quadm (p42, negative)

Percentage Accurate: 51.9% → 87.6%

Time: 8.3s

Alternatives: 9

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: 51.9% 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: 87.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}\\ \mathbf{if}\;b \leq -2.9 \cdot 10^{+77}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq -3.05 \cdot 10^{-84}:\\ \;\;\;\;\left(-4 \cdot c\right) \cdot \frac{a}{\left(t\_0 - b\right) \cdot \left(-2 \cdot a\right)}\\ \mathbf{elif}\;b \leq 10^{+105}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{t\_0}{a}, -2 \cdot a, -2 \cdot b\right)}{-2 \cdot \left(-2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* a -4.0) c (* b b)))))
   (if (<= b -2.9e+77)
     (/ c (- b))
     (if (<= b -3.05e-84)
       (* (* -4.0 c) (/ a (* (- t_0 b) (* -2.0 a))))
       (if (<= b 1e+105)
         (/ (fma (/ t_0 a) (* -2.0 a) (* -2.0 b)) (* -2.0 (* -2.0 a)))
         (fma (/ b a) -1.0 (/ c b)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((a * -4.0), c, (b * b)));
	double tmp;
	if (b <= -2.9e+77) {
		tmp = c / -b;
	} else if (b <= -3.05e-84) {
		tmp = (-4.0 * c) * (a / ((t_0 - b) * (-2.0 * a)));
	} else if (b <= 1e+105) {
		tmp = fma((t_0 / a), (-2.0 * a), (-2.0 * b)) / (-2.0 * (-2.0 * a));
	} else {
		tmp = fma((b / a), -1.0, (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(a * -4.0), c, Float64(b * b)))
	tmp = 0.0
	if (b <= -2.9e+77)
		tmp = Float64(c / Float64(-b));
	elseif (b <= -3.05e-84)
		tmp = Float64(Float64(-4.0 * c) * Float64(a / Float64(Float64(t_0 - b) * Float64(-2.0 * a))));
	elseif (b <= 1e+105)
		tmp = Float64(fma(Float64(t_0 / a), Float64(-2.0 * a), Float64(-2.0 * b)) / Float64(-2.0 * Float64(-2.0 * a)));
	else
		tmp = fma(Float64(b / a), -1.0, Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(a * -4.0), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2.9e+77], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, -3.05e-84], N[(N[(-4.0 * c), $MachinePrecision] * N[(a / N[(N[(t$95$0 - b), $MachinePrecision] * N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e+105], N[(N[(N[(t$95$0 / a), $MachinePrecision] * N[(-2.0 * a), $MachinePrecision] + N[(-2.0 * b), $MachinePrecision]), $MachinePrecision] / N[(-2.0 * N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b / a), $MachinePrecision] * -1.0 + N[(c / b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.9000000000000002e77

   1. Initial program 10.0%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 94.2

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

   5. Applied rewrites 94.2%

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

   if -2.9000000000000002e77 < b < -3.0499999999999998e-84

   1. Initial program 30.8%

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

   4. Applied rewrites 30.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

   6. Applied rewrites 77.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 90.7

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

   8. Applied rewrites 90.7%

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

   if -3.0499999999999998e-84 < b < 9.9999999999999994e104

   1. Initial program 86.3%

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

   4. Applied rewrites 86.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. frac-add N/A

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

      8. lower-/.f64 N/A

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

   6. Applied rewrites 86.4%

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

   if 9.9999999999999994e104 < b

   1. Initial program 42.9%

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

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 91.9

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

   5. Applied rewrites 91.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{+77}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq -3.05 \cdot 10^{-84}:\\ \;\;\;\;\left(-4 \cdot c\right) \cdot \frac{a}{\left(\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)} - b\right) \cdot \left(-2 \cdot a\right)}\\ \mathbf{elif}\;b \leq 10^{+105}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}}{a}, -2 \cdot a, -2 \cdot b\right)}{-2 \cdot \left(-2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-82}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 10^{+105}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}}{a}, -2 \cdot a, -2 \cdot b\right)}{-2 \cdot \left(-2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.1e-82)
   (/ c (- b))
   (if (<= b 1e+105)
     (/
      (fma (/ (sqrt (fma (* a -4.0) c (* b b))) a) (* -2.0 a) (* -2.0 b))
      (* -2.0 (* -2.0 a)))
     (fma (/ b a) -1.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.1e-82) {
		tmp = c / -b;
	} else if (b <= 1e+105) {
		tmp = fma((sqrt(fma((a * -4.0), c, (b * b))) / a), (-2.0 * a), (-2.0 * b)) / (-2.0 * (-2.0 * a));
	} else {
		tmp = fma((b / a), -1.0, (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.1e-82)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 1e+105)
		tmp = Float64(fma(Float64(sqrt(fma(Float64(a * -4.0), c, Float64(b * b))) / a), Float64(-2.0 * a), Float64(-2.0 * b)) / Float64(-2.0 * Float64(-2.0 * a)));
	else
		tmp = fma(Float64(b / a), -1.0, Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.1e-82], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 1e+105], N[(N[(N[(N[Sqrt[N[(N[(a * -4.0), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision] * N[(-2.0 * a), $MachinePrecision] + N[(-2.0 * b), $MachinePrecision]), $MachinePrecision] / N[(-2.0 * N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b / a), $MachinePrecision] * -1.0 + N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.09999999999999993e-82

