Quadratic roots, narrow range

Percentage Accurate: 55.8% → 99.5%

Time: 16.0s

Alternatives: 11

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: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(N[(2.0 * N[(N[(c * a), $MachinePrecision] / a), $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 53.1%

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

   1. *-commutative 53.1%

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

3. Simplified 53.1%

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

   1. add-cube-cbrt 53.1%

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

   2. pow3 53.1%

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

6. Applied egg-rr 53.1%

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

   1. flip-+ 53.1%

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

8. Applied egg-rr 54.7%

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

   1. associate--r- 99.2%

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

   2. associate-*r* 99.2%

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

   3. rem-cube-cbrt 98.1%

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

   4. *-commutative 98.1%

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

   5. rem-cube-cbrt 99.2%

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

   6. associate-*r* 99.2%

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

   7. rem-cube-cbrt 99.2%

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

   8. *-commutative 99.2%

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

   9. rem-cube-cbrt 99.2%

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

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. *-un-lft-identity 99.2%

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

    2. associate-/l/ 99.3%

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

    3. +-commutative 99.3%

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

    4. fma-define 99.3%

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

    5. neg-mul-1 99.3%

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

    6. unpow-prod-down 99.3%

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

    7. metadata-eval 99.3%

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

    8. *-un-lft-identity 99.3%

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

    9. *-commutative 99.3%

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

12. Applied egg-rr 99.3%

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

    1. *-un-lft-identity 99.3%

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

    2. *-commutative 99.3%

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

    3. associate-/r* 99.5%

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

    4. +-inverses 99.5%

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

    5. *-commutative 99.5%

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

14. Applied egg-rr 99.5%

    \[\leadsto 1 \cdot \color{blue}{\left(\frac{\frac{\mathsf{fma}\left(a, c \cdot 4, 0\right)}{a \cdot 2}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot 1\right)} \]
15. Step-by-step derivation

    1. *-rgt-identity 99.5%

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

    2. fma-define 99.5%

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

    3. +-rgt-identity 99.5%

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

    4. associate-*r* 99.5%

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

    5. *-commutative 99.5%

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

    6. *-commutative 99.5%

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

    7. times-frac 99.5%

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

    8. metadata-eval 99.5%

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

    9. *-commutative 99.5%

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

16. Simplified 99.5%

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

17. Final simplification 99.5%

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

Alternative 2: 89.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -6:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{-0.5 \cdot \frac{b}{c} + a \cdot \left(0.5 \cdot \frac{c \cdot a}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}{a}}}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a)) -6.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* 2.0 a))
   (/
    (/
     1.0
     (/
      (+
       (* -0.5 (/ b c))
       (* a (+ (* 0.5 (/ (* c a) (pow b 3.0))) (* 0.5 (/ 1.0 b)))))
      a))
    (* 2.0 a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)) <= -6.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (2.0 * a);
	} else {
		tmp = (1.0 / (((-0.5 * (b / c)) + (a * ((0.5 * ((c * a) / pow(b, 3.0))) + (0.5 * (1.0 / b))))) / a)) / (2.0 * a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a)) <= -6.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(-0.5 * Float64(b / c)) + Float64(a * Float64(Float64(0.5 * Float64(Float64(c * a) / (b ^ 3.0))) + Float64(0.5 * Float64(1.0 / b))))) / a)) / Float64(2.0 * a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -6.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(N[(-0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(0.5 * N[(N[(c * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -6

