.125}{b\_2}, \frac{\left(c \cdot c\right) \cdot a}{b\_2}, -0.5 \cdot c\right)}{b\_2}} \]

   6. Taylor expanded in c around 0

      \[\leadsto \frac{c \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}} - \frac{1}{2}\right)}{b\_2} \]
   7. Step-by-step derivation

   8. Applied rewrites 91.8%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-0.125}{b\_2}, a \cdot \frac{c}{b\_2}, -0.5\right) \cdot c}{b\_2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 84.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{+146}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 510000:\\ \;\;\;\;\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e+146)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 510000.0)
     (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
     (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e+146) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 510000.0) {
		tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = (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 <= (-1d+146)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 510000.0d0) then
        tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a
    else
        tmp = (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 <= -1e+146) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 510000.0) {
		tmp = (-b_2 + Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e+146:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 510000.0:
		tmp = (-b_2 + math.sqrt(((b_2 * b_2) - (a * c)))) / a
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e+146)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 510000.0)
		tmp = Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	else
		tmp = 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 <= -1e+146)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 510000.0)
		tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e+146], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 510000.0], 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], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -9.99999999999999934e145

   1. Initial program 53.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   4. Step-by-step derivation

      1. lower-*.f64 95.0

         \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]

   5. Applied rewrites 95.0%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]

   if -9.99999999999999934e145 < b_2 < 5.1e5

   1. Initial program 82.8%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   if 5.1e5 < b_2

   1. Initial program 13.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      3. lower-/.f64 91.5

         \[\leadsto \color{blue}{\frac{c}{b\_2}} \cdot -0.5 \]

   5. Applied rewrites 91.5%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 84.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{+146}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 510000:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, \left(-a\right) \cdot c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e+146)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 510000.0)
     (/ (- (sqrt (fma b_2 b_2 (* (- a) c))) b_2) a)
     (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e+146) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 510000.0) {
		tmp = (sqrt(fma(b_2, b_2, (-a * c))) - b_2) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e+146)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 510000.0)
		tmp = Float64(Float64(sqrt(fma(b_2, b_2, Float64(Float64(-a) * c))) - b_2) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e+146], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 510000.0], N[(N[(N[Sqrt[N[(b$95$2 * b$95$2 + N[((-a) * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -9.99999999999999934e145

   1. Initial program 53.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   4. Step-by-step derivation

      1. lower-*.f64 95.0

         \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]

   5. Applied rewrites 95.0%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]

   if -9.99999999999999934e145 < b_2 < 5.1e5

   1. Initial program 82.8%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}}}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}}}{a} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot c}}{a} \]

      4. lower-neg.f64 53.7

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\left(-a\right)} \cdot c}}{a} \]

   5. Applied rewrites 53.7%

      \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\left(-a\right) \cdot c}}}{a} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-b\_2\right) + \sqrt{\left(-a\right) \cdot c}}}{a} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-a\right) \cdot c} + \left(-b\_2\right)}}{a} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-a\right) \cdot c} + \color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)}}{a} \]

      4. unsub-neg N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-a\right) \cdot c} - b\_2}}{a} \]

      5. lower--.f64 53.7

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-a\right) \cdot c} - b\_2}}{a} \]

   7. Applied rewrites 53.7%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(-a\right) \cdot c} - b\_2}}{a} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{\sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right) + {b\_2}^{2}}} - b\_2}{a} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c} + {b\_2}^{2}} - b\_2}{a} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(-1 \cdot a, c, {b\_2}^{2}\right)}} - b\_2}{a} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, c, {b\_2}^{2}\right)} - b\_2}{a} \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\color{blue}{-a}, c, {b\_2}^{2}\right)} - b\_2}{a} \]

      5. unpow2 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(-a, c, \color{blue}{b\_2 \cdot b\_2}\right)} - b\_2}{a} \]

      6. lower-*.f64 82.8

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(-a, c, \color{blue}{b\_2 \cdot b\_2}\right)} - b\_2}{a} \]

   10. Applied rewrites 82.8%

       \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(-a, c, b\_2 \cdot b\_2\right)}} - b\_2}{a} \]
   11. Step-by-step derivation

   12. Applied rewrites 82.8%

       \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b\_2, \color{blue}{b\_2}, \left(-a\right) \cdot c\right)} - b\_2}{a} \]

   if 5.1e5 < b_2

   1. Initial program 13.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      3. lower-/.f64 91.5

         \[\leadsto \color{blue}{\frac{c}{b\_2}} \cdot -0.5 \]

