Quadratic roots, narrow range

Percentage Accurate: 55.8% → 99.6%

Time: 14.5s

Alternatives: 7

Speedup: 29.0×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 55.8% 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: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

   1. *-commutative 55.9%

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

3. Simplified 55.9%

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

   1. add-cube-cbrt 55.9%

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

   2. pow3 55.8%

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

6. Applied egg-rr 55.8%

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

   1. flip-+ 55.7%

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

8. Applied egg-rr 57.2%

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

   1. associate--r- 99.2%

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

   2. associate-*r* 99.2%

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

   3. *-commutative 99.2%

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

   4. associate-*r* 99.2%

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

   5. associate-*r* 99.2%

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

   6. *-commutative 99.2%

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

   7. associate-*r* 99.2%

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

10. Simplified 99.2%

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

11. Taylor expanded in b around 0 99.2%

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

    1. *-un-lft-identity 99.2%

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

    2. associate-/l/ 99.2%

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

    3. *-commutative 99.2%

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

    4. associate-*r* 99.2%

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

    5. *-commutative 99.2%

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

    6. *-commutative 99.2%

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

13. Applied egg-rr 99.2%

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

    1. *-lft-identity 99.2%

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

    2. associate-/r* 99.4%

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

    3. times-frac 99.5%

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

    4. *-commutative 99.5%

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

    5. *-commutative 99.5%

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

15. Simplified 99.5%

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

Alternative 2: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

   1. *-commutative 55.9%

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

3. Simplified 55.9%

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

   1. add-cube-cbrt 55.9%

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

   2. pow3 55.8%

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

6. Applied egg-rr 55.8%

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

   1. flip-+ 55.7%

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

8. Applied egg-rr 57.2%

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

   1. associate--r- 99.2%

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

   2. associate-*r* 99.2%

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

   3. *-commutative 99.2%

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

   4. associate-*r* 99.2%

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

   5. associate-*r* 99.2%

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

   6. *-commutative 99.2%

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

   7. associate-*r* 99.2%

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

10. Simplified 99.2%

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

11. Taylor expanded in b around 0 99.2%

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

    1. div-inv 99.1%

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

    2. associate-/l* 99.1%

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

    3. *-commutative 99.1%

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

13. Applied egg-rr 99.1%

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

    1. associate-*r/ 99.2%

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

    2. *-rgt-identity 99.2%

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

    3. *-commutative 99.2%

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

    4. times-frac 99.2%

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

    5. metadata-eval 99.2%

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

    6. associate-/l* 99.4%

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

    7. *-commutative 99.4%

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

15. Simplified 99.4%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

   1. *-commutative 55.9%

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

3. Simplified 55.9%

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

   1. add-cube-cbrt 55.9%

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

   2. pow3 55.8%

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

6. Applied egg-rr 55.8%

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

   1. flip-+ 55.7%

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

8. Applied egg-rr 57.2%

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

   1. associate--r- 99.2%

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

   2. associate-*r* 99.2%

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

   3. *-commutative 99.2%

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

   4. associate-*r* 99.2%

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

   5. associate-*r* 99.2%

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

   6. *-commutative 99.2%

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

   7. associate-*r* 99.2%

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

10. Simplified 99.2%

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

11. Taylor expanded in b around 0 99.2%

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

    1. unpow2 99.2%

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

13. Applied egg-rr 99.2%

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

Alternative 4: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 19.5:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 19.5)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
   (/ (/ (* 4.0 (* a c)) (* 2.0 (- (* a (/ c b)) b))) (* a 2.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 19.5) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = ((4.0 * (a * c)) / (2.0 * ((a * (c / b)) - b))) / (a * 2.0);
	}
	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 <= 19.5d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = ((4.0d0 * (a * c)) / (2.0d0 * ((a * (c / b)) - b))) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 19.5) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = ((4.0 * (a * c)) / (2.0 * ((a * (c / b)) - b))) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 19.5:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = ((4.0 * (a * c)) / (2.0 * ((a * (c / b)) - b))) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 19.5)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(4.0 * Float64(a * c)) / Float64(2.0 * Float64(Float64(a * Float64(c / b)) - b))) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 19.5)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = ((4.0 * (a * c)) / (2.0 * ((a * (c / b)) - b))) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 19.5], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 19.5

