The quadratic formula (r2)

Percentage Accurate: 51.1% → 86.0%

Time: 12.8s

Alternatives: 8

Speedup: 2.5×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b - Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b - math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 51.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b - Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b - math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 86.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{-74}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+112}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{a}{b} - b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.2e-74)
   (/ c (- b))
   (if (<= b 5e+112)
     (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* a 2.0))
     (/ (- (* c (/ a b)) b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.2e-74) {
		tmp = c / -b;
	} else if (b <= 5e+112) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (a * 2.0);
	} else {
		tmp = ((c * (a / b)) - b) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-6.2d-74)) then
        tmp = c / -b
    else if (b <= 5d+112) then
        tmp = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (a * 2.0d0)
    else
        tmp = ((c * (a / b)) - b) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.2e-74) {
		tmp = c / -b;
	} else if (b <= 5e+112) {
		tmp = (-b - Math.sqrt(((b * b) - (4.0 * (a * c))))) / (a * 2.0);
	} else {
		tmp = ((c * (a / b)) - b) / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.2e-74:
		tmp = c / -b
	elif b <= 5e+112:
		tmp = (-b - math.sqrt(((b * b) - (4.0 * (a * c))))) / (a * 2.0)
	else:
		tmp = ((c * (a / b)) - b) / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.2e-74)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 5e+112)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(c * Float64(a / b)) - b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.2e-74)
		tmp = c / -b;
	elseif (b <= 5e+112)
		tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (a * 2.0);
	else
		tmp = ((c * (a / b)) - b) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.2e-74], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 5e+112], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.2000000000000003e-74

   1. Initial program 18.7%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{c}{\color{blue}{\mathsf{neg}\left(b\right)}} \]

      6. lower-neg.f64 89.1

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

   5. Applied rewrites 89.1%

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

   if -6.2000000000000003e-74 < b < 5e112

   1. Initial program 80.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   if 5e112 < b

   1. Initial program 56.6%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 99.4

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

   5. Applied rewrites 99.4%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{-74}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+112}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{a}{b} - b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{-74}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.4e-74)
   (/ c (- b))
   (if (<= b 6.2e-18)
     (/ (- (- b) (sqrt (* c (* a -4.0)))) (* a 2.0))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.4e-74) {
		tmp = c / -b;
	} else if (b <= 6.2e-18) {
		tmp = (-b - sqrt((c * (a * -4.0)))) / (a * 2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.4d-74)) then
        tmp = c / -b
    else if (b <= 6.2d-18) then
        tmp = (-b - sqrt((c * (a * (-4.0d0))))) / (a * 2.0d0)
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.4e-74) {
		tmp = c / -b;
	} else if (b <= 6.2e-18) {
		tmp = (-b - Math.sqrt((c * (a * -4.0)))) / (a * 2.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.4e-74:
		tmp = c / -b
	elif b <= 6.2e-18:
		tmp = (-b - math.sqrt((c * (a * -4.0)))) / (a * 2.0)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.4e-74)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 6.2e-18)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(c * Float64(a * -4.0)))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.4e-74)
		tmp = c / -b;
	elseif (b <= 6.2e-18)
		tmp = (-b - sqrt((c * (a * -4.0)))) / (a * 2.0);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.4e-74], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 6.2e-18], N[(N[((-b) - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.4000000000000001e-74

   1. Initial program 17.3%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{c}{\color{blue}{\mathsf{neg}\left(b\right)}} \]

      6. lower-neg.f64 86.1

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

   5. Applied rewrites 86.1%

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

   if -3.4000000000000001e-74 < b < 6.20000000000000014e-18

   1. Initial program 76.7%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) - \sqrt{\color{blue}{\left(-4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) - \sqrt{\color{blue}{c \cdot \left(-4 \cdot a\right)}}}{2 \cdot a} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) - \sqrt{\color{blue}{c \cdot \left(-4 \cdot a\right)}}}{2 \cdot a} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) - \sqrt{c \cdot \color{blue}{\left(a \cdot -4\right)}}}{2 \cdot a} \]

      5. lower-*.f64 65.3

         \[\leadsto \frac{\left(-b\right) - \sqrt{c \cdot \color{blue}{\left(a \cdot -4\right)}}}{2 \cdot a} \]

   5. Applied rewrites 65.3%

      \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}}{2 \cdot a} \]

   if 6.20000000000000014e-18 < b

   1. Initial program 64.4%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{c}{b}} - \frac{b}{a} \]

      6. lower-/.f64 90.5

         \[\leadsto \frac{c}{b} - \color{blue}{\frac{b}{a}} \]

   5. Applied rewrites 90.5%

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{-74}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 70.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024223 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((d (sqrt (- (* b b) (* 4 (* a c)))))) (let ((r1 (/ (+ (- b) d) (* 2 a)))) (let ((r2 (/ (- (- b) d) (* 2 a)))) (if (< b 0) (/ c (* a r1)) r2)))))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))