   1. Initial program 17.0%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 86.6

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

   5. Applied rewrites 86.6%

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

   if -1.09999999999999993e-82 < b < 9.9999999999999994e104

   1. Initial program 86.3%

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

   4. Applied rewrites 86.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. frac-add N/A

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

      8. lower-/.f64 N/A

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

   6. Applied rewrites 86.4%

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

   if 9.9999999999999994e104 < b

   1. Initial program 42.9%

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

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 91.9

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

   5. Applied rewrites 91.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-82}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 10^{+105}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} + b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.1e-82)
   (/ c (- b))
   (if (<= b 1e+105)
     (/ (+ (sqrt (fma (* c -4.0) a (* b b))) b) (* -2.0 a))
     (fma (/ b a) -1.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.1e-82) {
		tmp = c / -b;
	} else if (b <= 1e+105) {
		tmp = (sqrt(fma((c * -4.0), a, (b * b))) + b) / (-2.0 * a);
	} else {
		tmp = fma((b / a), -1.0, (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.1e-82)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 1e+105)
		tmp = Float64(Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) + b) / Float64(-2.0 * a));
	else
		tmp = fma(Float64(b / a), -1.0, Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.1e-82], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 1e+105], N[(N[(N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(b / a), $MachinePrecision] * -1.0 + N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.09999999999999993e-82

   1. Initial program 17.0%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 86.6

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

   5. Applied rewrites 86.6%

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

   if -1.09999999999999993e-82 < b < 9.9999999999999994e104

   1. Initial program 86.3%

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

   4. Applied rewrites 86.3%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 86.4

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

   6. Applied rewrites 86.4%

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

   if 9.9999999999999994e104 < b

   1. Initial program 42.9%

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

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 91.9

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

   5. Applied rewrites 91.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-82}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 10^{+105}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.1e-82)
   (/ c (- b))
   (if (<= b 1e+105)
     (/ (+ (sqrt (fma -4.0 (* c a) (* b b))) b) (* -2.0 a))
     (fma (/ b a) -1.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.1e-82) {
		tmp = c / -b;
	} else if (b <= 1e+105) {
		tmp = (sqrt(fma(-4.0, (c * a), (b * b))) + b) / (-2.0 * a);
	} else {
		tmp = fma((b / a), -1.0, (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.1e-82)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 1e+105)
		tmp = Float64(Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) + b) / Float64(-2.0 * a));
	else
		tmp = fma(Float64(b / a), -1.0, Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.1e-82], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 1e+105], N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(b / a), $MachinePrecision] * -1.0 + N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.09999999999999993e-82

   1. Initial program 17.0%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 86.6

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

   5. Applied rewrites 86.6%

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

   if -1.09999999999999993e-82 < b < 9.9999999999999994e104

   1. Initial program 86.3%

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

   4. Applied rewrites 86.3%

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

   if 9.9999999999999994e104 < b

   1. Initial program 42.9%

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

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 91.9

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

   5. Applied rewrites 91.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-82}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 620000:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)} + b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.1e-82)
   (/ c (- b))
   (if (<= b 620000.0)
     (/ (+ (sqrt (* -4.0 (* a c))) b) (* -2.0 a))
     (fma (/ b a) -1.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.1e-82) {
		tmp = c / -b;
	} else if (b <= 620000.0) {
		tmp = (sqrt((-4.0 * (a * c))) + b) / (-2.0 * a);
	} else {
		tmp = fma((b / a), -1.0, (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.1e-82)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 620000.0)
		tmp = Float64(Float64(sqrt(Float64(-4.0 * Float64(a * c))) + b) / Float64(-2.0 * a));
	else
		tmp = fma(Float64(b / a), -1.0, Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.1e-82], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 620000.0], N[(N[(N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(b / a), $MachinePrecision] * -1.0 + N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.09999999999999993e-82

   1. Initial program 17.0%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 86.6

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

   5. Applied rewrites 86.6%

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

   if -1.09999999999999993e-82 < b < 6.2e5

   1. Initial program 84.6%

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

   4. Applied rewrites 84.6%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 79.3

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

   7. Applied rewrites 79.3%

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

   if 6.2e5 < b

   1. Initial program 57.5%

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

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 87.4

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

   5. Applied rewrites 87.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 67.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (/ c (- b)) (fma (/ b a) -1.0 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = c / -b;
	} else {
		tmp = fma((b / a), -1.0, (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(c / Float64(-b));
	else
		tmp = fma(Float64(b / a), -1.0, Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[(c / (-b)), $MachinePrecision], N[(N[(b / a), $MachinePrecision] * -1.0 + N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 32.3%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 67.7