   1. Initial program 88.4%

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

      1. *-commutative 88.4%

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

      2. +-commutative 88.4%

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

      3. sqr-neg 88.4%

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

      4. unsub-neg 88.4%

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

      5. sqr-neg 88.4%

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

      6. fma-neg 88.5%

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

      7. distribute-lft-neg-in 88.5%

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

      8. *-commutative 88.5%

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

      9. *-commutative 88.5%

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

      10. distribute-rgt-neg-in 88.5%

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

      11. metadata-eval 88.5%

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

   3. Simplified 88.5%

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

   if -6 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 48.4%

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

      1. *-commutative 48.4%

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

   3. Simplified 48.4%

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

      1. add-cube-cbrt 48.5%

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

      2. pow3 48.5%

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

   6. Applied egg-rr 48.5%

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

      1. flip-+ 48.4%

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

   8. Applied egg-rr 50.0%

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

      1. associate--r- 99.2%

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

      2. associate-*r* 99.2%

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

      3. rem-cube-cbrt 98.2%

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

      4. *-commutative 98.2%

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

      5. rem-cube-cbrt 99.2%

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

      6. associate-*r* 99.2%

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

      7. rem-cube-cbrt 99.2%

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

      8. *-commutative 99.2%

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

      9. rem-cube-cbrt 99.2%

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

   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 99.1%

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

       2. inv-pow 99.1%

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

       3. +-commutative 99.1%

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

       4. fma-define 99.1%

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

       5. neg-mul-1 99.1%

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

       6. unpow-prod-down 99.1%

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

       7. metadata-eval 99.1%

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

       8. *-un-lft-identity 99.1%

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

   12. Applied egg-rr 99.1%

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

       1. unpow-1 99.1%

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

       2. associate-*r* 99.1%

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

       3. *-commutative 99.1%

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

       4. sub-neg 99.1%

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

       5. +-commutative 99.1%

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

       6. distribute-lft-neg-in 99.1%

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

       7. metadata-eval 99.1%

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

       8. fma-define 99.1%

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

       9. *-commutative 99.1%

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

       10. fma-undefine 99.1%

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

       11. +-inverses 99.1%

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

       12. +-rgt-identity 99.1%

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

   14. Simplified 99.1%

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

   15. Taylor expanded in a around 0 93.0%

       \[\leadsto \frac{\frac{1}{\color{blue}{\frac{-0.5 \cdot \frac{b}{c} + a \cdot \left(0.5 \cdot \frac{a \cdot c}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}{a}}}}{a \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -6:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{-0.5 \cdot \frac{b}{c} + a \cdot \left(0.5 \cdot \frac{c \cdot a}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}{a}}}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{if}\;t\_0 \leq -6:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{-0.5 \cdot \frac{b}{c} + a \cdot \left(0.5 \cdot \frac{c \cdot a}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}{a}}}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a))))
   (if (<= t_0 -6.0)
     t_0
     (/
      (/
       1.0
       (/
        (+
         (* -0.5 (/ b c))
         (* a (+ (* 0.5 (/ (* c a) (pow b 3.0))) (* 0.5 (/ 1.0 b)))))
        a))
      (* 2.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	double tmp;
	if (t_0 <= -6.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 / (((-0.5 * (b / c)) + (a * ((0.5 * ((c * a) / pow(b, 3.0))) + (0.5 * (1.0 / b))))) / a)) / (2.0 * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (2.0d0 * a)
    if (t_0 <= (-6.0d0)) then
        tmp = t_0
    else
        tmp = (1.0d0 / ((((-0.5d0) * (b / c)) + (a * ((0.5d0 * ((c * a) / (b ** 3.0d0))) + (0.5d0 * (1.0d0 / b))))) / a)) / (2.0d0 * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	double tmp;
	if (t_0 <= -6.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 / (((-0.5 * (b / c)) + (a * ((0.5 * ((c * a) / Math.pow(b, 3.0))) + (0.5 * (1.0 / b))))) / a)) / (2.0 * a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)
	tmp = 0
	if t_0 <= -6.0:
		tmp = t_0
	else:
		tmp = (1.0 / (((-0.5 * (b / c)) + (a * ((0.5 * ((c * a) / math.pow(b, 3.0))) + (0.5 * (1.0 / b))))) / a)) / (2.0 * a)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a))
	tmp = 0.0
	if (t_0 <= -6.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(-0.5 * Float64(b / c)) + Float64(a * Float64(Float64(0.5 * Float64(Float64(c * a) / (b ^ 3.0))) + Float64(0.5 * Float64(1.0 / b))))) / a)) / Float64(2.0 * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	tmp = 0.0;
	if (t_0 <= -6.0)
		tmp = t_0;
	else
		tmp = (1.0 / (((-0.5 * (b / c)) + (a * ((0.5 * ((c * a) / (b ^ 3.0))) + (0.5 * (1.0 / b))))) / a)) / (2.0 * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -6.0], t$95$0, N[(N[(1.0 / N[(N[(N[(-0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(0.5 * N[(N[(c * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -6

