Quadratic roots, wide range

Percentage Accurate: 18.2% → 97.5%

Time: 11.7s

Alternatives: 7

Speedup: 29.0×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

Math

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

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(Float64(4.0 * 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 18.2% accurate, 1.0× speedup

Math

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

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(Float64(4.0 * 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right) - {c}^{2} \cdot {b}^{-3}\right) - \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (*
   a
   (-
    (*
     a
     (+
      (* -2.0 (/ (pow c 3.0) (pow b 5.0)))
      (* -5.0 (/ (* a (pow c 4.0)) (pow b 7.0)))))
    (* (pow c 2.0) (pow b -3.0))))
  (/ c b)))

C

double code(double a, double b, double c) {
	return (a * ((a * ((-2.0 * (pow(c, 3.0) / pow(b, 5.0))) + (-5.0 * ((a * pow(c, 4.0)) / pow(b, 7.0))))) - (pow(c, 2.0) * pow(b, -3.0)))) - (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 = (a * ((a * (((-2.0d0) * ((c ** 3.0d0) / (b ** 5.0d0))) + ((-5.0d0) * ((a * (c ** 4.0d0)) / (b ** 7.0d0))))) - ((c ** 2.0d0) * (b ** (-3.0d0))))) - (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return (a * ((a * ((-2.0 * (Math.pow(c, 3.0) / Math.pow(b, 5.0))) + (-5.0 * ((a * Math.pow(c, 4.0)) / Math.pow(b, 7.0))))) - (Math.pow(c, 2.0) * Math.pow(b, -3.0)))) - (c / b);
}

Python

def code(a, b, c):
	return (a * ((a * ((-2.0 * (math.pow(c, 3.0) / math.pow(b, 5.0))) + (-5.0 * ((a * math.pow(c, 4.0)) / math.pow(b, 7.0))))) - (math.pow(c, 2.0) * math.pow(b, -3.0)))) - (c / b)

Julia

function code(a, b, c)
	return Float64(Float64(a * Float64(Float64(a * Float64(Float64(-2.0 * Float64((c ^ 3.0) / (b ^ 5.0))) + Float64(-5.0 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 7.0))))) - Float64((c ^ 2.0) * (b ^ -3.0)))) - Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (a * ((a * ((-2.0 * ((c ^ 3.0) / (b ^ 5.0))) + (-5.0 * ((a * (c ^ 4.0)) / (b ^ 7.0))))) - ((c ^ 2.0) * (b ^ -3.0)))) - (c / b);
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[(a * N[(N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-5.0 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] * N[Power[b, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 22.0%

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

   1. *-commutative 22.0%

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

3. Simplified 22.0%

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

5. Taylor expanded in a around 0 96.6%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

6. Taylor expanded in c around 0 96.6%

   \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \color{blue}{-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}}\right)\right) \]
7. Step-by-step derivation

   1. associate-*r/ 96.6%

      \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]

   2. neg-mul-1 96.6%

      \[\leadsto \frac{\color{blue}{-c}}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]

8. Applied egg-rr 96.6%

   \[\leadsto \color{blue}{\frac{-c}{b}} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]
9. Step-by-step derivation

   1. pow1 96.6%

      \[\leadsto \frac{-c}{b} + a \cdot \left(\color{blue}{{\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{1}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]

   2. mul-1-neg 96.6%

      \[\leadsto \frac{-c}{b} + a \cdot \left({\color{blue}{\left(-\frac{{c}^{2}}{{b}^{3}}\right)}}^{1} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]

   3. div-inv 96.6%

      \[\leadsto \frac{-c}{b} + a \cdot \left({\left(-\color{blue}{{c}^{2} \cdot \frac{1}{{b}^{3}}}\right)}^{1} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]

   4. pow-flip 96.6%

      \[\leadsto \frac{-c}{b} + a \cdot \left({\left(-{c}^{2} \cdot \color{blue}{{b}^{\left(-3\right)}}\right)}^{1} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]

   5. metadata-eval 96.6%

      \[\leadsto \frac{-c}{b} + a \cdot \left({\left(-{c}^{2} \cdot {b}^{\color{blue}{-3}}\right)}^{1} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]

10. Applied egg-rr 96.6%

    \[\leadsto \frac{-c}{b} + a \cdot \left(\color{blue}{{\left(-{c}^{2} \cdot {b}^{-3}\right)}^{1}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]
11. Step-by-step derivation

    1. unpow1 96.6%

       \[\leadsto \frac{-c}{b} + a \cdot \left(\color{blue}{\left(-{c}^{2} \cdot {b}^{-3}\right)} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]

    2. distribute-rgt-neg-in 96.6%

       \[\leadsto \frac{-c}{b} + a \cdot \left(\color{blue}{{c}^{2} \cdot \left(-{b}^{-3}\right)} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]