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

   5. Applied rewrites 67.7%

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

   if -3.999999999999988e-310 < b

   1. Initial program 68.3%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 62.8

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

   5. Applied rewrites 62.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 67.7% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (/ c (- b)) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = c / -b;
	} else {
		tmp = -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 <= (-4d-310)) then
        tmp = c / -b
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = c / -b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = c / -b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(c / Float64(-b));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e-310)
		tmp = c / -b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[(c / (-b)), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 32.3%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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 67.7

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

   5. Applied rewrites 67.7%

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

   if -3.999999999999988e-310 < b

   1. Initial program 68.3%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 62.7

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

   5. Applied rewrites 62.7%

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

Alternative 8: 35.3% accurate, 3.6× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return c / -b;
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(c / (-b)), $MachinePrecision]

Derivation

1. Initial program 50.1%

   \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
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 35.2

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

5. Applied rewrites 35.2%

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

Alternative 9: 11.1% accurate, 50.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 50.1%

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

4. Applied rewrites 31.7%

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

5. Taylor expanded in b around -inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt1-in N/A

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

   5. metadata-eval N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot \left(\color{blue}{0} \cdot a\right)\right) \cdot b}{{a}^{2}} \]

   6. mul0-lft N/A

      \[\leadsto \frac{\left(\frac{1}{2} \cdot \color{blue}{0}\right) \cdot b}{{a}^{2}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0} \cdot b}{{a}^{2}} \]

   8. mul0-lft N/A

      \[\leadsto \frac{\color{blue}{\left(0 \cdot a\right)} \cdot b}{{a}^{2}} \]

   9. metadata-eval N/A

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

   10. distribute-rgt1-in N/A

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

   11. *-commutative N/A

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

   12. associate-/l* N/A

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

   13. distribute-rgt1-in N/A

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

   14. metadata-eval N/A

       \[\leadsto b \cdot \frac{\color{blue}{0} \cdot a}{{a}^{2}} \]

   15. mul0-lft N/A

       \[\leadsto b \cdot \frac{\color{blue}{0}}{{a}^{2}} \]

   16. metadata-eval N/A

       \[\leadsto b \cdot \frac{\color{blue}{\frac{-1}{2} \cdot 0}}{{a}^{2}} \]

   17. mul0-lft N/A

       \[\leadsto b \cdot \frac{\frac{-1}{2} \cdot \color{blue}{\left(0 \cdot a\right)}}{{a}^{2}} \]

   18. metadata-eval N/A

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

   19. distribute-rgt1-in N/A

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

   20. associate-*r/ N/A

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

   21. lower-*.f64 N/A

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

   22. associate-*r/ N/A

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

7. Applied rewrites 9.2%

   \[\leadsto \color{blue}{b \cdot \frac{0}{a \cdot a}} \]
8. Step-by-step derivation

9. Applied rewrites 13.7%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t\_0 - t\_1} \cdot \sqrt{t\_0 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t\_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{t\_2 - \frac{b}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{2} + t\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fabs (/ b 2.0)))
        (t_1 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_2
         (if (== (copysign a c) a)
           (* (sqrt (- t_0 t_1)) (sqrt (+ t_0 t_1)))
           (hypot (/ b 2.0) t_1))))
   (if (< b 0.0) (/ c (- t_2 (/ b 2.0))) (/ (+ (/ b 2.0) t_2) (- a)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs((b / 2.0));
	double t_1 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	} else {
		tmp = hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs((b / 2.0));
	double t_1 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((t_0 - t_1)) * Math.sqrt((t_0 + t_1));
	} else {
		tmp = Math.hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.fabs((b / 2.0))
	t_1 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((t_0 - t_1)) * math.sqrt((t_0 + t_1))
	else:
		tmp = math.hypot((b / 2.0), t_1)
	t_2 = tmp
	tmp_1 = 0
	if b < 0.0:
		tmp_1 = c / (t_2 - (b / 2.0))
	else:
		tmp_1 = ((b / 2.0) + t_2) / -a
	return tmp_1

Julia

function code(a, b, c)
	t_0 = abs(Float64(b / 2.0))
	t_1 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(t_0 - t_1)) * sqrt(Float64(t_0 + t_1)));
	else
		tmp = hypot(Float64(b / 2.0), t_1);
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b < 0.0)
		tmp_1 = Float64(c / Float64(t_2 - Float64(b / 2.0)));
	else
		tmp_1 = Float64(Float64(Float64(b / 2.0) + t_2) / Float64(-a));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = abs((b / 2.0));
	t_1 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	else
		tmp = hypot((b / 2.0), t_1);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b < 0.0)
		tmp_2 = c / (t_2 - (b / 2.0));
	else
		tmp_2 = ((b / 2.0) + t_2) / -a;
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Abs[N[(b / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(b / 2.0), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]]}, If[Less[b, 0.0], N[(c / N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision] / (-a)), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024340 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :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 0) (/ c (- sqtD (/ b 2))) (/ (+ (/ b 2) sqtD) (- a)))))

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