   1. Initial program 88.4%

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

   if -6 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 48.4%

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

      1. *-commutative 48.4%

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

   3. Simplified 48.4%

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

      1. add-cube-cbrt 48.5%

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

      2. pow3 48.5%

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

   6. Applied egg-rr 48.5%

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

      1. flip-+ 48.4%

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

   8. Applied egg-rr 50.0%

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

      1. associate--r- 99.2%

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

      2. associate-*r* 99.2%

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

      3. rem-cube-cbrt 98.2%

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

      4. *-commutative 98.2%

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

      5. rem-cube-cbrt 99.2%

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

      6. associate-*r* 99.2%

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

      7. rem-cube-cbrt 99.2%

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

      8. *-commutative 99.2%

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

      9. rem-cube-cbrt 99.2%

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

   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 99.1%

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

       2. inv-pow 99.1%

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

       3. +-commutative 99.1%

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

       4. fma-define 99.1%

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

       5. neg-mul-1 99.1%

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

       6. unpow-prod-down 99.1%

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

       7. metadata-eval 99.1%

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

       8. *-un-lft-identity 99.1%

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

   12. Applied egg-rr 99.1%

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

       1. unpow-1 99.1%

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

       2. associate-*r* 99.1%

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

       3. *-commutative 99.1%

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

       4. sub-neg 99.1%

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

       5. +-commutative 99.1%

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

       6. distribute-lft-neg-in 99.1%

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

       7. metadata-eval 99.1%

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

       8. fma-define 99.1%

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

       9. *-commutative 99.1%

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

       10. fma-undefine 99.1%

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

       11. +-inverses 99.1%

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

       12. +-rgt-identity 99.1%

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

   14. Simplified 99.1%

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

   15. Taylor expanded in a around 0 93.0%

       \[\leadsto \frac{\frac{1}{\color{blue}{\frac{-0.5 \cdot \frac{b}{c} + a \cdot \left(0.5 \cdot \frac{a \cdot c}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}{a}}}}{a \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -6:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{-0.5 \cdot \frac{b}{c} + a \cdot \left(0.5 \cdot \frac{c \cdot a}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}{a}}}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{if}\;t\_0 \leq -6:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\frac{c \cdot \left(0.5 \cdot \frac{-1}{b} - 0.5 \cdot \frac{c \cdot a}{{b}^{3}}\right) - -0.5 \cdot \frac{b}{a}}{c}}}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a))))
   (if (<= t_0 -6.0)
     t_0
     (/
      (/
       -1.0
       (/
        (-
         (* c (- (* 0.5 (/ -1.0 b)) (* 0.5 (/ (* c a) (pow b 3.0)))))
         (* -0.5 (/ b a)))
        c))
      (* 2.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	double tmp;
	if (t_0 <= -6.0) {
		tmp = t_0;
	} else {
		tmp = (-1.0 / (((c * ((0.5 * (-1.0 / b)) - (0.5 * ((c * a) / pow(b, 3.0))))) - (-0.5 * (b / a))) / c)) / (2.0 * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (2.0d0 * a)
    if (t_0 <= (-6.0d0)) then
        tmp = t_0
    else
        tmp = ((-1.0d0) / (((c * ((0.5d0 * ((-1.0d0) / b)) - (0.5d0 * ((c * a) / (b ** 3.0d0))))) - ((-0.5d0) * (b / a))) / c)) / (2.0d0 * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	double tmp;
	if (t_0 <= -6.0) {
		tmp = t_0;
	} else {
		tmp = (-1.0 / (((c * ((0.5 * (-1.0 / b)) - (0.5 * ((c * a) / Math.pow(b, 3.0))))) - (-0.5 * (b / a))) / c)) / (2.0 * a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)
	tmp = 0
	if t_0 <= -6.0:
		tmp = t_0
	else:
		tmp = (-1.0 / (((c * ((0.5 * (-1.0 / b)) - (0.5 * ((c * a) / math.pow(b, 3.0))))) - (-0.5 * (b / a))) / c)) / (2.0 * a)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a))
	tmp = 0.0
	if (t_0 <= -6.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(-1.0 / Float64(Float64(Float64(c * Float64(Float64(0.5 * Float64(-1.0 / b)) - Float64(0.5 * Float64(Float64(c * a) / (b ^ 3.0))))) - Float64(-0.5 * Float64(b / a))) / c)) / Float64(2.0 * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	tmp = 0.0;
	if (t_0 <= -6.0)
		tmp = t_0;
	else
		tmp = (-1.0 / (((c * ((0.5 * (-1.0 / b)) - (0.5 * ((c * a) / (b ^ 3.0))))) - (-0.5 * (b / a))) / c)) / (2.0 * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -6.0], t$95$0, N[(N[(-1.0 / N[(N[(N[(c * N[(N[(0.5 * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(N[(c * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -6