   5. Applied rewrites 91.5%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 78.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.8 \cdot 10^{-39}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 510000:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7.8e-39)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 510000.0) (/ (- (sqrt (* (- a) c)) b_2) a) (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.8e-39) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 510000.0) {
		tmp = (sqrt((-a * c)) - b_2) / a;
	} else {
		tmp = (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 <= (-7.8d-39)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 510000.0d0) then
        tmp = (sqrt((-a * c)) - b_2) / a
    else
        tmp = (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 <= -7.8e-39) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 510000.0) {
		tmp = (Math.sqrt((-a * c)) - b_2) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -7.8e-39:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 510000.0:
		tmp = (math.sqrt((-a * c)) - b_2) / a
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -7.8e-39)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 510000.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(-a) * c)) - b_2) / a);
	else
		tmp = 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 <= -7.8e-39)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 510000.0)
		tmp = (sqrt((-a * c)) - b_2) / a;
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7.8e-39], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 510000.0], N[(N[(N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -7.80000000000000059e-39

   1. Initial program 76.5%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   4. Step-by-step derivation

      1. lower-*.f64 95.2

         \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]

   5. Applied rewrites 95.2%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]

   if -7.80000000000000059e-39 < b_2 < 5.1e5

   1. Initial program 77.1%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}}}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}}}{a} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot c}}{a} \]

      4. lower-neg.f64 67.6

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\left(-a\right)} \cdot c}}{a} \]

   5. Applied rewrites 67.6%

      \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\left(-a\right) \cdot c}}}{a} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-b\_2\right) + \sqrt{\left(-a\right) \cdot c}}}{a} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-a\right) \cdot c} + \left(-b\_2\right)}}{a} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-a\right) \cdot c} + \color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)}}{a} \]

      4. unsub-neg N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-a\right) \cdot c} - b\_2}}{a} \]

      5. lower--.f64 67.6

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-a\right) \cdot c} - b\_2}}{a} \]

   7. Applied rewrites 67.6%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(-a\right) \cdot c} - b\_2}}{a} \]

   if 5.1e5 < b_2

   1. Initial program 13.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      3. lower-/.f64 91.5

         \[\leadsto \color{blue}{\frac{c}{b\_2}} \cdot -0.5 \]

   5. Applied rewrites 91.5%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 67.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310) (/ (* -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 = (-2.0 * b_2) / a;
	} else {
		tmp = (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 = ((-2.0d0) * b_2) / a
    else
        tmp = (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 = (-2.0 * b_2) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = (-2.0 * b_2) / a
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	else
		tmp = 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 = (-2.0 * b_2) / a;
	else
		tmp = (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[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 81.4%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   4. Step-by-step derivation

      1. lower-*.f64 72.7

         \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]

   5. Applied rewrites 72.7%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]

   if -4.999999999999985e-310 < b_2

   1. Initial program 38.9%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      3. lower-/.f64 60.2

         \[\leadsto \color{blue}{\frac{c}{b\_2}} \cdot -0.5 \]

   5. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 34.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b\_2} \cdot -0.5 \end{array} \]

FPCore

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

C

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

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 = (c / b_2) * (-0.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.3%

   \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
2. Add Preprocessing

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

   3. lower-/.f64 33.8

      \[\leadsto \color{blue}{\frac{c}{b\_2}} \cdot -0.5 \]

5. Applied rewrites 33.8%

   \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
6. Add Preprocessing

Alternative 7: 34.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{-0.5}{b\_2} \end{array} \]

FPCore

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

C

double code(double a, double b_2, double c) {
	return c * (-0.5 / 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 = c * ((-0.5d0) / b_2)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.3%

   \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
2. Add Preprocessing

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

   3. lower-/.f64 33.8

      \[\leadsto \color{blue}{\frac{c}{b\_2}} \cdot -0.5 \]

5. Applied rewrites 33.8%

   \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
6. Step-by-step derivation

7. Applied rewrites 33.7%

   \[\leadsto c \cdot \color{blue}{\frac{-0.5}{b\_2}} \]
8. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_1 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{\left|b\_2\right| - t\_0} \cdot \sqrt{\left|b\_2\right| + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b\_2, t\_0\right)\\ \end{array}\\ \mathbf{if}\;b\_2 < 0:\\ \;\;\;\;\frac{t\_1 - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b\_2 + t\_1}\\ \end{array} \end{array} \]

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) (/ (- t_1 b_2) a) (/ (- c) (+ b_2 t_1)))))

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 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	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 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	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 = (t_1 - b_2) / a
	else:
		tmp_1 = -c / (b_2 + t_1)
	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(Float64(t_1 - b_2) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(b_2 + t_1));
	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 = (t_1 - b_2) / a;
	else
		tmp_2 = -c / (b_2 + t_1);
	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[(N[(t$95$1 - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(b$95$2 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2024340 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) x)) (sqrt (+ (fabs b_2) x))) (hypot b_2 x))))) (if (< b_2 0) (/ (- sqtD b_2) a) (/ (- c) (+ b_2 sqtD)))))

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