12. Simplified 96.6%

    \[\leadsto \frac{-c}{b} + a \cdot \left(\color{blue}{{c}^{2} \cdot \left(-{b}^{-3}\right)} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right)\right) \]

13. Final simplification 96.6%

    \[\leadsto a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}}\right) - {c}^{2} \cdot {b}^{-3}\right) - \frac{c}{b} \]
14. Add Preprocessing

Alternative 2: 96.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ a \cdot \left({c}^{3} \cdot \left(-2 \cdot \frac{a}{{b}^{5}} - \frac{1}{c \cdot {b}^{3}}\right)\right) - \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (*
   a
   (* (pow c 3.0) (- (* -2.0 (/ a (pow b 5.0))) (/ 1.0 (* c (pow b 3.0))))))
  (/ c b)))

C

double code(double a, double b, double c) {
	return (a * (pow(c, 3.0) * ((-2.0 * (a / pow(b, 5.0))) - (1.0 / (c * pow(b, 3.0)))))) - (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 = (a * ((c ** 3.0d0) * (((-2.0d0) * (a / (b ** 5.0d0))) - (1.0d0 / (c * (b ** 3.0d0)))))) - (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return (a * (Math.pow(c, 3.0) * ((-2.0 * (a / Math.pow(b, 5.0))) - (1.0 / (c * Math.pow(b, 3.0)))))) - (c / b);
}

Python

def code(a, b, c):
	return (a * (math.pow(c, 3.0) * ((-2.0 * (a / math.pow(b, 5.0))) - (1.0 / (c * math.pow(b, 3.0)))))) - (c / b)

Julia

function code(a, b, c)
	return Float64(Float64(a * Float64((c ^ 3.0) * Float64(Float64(-2.0 * Float64(a / (b ^ 5.0))) - Float64(1.0 / Float64(c * (b ^ 3.0)))))) - Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (a * ((c ^ 3.0) * ((-2.0 * (a / (b ^ 5.0))) - (1.0 / (c * (b ^ 3.0)))))) - (c / b);
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[Power[c, 3.0], $MachinePrecision] * N[(N[(-2.0 * N[(a / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(c * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 22.0%

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

   1. *-commutative 22.0%

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

3. Simplified 22.0%

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

5. Taylor expanded in a around 0 95.8%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -1 \cdot \frac{{c}^{2}}{{b}^{3}}\right)} \]

6. Taylor expanded in c around inf 95.8%

   \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \color{blue}{\left({c}^{3} \cdot \left(-2 \cdot \frac{a}{{b}^{5}} - \frac{1}{{b}^{3} \cdot c}\right)\right)} \]

7. Final simplification 95.8%

   \[\leadsto a \cdot \left({c}^{3} \cdot \left(-2 \cdot \frac{a}{{b}^{5}} - \frac{1}{c \cdot {b}^{3}}\right)\right) - \frac{c}{b} \]
8. Add Preprocessing

Alternative 3: 96.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(c \cdot \left(a \cdot \left(\frac{-2 \cdot \left(c \cdot a\right)}{{b}^{5}} - \frac{1}{{b}^{3}}\right)\right) + \frac{-1}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (* c (* a (- (/ (* -2.0 (* c a)) (pow b 5.0)) (/ 1.0 (pow b 3.0)))))
   (/ -1.0 b))))

C

double code(double a, double b, double c) {
	return c * ((c * (a * (((-2.0 * (c * a)) / pow(b, 5.0)) - (1.0 / pow(b, 3.0))))) + (-1.0 / 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 * ((c * (a * ((((-2.0d0) * (c * a)) / (b ** 5.0d0)) - (1.0d0 / (b ** 3.0d0))))) + ((-1.0d0) / b))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((c * (a * (((-2.0 * (c * a)) / Math.pow(b, 5.0)) - (1.0 / Math.pow(b, 3.0))))) + (-1.0 / b));
}

Python

def code(a, b, c):
	return c * ((c * (a * (((-2.0 * (c * a)) / math.pow(b, 5.0)) - (1.0 / math.pow(b, 3.0))))) + (-1.0 / b))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(c * Float64(a * Float64(Float64(Float64(-2.0 * Float64(c * a)) / (b ^ 5.0)) - Float64(1.0 / (b ^ 3.0))))) + Float64(-1.0 / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((c * (a * (((-2.0 * (c * a)) / (b ^ 5.0)) - (1.0 / (b ^ 3.0))))) + (-1.0 / b));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(c * N[(a * N[(N[(N[(-2.0 * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 22.0%