   1. Initial program 88.4%

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

   if -6 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 48.4%

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

      1. *-commutative 48.4%

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

   3. Simplified 48.4%

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

      1. add-cube-cbrt 48.5%

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

      2. pow3 48.5%

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

   6. Applied egg-rr 48.5%

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

      1. flip-+ 48.4%

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

   8. Applied egg-rr 50.0%

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

      1. associate--r- 99.2%

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

      2. associate-*r* 99.2%

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

      3. rem-cube-cbrt 98.2%

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

      4. *-commutative 98.2%

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

      5. rem-cube-cbrt 99.2%

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

      6. associate-*r* 99.2%

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

      7. rem-cube-cbrt 99.2%

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

      8. *-commutative 99.2%

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

      9. rem-cube-cbrt 99.2%

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

   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 99.1%

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

       2. inv-pow 99.1%

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

       3. +-commutative 99.1%

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

       4. fma-define 99.1%

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

       5. neg-mul-1 99.1%

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

       6. unpow-prod-down 99.1%

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

       7. metadata-eval 99.1%

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

       8. *-un-lft-identity 99.1%

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

   12. Applied egg-rr 99.1%

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

       1. unpow-1 99.1%

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

       2. associate-*r* 99.1%

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

       3. *-commutative 99.1%

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

       4. sub-neg 99.1%

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

       5. +-commutative 99.1%

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

       6. distribute-lft-neg-in 99.1%

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

       7. metadata-eval 99.1%

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

       8. fma-define 99.1%

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

       9. *-commutative 99.1%

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

       10. fma-undefine 99.1%

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

       11. +-inverses 99.1%

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

       12. +-rgt-identity 99.1%

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

   14. Simplified 99.1%

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

   15. Taylor expanded in c around 0 93.0%

       \[\leadsto \frac{\frac{1}{\color{blue}{\frac{-0.5 \cdot \frac{b}{a} + c \cdot \left(0.5 \cdot \frac{a \cdot c}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}{c}}}}{a \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -6:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\frac{c \cdot \left(0.5 \cdot \frac{-1}{b} - 0.5 \cdot \frac{c \cdot a}{{b}^{3}}\right) - -0.5 \cdot \frac{b}{a}}{c}}}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{if}\;t\_0 \leq -0.7:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{2 \cdot \left(\frac{c \cdot a}{b} - b\right)}{a \cdot \left(c \cdot 4\right)}}}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a))))
   (if (<= t_0 -0.7)
     t_0
     (/ (/ 1.0 (/ (* 2.0 (- (/ (* c a) b) b)) (* a (* c 4.0)))) (* 2.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	double tmp;
	if (t_0 <= -0.7) {
		tmp = t_0;
	} else {
		tmp = (1.0 / ((2.0 * (((c * a) / b) - b)) / (a * (c * 4.0)))) / (2.0 * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (2.0d0 * a)
    if (t_0 <= (-0.7d0)) then
        tmp = t_0
    else
        tmp = (1.0d0 / ((2.0d0 * (((c * a) / b) - b)) / (a * (c * 4.0d0)))) / (2.0d0 * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	double tmp;
	if (t_0 <= -0.7) {
		tmp = t_0;
	} else {
		tmp = (1.0 / ((2.0 * (((c * a) / b) - b)) / (a * (c * 4.0)))) / (2.0 * a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)
	tmp = 0
	if t_0 <= -0.7:
		tmp = t_0
	else:
		tmp = (1.0 / ((2.0 * (((c * a) / b) - b)) / (a * (c * 4.0)))) / (2.0 * a)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a))
	tmp = 0.0
	if (t_0 <= -0.7)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(2.0 * Float64(Float64(Float64(c * a) / b) - b)) / Float64(a * Float64(c * 4.0)))) / Float64(2.0 * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	tmp = 0.0;
	if (t_0 <= -0.7)
		tmp = t_0;
	else
		tmp = (1.0 / ((2.0 * (((c * a) / b) - b)) / (a * (c * 4.0)))) / (2.0 * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.7], t$95$0, N[(N[(1.0 / N[(N[(2.0 * N[(N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / N[(a * N[(c * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.69999999999999996