   1. Initial program 79.1%

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

   if 19.5 < b

   1. Initial program 47.8%

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

      1. *-commutative 47.8%

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

   3. Simplified 47.8%

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

      1. add-cube-cbrt 47.8%

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

      2. pow3 47.8%

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

   6. Applied egg-rr 47.8%

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

      1. flip-+ 47.7%

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

   8. Applied egg-rr 49.1%

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

      1. associate--r- 99.2%

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

      2. associate-*r* 99.2%

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

      3. *-commutative 99.2%

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

      4. associate-*r* 99.2%

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

      5. associate-*r* 99.2%

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

      6. *-commutative 99.2%

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

      7. associate-*r* 99.2%

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

   10. Simplified 99.2%

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

   11. Taylor expanded in b around 0 99.2%

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

   12. Taylor expanded in a around 0 89.1%

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

       1. distribute-lft-out-- 89.1%

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

       2. associate-/l* 89.1%

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

   14. Simplified 89.1%

       \[\leadsto \frac{\frac{4 \cdot \left(a \cdot c\right)}{\color{blue}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}}}{a \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 19.5:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.7% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

   1. *-commutative 55.9%

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

3. Simplified 55.9%

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

   1. add-cube-cbrt 55.9%

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

   2. pow3 55.8%

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

6. Applied egg-rr 55.8%

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

   1. flip-+ 55.7%

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

8. Applied egg-rr 57.2%

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

   1. associate--r- 99.2%

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

   2. associate-*r* 99.2%

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

   3. *-commutative 99.2%

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

   4. associate-*r* 99.2%

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

   5. associate-*r* 99.2%

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

   6. *-commutative 99.2%

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

   7. associate-*r* 99.2%

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

10. Simplified 99.2%

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

11. Taylor expanded in b around 0 99.2%

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

12. Taylor expanded in a around 0 82.9%

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

    1. distribute-lft-out-- 82.9%

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

    2. associate-/l* 82.9%

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

14. Simplified 82.9%

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

Alternative 6: 81.6% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\frac{-0.5 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5}{c}}}{a \cdot 2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (/ 1.0 (/ (+ (* -0.5 (/ b a)) (* (/ c b) 0.5)) c)) (* a 2.0)))

C

double code(double a, double b, double c) {
	return (1.0 / (((-0.5 * (b / a)) + ((c / b) * 0.5)) / c)) / (a * 2.0);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (1.0d0 / ((((-0.5d0) * (b / a)) + ((c / b) * 0.5d0)) / c)) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return (1.0 / (((-0.5 * (b / a)) + ((c / b) * 0.5)) / c)) / (a * 2.0);
}

Python

def code(a, b, c):
	return (1.0 / (((-0.5 * (b / a)) + ((c / b) * 0.5)) / c)) / (a * 2.0)

Julia

function code(a, b, c)
	return Float64(Float64(1.0 / Float64(Float64(Float64(-0.5 * Float64(b / a)) + Float64(Float64(c / b) * 0.5)) / c)) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (1.0 / (((-0.5 * (b / a)) + ((c / b) * 0.5)) / c)) / (a * 2.0);
end

Wolfram

code[a_, b_, c_] := N[(N[(1.0 / N[(N[(N[(-0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.9%

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

   1. *-commutative 55.9%

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

3. Simplified 55.9%

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

   1. add-cube-cbrt 55.9%

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

   2. pow3 55.8%

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

6. Applied egg-rr 55.8%

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

   1. flip-+ 55.7%

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

8. Applied egg-rr 57.2%

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

   1. associate--r- 99.2%

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

   2. associate-*r* 99.2%

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

   3. *-commutative 99.2%

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

   4. associate-*r* 99.2%

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

   5. associate-*r* 99.2%

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

   6. *-commutative 99.2%

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

   7. associate-*r* 99.2%

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

10. Simplified 99.2%

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

    1. clear-num 99.2%

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

    2. inv-pow 99.2%

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

    3. +-commutative 99.2%

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

    4. fma-define 99.2%

       \[\leadsto \frac{{\left(\frac{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}}{\color{blue}{\mathsf{fma}\left(a, c \cdot 4, {\left(-b\right)}^{2} - {b}^{2}\right)}}\right)}^{-1}}{a \cdot 2} \]

    5. neg-mul-1 99.2%

       \[\leadsto \frac{{\left(\frac{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}}{\mathsf{fma}\left(a, c \cdot 4, {\color{blue}{\left(-1 \cdot b\right)}}^{2} - {b}^{2}\right)}\right)}^{-1}}{a \cdot 2} \]

    6. unpow-prod-down 99.2%

       \[\leadsto \frac{{\left(\frac{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}}{\mathsf{fma}\left(a, c \cdot 4, \color{blue}{{-1}^{2} \cdot {b}^{2}} - {b}^{2}\right)}\right)}^{-1}}{a \cdot 2} \]

    7. metadata-eval 99.2%

       \[\leadsto \frac{{\left(\frac{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}}{\mathsf{fma}\left(a, c \cdot 4, \color{blue}{1} \cdot {b}^{2} - {b}^{2}\right)}\right)}^{-1}}{a \cdot 2} \]