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

   1. *-commutative 22.0%

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

3. Simplified 22.0%

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

5. Taylor expanded in c around 0 95.5%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)} \]

6. Taylor expanded in a around 0 95.5%

   \[\leadsto c \cdot \left(c \cdot \color{blue}{\left(a \cdot \left(-2 \cdot \frac{a \cdot c}{{b}^{5}} - \frac{1}{{b}^{3}}\right)\right)} - \frac{1}{b}\right) \]
7. Step-by-step derivation

   1. associate-*r/ 95.5%

      \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(\color{blue}{\frac{-2 \cdot \left(a \cdot c\right)}{{b}^{5}}} - \frac{1}{{b}^{3}}\right)\right) - \frac{1}{b}\right) \]

8. Simplified 95.5%

   \[\leadsto c \cdot \left(c \cdot \color{blue}{\left(a \cdot \left(\frac{-2 \cdot \left(a \cdot c\right)}{{b}^{5}} - \frac{1}{{b}^{3}}\right)\right)} - \frac{1}{b}\right) \]

9. Final simplification 95.5%

   \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(\frac{-2 \cdot \left(c \cdot a\right)}{{b}^{5}} - \frac{1}{{b}^{3}}\right)\right) + \frac{-1}{b}\right) \]
10. Add Preprocessing

Alternative 4: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(a, \frac{\frac{c}{b}}{\frac{b}{c}}, c\right)}{-b} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 22.0%

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

   1. *-commutative 22.0%

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

3. Simplified 22.0%

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

5. Taylor expanded in c around 0 93.8%

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

   1. associate-*r/ 93.8%

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

   2. neg-mul-1 93.8%

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

   3. distribute-rgt-neg-in 93.8%

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

7. Simplified 93.8%

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

8. Taylor expanded in a around inf 93.5%

   \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \frac{c}{a \cdot b} + -1 \cdot \frac{{c}^{2}}{{b}^{3}}\right)} \]
9. Step-by-step derivation

   1. mul-1-neg 93.5%

      \[\leadsto a \cdot \left(-1 \cdot \frac{c}{a \cdot b} + \color{blue}{\left(-\frac{{c}^{2}}{{b}^{3}}\right)}\right) \]

   2. unsub-neg 93.5%

      \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \frac{c}{a \cdot b} - \frac{{c}^{2}}{{b}^{3}}\right)} \]

   3. mul-1-neg 93.5%

      \[\leadsto a \cdot \left(\color{blue}{\left(-\frac{c}{a \cdot b}\right)} - \frac{{c}^{2}}{{b}^{3}}\right) \]

   4. associate-/r* 93.5%

      \[\leadsto a \cdot \left(\left(-\color{blue}{\frac{\frac{c}{a}}{b}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) \]

   5. distribute-neg-frac 93.5%

      \[\leadsto a \cdot \left(\color{blue}{\frac{-\frac{c}{a}}{b}} - \frac{{c}^{2}}{{b}^{3}}\right) \]

   6. distribute-neg-frac2 93.5%

      \[\leadsto a \cdot \left(\frac{\color{blue}{\frac{c}{-a}}}{b} - \frac{{c}^{2}}{{b}^{3}}\right) \]

10. Simplified 93.5%

    \[\leadsto \color{blue}{a \cdot \left(\frac{\frac{c}{-a}}{b} - \frac{{c}^{2}}{{b}^{3}}\right)} \]

11. Taylor expanded in b around inf 94.0%

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

    1. distribute-lft-out 94.0%

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

    2. mul-1-neg 94.0%

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

    3. distribute-neg-frac 94.0%

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

    4. distribute-neg-frac2 94.0%

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

    5. +-commutative 94.0%

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

    6. associate-/l* 94.0%

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

    7. fma-define 94.0%

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

    8. unpow2 94.0%

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

    9. unpow2 94.0%

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

    10. times-frac 94.0%

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

    11. unpow2 94.0%

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

13. Simplified 94.0%

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

    1. unpow2 94.0%

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

    2. clear-num 94.0%

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

    3. un-div-inv 94.0%

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

15. Applied egg-rr 94.0%

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

Alternative 5: 95.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c) :precision binary64 (/ (+ c (* a (pow (/ b c) -2.0))) (- b)))

C

double code(double a, double b, double c) {
	return (c + (a * pow((b / c), -2.0))) / -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 + (a * ((b / c) ** (-2.0d0)))) / -b
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(Float64(c + Float64(a * (Float64(b / c) ^ -2.0))) / Float64(-b))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.0%

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

   1. *-commutative 22.0%

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

3. Simplified 22.0%

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

5. Taylor expanded in c around 0 93.8%

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

   1. associate-*r/ 93.8%

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

   2. neg-mul-1 93.8%

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

   3. distribute-rgt-neg-in 93.8%

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

7. Simplified 93.8%

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

8. Taylor expanded in a around inf 93.5%

   \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \frac{c}{a \cdot b} + -1 \cdot \frac{{c}^{2}}{{b}^{3}}\right)} \]
9. Step-by-step derivation