   1. Initial program 85.8%

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

   if -0.69999999999999996 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 47.1%

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

      1. *-commutative 47.1%

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

   3. Simplified 47.1%

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

      1. add-cube-cbrt 47.1%

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

      2. pow3 47.1%

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

   6. Applied egg-rr 47.1%

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

      1. flip-+ 47.0%

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

   8. Applied egg-rr 48.6%

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

      1. associate--r- 99.2%

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

      2. associate-*r* 99.2%

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

      3. rem-cube-cbrt 98.2%

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

      4. *-commutative 98.2%

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

      5. rem-cube-cbrt 99.2%

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

      6. associate-*r* 99.2%

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

      7. rem-cube-cbrt 99.2%

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

      8. *-commutative 99.2%

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

      9. rem-cube-cbrt 99.2%

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

   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 99.1%

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

       2. inv-pow 99.1%

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

       3. +-commutative 99.1%

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

       4. fma-define 99.1%

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

       5. neg-mul-1 99.1%

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

       6. unpow-prod-down 99.1%

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

       7. metadata-eval 99.1%

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

       8. *-un-lft-identity 99.1%

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

   12. Applied egg-rr 99.1%

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

       1. unpow-1 99.1%

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

       2. associate-*r* 99.1%

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

       3. *-commutative 99.1%

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

       4. sub-neg 99.1%

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

       5. +-commutative 99.1%

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

       6. distribute-lft-neg-in 99.1%

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

       7. metadata-eval 99.1%

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

       8. fma-define 99.1%

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

       9. *-commutative 99.1%

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

       10. fma-undefine 99.1%

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

       11. +-inverses 99.1%

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

       12. +-rgt-identity 99.1%

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

   14. Simplified 99.1%

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

   15. Taylor expanded in c around 0 89.5%

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

       1. distribute-lft-out-- 89.5%

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

       2. *-commutative 89.5%

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

   17. Simplified 89.5%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.7:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{2 \cdot \left(\frac{c \cdot a}{b} - b\right)}{a \cdot \left(c \cdot 4\right)}}}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.1%

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

   1. *-commutative 53.1%

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

3. Simplified 53.1%

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

   1. add-cube-cbrt 53.1%

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

   2. pow3 53.1%

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

6. Applied egg-rr 53.1%

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

   1. flip-+ 53.1%

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

8. Applied egg-rr 54.7%

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

   1. associate--r- 99.2%

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

   2. associate-*r* 99.2%

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

   3. rem-cube-cbrt 98.1%

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

   4. *-commutative 98.1%

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

   5. rem-cube-cbrt 99.2%

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

   6. associate-*r* 99.2%

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

   7. rem-cube-cbrt 99.2%

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

   8. *-commutative 99.2%

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

   9. rem-cube-cbrt 99.2%

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

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. *-un-lft-identity 99.2%

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

    2. associate-/l/ 99.3%

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

    3. +-commutative 99.3%

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

    4. fma-define 99.3%

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

    5. neg-mul-1 99.3%

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

    6. unpow-prod-down 99.3%

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

    7. metadata-eval 99.3%

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

    8. *-un-lft-identity 99.3%

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

    9. *-commutative 99.3%

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

12. Applied egg-rr 99.3%

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

    1. div-inv 99.2%

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

    2. +-inverses 99.2%

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

    3. associate-*l* 99.2%

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

14. Applied egg-rr 99.2%

    \[\leadsto 1 \cdot \color{blue}{\left(\mathsf{fma}\left(a, c \cdot 4, 0\right) \cdot \frac{1}{2 \cdot \left(a \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}\right)\right)}\right)} \]
15. Step-by-step derivation