    8. *-un-lft-identity 99.2%

       \[\leadsto \frac{{\left(\frac{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}}{\mathsf{fma}\left(a, c \cdot 4, \color{blue}{{b}^{2}} - {b}^{2}\right)}\right)}^{-1}}{a \cdot 2} \]

12. Applied egg-rr 99.2%

    \[\leadsto \frac{\color{blue}{{\left(\frac{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}}{\mathsf{fma}\left(a, c \cdot 4, {b}^{2} - {b}^{2}\right)}\right)}^{-1}}}{a \cdot 2} \]
13. Step-by-step derivation

    1. unpow-1 99.2%

       \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}}{\mathsf{fma}\left(a, c \cdot 4, {b}^{2} - {b}^{2}\right)}}}}{a \cdot 2} \]

    2. associate-*r* 99.2%

       \[\leadsto \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right) \cdot 4}}}{\mathsf{fma}\left(a, c \cdot 4, {b}^{2} - {b}^{2}\right)}}}{a \cdot 2} \]

    3. *-commutative 99.2%

       \[\leadsto \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}}{\mathsf{fma}\left(a, c \cdot 4, {b}^{2} - {b}^{2}\right)}}}{a \cdot 2} \]

    4. sub-neg 99.2%

       \[\leadsto \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{\color{blue}{{b}^{2} + \left(-4 \cdot \left(a \cdot c\right)\right)}}}{\mathsf{fma}\left(a, c \cdot 4, {b}^{2} - {b}^{2}\right)}}}{a \cdot 2} \]

    5. +-commutative 99.2%

       \[\leadsto \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right) + {b}^{2}}}}{\mathsf{fma}\left(a, c \cdot 4, {b}^{2} - {b}^{2}\right)}}}{a \cdot 2} \]

    6. distribute-lft-neg-in 99.2%

       \[\leadsto \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{\color{blue}{\left(-4\right) \cdot \left(a \cdot c\right)} + {b}^{2}}}{\mathsf{fma}\left(a, c \cdot 4, {b}^{2} - {b}^{2}\right)}}}{a \cdot 2} \]

    7. metadata-eval 99.2%

       \[\leadsto \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{\color{blue}{-4} \cdot \left(a \cdot c\right) + {b}^{2}}}{\mathsf{fma}\left(a, c \cdot 4, {b}^{2} - {b}^{2}\right)}}}{a \cdot 2} \]

    8. fma-define 99.2%

       \[\leadsto \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(-4, a \cdot c, {b}^{2}\right)}}}{\mathsf{fma}\left(a, c \cdot 4, {b}^{2} - {b}^{2}\right)}}}{a \cdot 2} \]

    9. fma-undefine 99.2%

       \[\leadsto \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, a \cdot c, {b}^{2}\right)}}{\color{blue}{a \cdot \left(c \cdot 4\right) + \left({b}^{2} - {b}^{2}\right)}}}}{a \cdot 2} \]

    10. +-inverses 99.2%

        \[\leadsto \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, a \cdot c, {b}^{2}\right)}}{a \cdot \left(c \cdot 4\right) + \color{blue}{0}}}}{a \cdot 2} \]

    11. +-rgt-identity 99.2%

        \[\leadsto \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, a \cdot c, {b}^{2}\right)}}{\color{blue}{a \cdot \left(c \cdot 4\right)}}}}{a \cdot 2} \]

    12. associate-*r* 99.2%

        \[\leadsto \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, a \cdot c, {b}^{2}\right)}}{\color{blue}{\left(a \cdot c\right) \cdot 4}}}}{a \cdot 2} \]

    13. *-commutative 99.2%

        \[\leadsto \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, a \cdot c, {b}^{2}\right)}}{\color{blue}{4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

14. Simplified 99.2%

    \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, a \cdot c, {b}^{2}\right)}}{4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

15. Taylor expanded in c around 0 82.8%

    \[\leadsto \frac{\frac{1}{\color{blue}{\frac{-0.5 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}}{c}}}}{a \cdot 2} \]

16. Final simplification 82.8%

    \[\leadsto \frac{\frac{1}{\frac{-0.5 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5}{c}}}{a \cdot 2} \]
17. Add Preprocessing

Alternative 7: 64.1% 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(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 55.9%

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

   1. *-commutative 55.9%

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

3. Simplified 56.0%

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

5. Taylor expanded in b around inf 64.3%

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

   1. associate-*r/ 64.3%

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

   2. mul-1-neg 64.3%

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

7. Simplified 64.3%

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

8. Final simplification 64.3%

   \[\leadsto \frac{c}{-b} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024165 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))