   1. mul-1-neg 93.5%

      \[\leadsto a \cdot \left(-1 \cdot \frac{c}{a \cdot b} + \color{blue}{\left(-\frac{{c}^{2}}{{b}^{3}}\right)}\right) \]

   2. unsub-neg 93.5%

      \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \frac{c}{a \cdot b} - \frac{{c}^{2}}{{b}^{3}}\right)} \]

   3. mul-1-neg 93.5%

      \[\leadsto a \cdot \left(\color{blue}{\left(-\frac{c}{a \cdot b}\right)} - \frac{{c}^{2}}{{b}^{3}}\right) \]

   4. associate-/r* 93.5%

      \[\leadsto a \cdot \left(\left(-\color{blue}{\frac{\frac{c}{a}}{b}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) \]

   5. distribute-neg-frac 93.5%

      \[\leadsto a \cdot \left(\color{blue}{\frac{-\frac{c}{a}}{b}} - \frac{{c}^{2}}{{b}^{3}}\right) \]

   6. distribute-neg-frac2 93.5%

      \[\leadsto a \cdot \left(\frac{\color{blue}{\frac{c}{-a}}}{b} - \frac{{c}^{2}}{{b}^{3}}\right) \]

10. Simplified 93.5%

    \[\leadsto \color{blue}{a \cdot \left(\frac{\frac{c}{-a}}{b} - \frac{{c}^{2}}{{b}^{3}}\right)} \]

11. Taylor expanded in b around inf 94.0%

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

    1. distribute-lft-out 94.0%

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

    2. mul-1-neg 94.0%

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

    3. distribute-neg-frac 94.0%

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

    4. distribute-neg-frac2 94.0%

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

    5. +-commutative 94.0%

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

    6. associate-/l* 94.0%

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

    7. fma-define 94.0%

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

    8. unpow2 94.0%

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

    9. unpow2 94.0%

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

    10. times-frac 94.0%

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

    11. unpow2 94.0%

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

13. Simplified 94.0%

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

    1. fma-undefine 94.0%

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

    2. unpow2 94.0%

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

    3. frac-times 94.0%

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

    4. sqr-neg 94.0%

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

    5. frac-times 94.0%

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

    6. clear-num 94.0%

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

    7. clear-num 94.0%

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

    8. inv-pow 94.0%

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

    9. inv-pow 94.0%

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

    10. pow-sqr 94.0%

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

    11. add-sqr-sqrt 94.0%

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

    12. sqrt-unprod 94.0%

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

    13. sqr-neg 94.0%

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

    14. sqrt-unprod 0.0%

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

    15. add-sqr-sqrt 94.0%

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

    16. frac-2neg 94.0%

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

    17. metadata-eval 94.0%

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

15. Applied egg-rr 94.0%

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

16. Final simplification 94.0%

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

Alternative 6: 90.2% accurate, 29.0× 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(Float64(-c) / 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 22.0%

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

   1. *-commutative 22.0%

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

3. Simplified 22.0%

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

5. Taylor expanded in b around inf 87.6%

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

   1. associate-*r/ 87.6%

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

   2. mul-1-neg 87.6%

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

7. Simplified 87.6%

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

Alternative 7: 1.7% accurate, 38.7× 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 / 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 22.0%

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

   1. *-commutative 22.0%

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

3. Simplified 22.0%

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

5. Taylor expanded in b around inf 87.6%

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

   1. associate-*r/ 87.6%

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

   2. mul-1-neg 87.6%

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

7. Simplified 87.6%

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

   1. clear-num 87.3%

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

   2. inv-pow 87.3%

      \[\leadsto \color{blue}{{\left(\frac{b}{-c}\right)}^{-1}} \]

9. Applied egg-rr 87.3%

   \[\leadsto \color{blue}{{\left(\frac{b}{-c}\right)}^{-1}} \]
10. Step-by-step derivation

    1. unpow-1 87.3%

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

11. Applied egg-rr 87.3%

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

    1. clear-num 87.6%

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

    2. add-sqr-sqrt 87.1%

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

    3. sqrt-unprod 87.6%

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

    4. sqr-neg 87.6%

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

    5. sqrt-unprod 0.0%

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

    6. add-sqr-sqrt 1.7%

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

    7. *-un-lft-identity 1.7%

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

    8. frac-2neg 1.7%

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

13. Applied egg-rr 1.7%

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

    1. *-lft-identity 1.7%

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

15. Simplified 1.7%

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

Reproduce

herbie shell --seed 2024174 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))