    1. associate-*r/ 99.3%

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

    2. *-rgt-identity 99.3%

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

    3. fma-define 99.3%

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

    4. +-rgt-identity 99.3%

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

    5. associate-*r* 99.3%

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

    6. *-commutative 99.3%

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

    7. times-frac 99.3%

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

    8. metadata-eval 99.3%

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

    9. *-commutative 99.3%

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

16. Simplified 99.3%

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

17. Final simplification 99.3%

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

Alternative 7: 81.7% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\frac{2 \cdot \left(\frac{c \cdot a}{b} - b\right)}{a \cdot \left(c \cdot 4\right)}}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (/ 1.0 (/ (* 2.0 (- (/ (* c a) b) b)) (* a (* c 4.0)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (1.0 / ((2.0 * (((c * a) / b) - b)) / (a * (c * 4.0)))) / (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 = (1.0d0 / ((2.0d0 * (((c * a) / b) - b)) / (a * (c * 4.0d0)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (1.0 / ((2.0 * (((c * a) / b) - b)) / (a * (c * 4.0)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (1.0 / ((2.0 * (((c * a) / b) - b)) / (a * (c * 4.0)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(1.0 / Float64(Float64(2.0 * Float64(Float64(Float64(c * a) / b) - b)) / Float64(a * Float64(c * 4.0)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (1.0 / ((2.0 * (((c * a) / b) - b)) / (a * (c * 4.0)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[(1.0 / N[(N[(2.0 * N[(N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / N[(a * N[(c * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.1%

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

   1. *-commutative 53.1%

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

3. Simplified 53.1%

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

   1. add-cube-cbrt 53.1%

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

   2. pow3 53.1%

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

6. Applied egg-rr 53.1%

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

   1. flip-+ 53.1%

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

8. Applied egg-rr 54.7%

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

   1. associate--r- 99.2%

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

   2. associate-*r* 99.2%

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

   3. rem-cube-cbrt 98.1%

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

   4. *-commutative 98.1%

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

   5. rem-cube-cbrt 99.2%

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

   6. associate-*r* 99.2%

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

   7. rem-cube-cbrt 99.2%

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

   8. *-commutative 99.2%

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

   9. rem-cube-cbrt 99.2%

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

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 99.1%

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

    2. inv-pow 99.1%

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

    3. +-commutative 99.1%

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

    4. fma-define 99.1%

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

    5. neg-mul-1 99.1%

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

    6. unpow-prod-down 99.1%

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

    7. metadata-eval 99.1%

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

    8. *-un-lft-identity 99.1%

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

12. Applied egg-rr 99.1%

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

    1. unpow-1 99.1%

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

    2. associate-*r* 99.1%

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

    3. *-commutative 99.1%

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

    4. sub-neg 99.1%

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

    5. +-commutative 99.1%

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

    6. distribute-lft-neg-in 99.1%

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

    7. metadata-eval 99.1%

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

    8. fma-define 99.1%

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

    9. *-commutative 99.1%

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

    10. fma-undefine 99.1%

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

    11. +-inverses 99.1%

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

    12. +-rgt-identity 99.1%

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

14. Simplified 99.1%

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

15. Taylor expanded in c around 0 83.8%

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

    1. distribute-lft-out-- 83.8%

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

    2. *-commutative 83.8%

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

17. Simplified 83.8%

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

18. Final simplification 83.8%

    \[\leadsto \frac{\frac{1}{\frac{2 \cdot \left(\frac{c \cdot a}{b} - b\right)}{a \cdot \left(c \cdot 4\right)}}}{2 \cdot a} \]
19. Add Preprocessing

Alternative 8: 81.7% accurate, 6.1× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (1.0 / (((-0.5 * (b / a)) + (0.5 * (c / b))) / 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 = (1.0d0 / ((((-0.5d0) * (b / a)) + (0.5d0 * (c / b))) / c)) / (2.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.1%

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

   1. *-commutative 53.1%

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

3. Simplified 53.1%

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

   1. add-cube-cbrt 53.1%

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

   2. pow3 53.1%

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

6. Applied egg-rr 53.1%

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

   1. flip-+ 53.1%

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

8. Applied egg-rr 54.7%

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

   1. associate--r- 99.2%

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

   2. associate-*r* 99.2%

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

   3. rem-cube-cbrt 98.1%

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

   4. *-commutative 98.1%

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

   5. rem-cube-cbrt 99.2%

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

   6. associate-*r* 99.2%

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

   7. rem-cube-cbrt 99.2%

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

   8. *-commutative 99.2%

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

   9. rem-cube-cbrt 99.2%

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

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 99.1%

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

    2. inv-pow 99.1%

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

    3. +-commutative 99.1%

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

    4. fma-define 99.1%

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

    5. neg-mul-1 99.1%

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

    6. unpow-prod-down 99.1%

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

    7. metadata-eval 99.1%

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

    8. *-un-lft-identity 99.1%

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

12. Applied egg-rr 99.1%

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

    1. unpow-1 99.1%

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

    2. associate-*r* 99.1%

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

    3. *-commutative 99.1%

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

    4. sub-neg 99.1%

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

    5. +-commutative 99.1%

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

    6. distribute-lft-neg-in 99.1%

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

    7. metadata-eval 99.1%

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

    8. fma-define 99.1%

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

    9. *-commutative 99.1%

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

    10. fma-undefine 99.1%

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

    11. +-inverses 99.1%

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

    12. +-rgt-identity 99.1%

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

14. Simplified 99.1%

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

15. Taylor expanded in c around 0 83.8%

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

16. Final simplification 83.8%

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

Alternative 9: 81.1% accurate, 7.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (a * ((c / -a) - ((c / b) * (c / b)))) / 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 / -a) - ((c / b) * (c / b)))) / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.1%

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

   1. *-commutative 53.1%

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

3. Simplified 53.1%

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

5. Taylor expanded in b around inf 83.5%

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

   1. mul-1-neg 83.5%

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

   2. unsub-neg 83.5%

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

   3. mul-1-neg 83.5%

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

7. Simplified 83.5%

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

8. Taylor expanded in a around inf 83.4%

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

   1. mul-1-neg 83.4%

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

   2. distribute-neg-frac2 83.4%

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

   3. unpow2 83.4%

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

   4. unpow2 83.4%

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

   5. times-frac 83.4%

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

   6. sqr-neg 83.4%

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

   7. distribute-frac-neg 83.4%

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

   8. distribute-frac-neg 83.4%

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

   9. unpow2 83.4%

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

   10. distribute-frac-neg 83.4%

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

   11. distribute-neg-frac2 83.4%

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

10. Simplified 83.4%

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

    1. unpow2 83.4%

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

12. Applied egg-rr 83.4%

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

13. Final simplification 83.4%

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

Alternative 10: 81.0% accurate, 7.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return a * (((c / -a) - ((c / b) * (c / b))) / 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 / -a) - ((c / b) * (c / b))) / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.1%

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

   1. *-commutative 53.1%

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

3. Simplified 53.1%

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

5. Taylor expanded in b around inf 83.5%

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

   1. mul-1-neg 83.5%

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

   2. unsub-neg 83.5%

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

   3. mul-1-neg 83.5%

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

7. Simplified 83.5%

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

8. Taylor expanded in a around inf 83.4%

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

   1. mul-1-neg 83.4%

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

   2. distribute-neg-frac2 83.4%

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

   3. unpow2 83.4%

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

   4. unpow2 83.4%

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

   5. times-frac 83.4%

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

   6. sqr-neg 83.4%

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

   7. distribute-frac-neg 83.4%

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

   8. distribute-frac-neg 83.4%

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

   9. unpow2 83.4%

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

   10. distribute-frac-neg 83.4%

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

   11. distribute-neg-frac2 83.4%

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

10. Simplified 83.4%

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

    1. associate-/l* 83.3%

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

12. Applied egg-rr 83.3%

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

    1. distribute-frac-neg2 83.3%

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

    2. distribute-frac-neg 83.3%

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

14. Simplified 83.3%

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

    1. unpow2 83.3%

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

16. Applied egg-rr 83.3%

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

17. Final simplification 83.3%

    \[\leadsto a \cdot \frac{\frac{c}{-a} - \frac{c}{b} \cdot \frac{c}{b}}{b} \]
18. Add Preprocessing

Alternative 11: 64.0% 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 53.1%

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

   1. *-commutative 53.1%

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

3. Simplified 53.1%

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

5. Taylor expanded in b around inf 66.4%

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

   1. associate-*r/ 66.4%

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

   2. mul-1-neg 66.4%

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

7. Simplified 66.4%

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

8. Final simplification 66.4%

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

Reproduce

herbie shell --seed 2024121 
(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)))