Quadratic roots, narrow range

Percentage Accurate: 55.2% → 99.5%

Time: 14.3s

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: 55.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

   1. *-commutative 55.8%

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

3. Simplified 55.8%

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

   1. neg-sub0 55.8%

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

   2. flip3-- 55.1%

      \[\leadsto \frac{\color{blue}{\frac{{0}^{3} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. metadata-eval 55.1%

      \[\leadsto \frac{\frac{\color{blue}{0} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. metadata-eval 55.1%

      \[\leadsto \frac{\frac{0 - {b}^{3}}{\color{blue}{0} + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. pow2 55.1%

      \[\leadsto \frac{\frac{0 - {b}^{3}}{0 + \left(\color{blue}{{b}^{2}} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 55.1%

   \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{3}}{0 + \left({b}^{2} + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. sub0-neg 55.1%

      \[\leadsto \frac{\frac{\color{blue}{-{b}^{3}}}{0 + \left({b}^{2} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   2. +-lft-identity 55.1%

      \[\leadsto \frac{\frac{-{b}^{3}}{\color{blue}{{b}^{2} + 0 \cdot b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. mul0-lft 55.1%

      \[\leadsto \frac{\frac{-{b}^{3}}{{b}^{2} + \color{blue}{0}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. +-rgt-identity 55.1%

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

8. Simplified 55.1%

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

   1. flip-+ 55.4%

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

10. Applied egg-rr 57.8%

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

    1. associate--r- 99.4%

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

    2. unpow2 99.4%

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

    3. unpow2 99.4%

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

    4. difference-of-squares 99.4%

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

    5. +-commutative 99.4%

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

    6. neg-mul-1 99.4%

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

    7. distribute-rgt1-in 99.4%

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

    8. metadata-eval 99.4%

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

    9. mul0-lft 99.4%

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

    10. *-commutative 99.4%

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

    11. *-commutative 99.4%

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

12. Simplified 99.4%

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

    1. div-inv 99.2%

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

    2. +-commutative 99.2%

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

    3. mul0-lft 99.2%

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

    4. div0 99.2%

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

    5. fma-define 99.2%

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

    6. div0 99.2%

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

14. Applied egg-rr 99.2%

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

    1. *-commutative 99.2%

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

    2. times-frac 99.3%

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

    3. *-lft-identity 99.3%

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

    4. associate-/r* 99.6%

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

    5. *-lft-identity 99.6%

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

    6. *-commutative 99.6%

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

    7. times-frac 99.6%

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

    8. metadata-eval 99.6%

       \[\leadsto \frac{\color{blue}{0.5} \cdot \frac{\mathsf{fma}\left(c, a \cdot 4, 0\right)}{a}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}} \]

    9. fma-undefine 99.6%

       \[\leadsto \frac{0.5 \cdot \frac{\color{blue}{c \cdot \left(a \cdot 4\right) + 0}}{a}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}} \]

    10. +-rgt-identity 99.6%

        \[\leadsto \frac{0.5 \cdot \frac{\color{blue}{c \cdot \left(a \cdot 4\right)}}{a}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}} \]

    11. associate-*r* 99.6%

        \[\leadsto \frac{0.5 \cdot \frac{\color{blue}{\left(c \cdot a\right) \cdot 4}}{a}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}} \]

    12. *-commutative 99.6%

        \[\leadsto \frac{0.5 \cdot \frac{\color{blue}{4 \cdot \left(c \cdot a\right)}}{a}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}} \]

    13. associate-*r* 99.6%

        \[\leadsto \frac{0.5 \cdot \frac{4 \cdot \left(c \cdot a\right)}{a}}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(c \cdot a\right) \cdot 4}}} \]

    14. *-commutative 99.6%

        \[\leadsto \frac{0.5 \cdot \frac{4 \cdot \left(c \cdot a\right)}{a}}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right)} \cdot 4}} \]

    15. *-commutative 99.6%

        \[\leadsto \frac{0.5 \cdot \frac{4 \cdot \left(c \cdot a\right)}{a}}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}} \]

    16. cancel-sign-sub-inv 99.6%

        \[\leadsto \frac{0.5 \cdot \frac{4 \cdot \left(c \cdot a\right)}{a}}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2} + \left(-4\right) \cdot \left(a \cdot c\right)}}} \]

    17. metadata-eval 99.6%

        \[\leadsto \frac{0.5 \cdot \frac{4 \cdot \left(c \cdot a\right)}{a}}{\left(-b\right) - \sqrt{{b}^{2} + \color{blue}{-4} \cdot \left(a \cdot c\right)}} \]

16. Simplified 99.6%

    \[\leadsto \color{blue}{\frac{0.5 \cdot \frac{4 \cdot \left(c \cdot a\right)}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}}} \]

17. Taylor expanded in c around 0 99.6%

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

Alternative 2: 90.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -2.6:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{-0.5 \cdot \frac{b}{c} + a \cdot \left(0.5 \cdot \frac{c \cdot a}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}{a}}}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a)) -2.6)
   (/ (- (sqrt (fma b b (* c (* -4.0 a)))) 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)) <= -2.6) {
		tmp = (sqrt(fma(b, b, (c * (-4.0 * a)))) - 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)) <= -2.6)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(-4.0 * a)))) - 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], -2.6], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(-4.0 * a), $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)) < -2.60000000000000009

   1. Initial program 86.3%

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

      1. *-commutative 86.3%

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

      2. +-commutative 86.3%

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

      3. sqr-neg 86.3%

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

      4. unsub-neg 86.3%

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

      5. sqr-neg 86.3%

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

      6. fma-neg 86.5%

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

      7. distribute-lft-neg-in 86.5%

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

      8. *-commutative 86.5%

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

      9. *-commutative 86.5%

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

      10. distribute-rgt-neg-in 86.5%

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

      11. metadata-eval 86.5%

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

   3. Simplified 86.5%

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

   if -2.60000000000000009 < (/.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 52.4%

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

      1. *-commutative 52.4%

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

   3. Simplified 52.4%

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

      1. neg-sub0 52.4%

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

      2. flip3-- 51.7%

         \[\leadsto \frac{\color{blue}{\frac{{0}^{3} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. metadata-eval 51.7%

         \[\leadsto \frac{\frac{\color{blue}{0} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. metadata-eval 51.7%

         \[\leadsto \frac{\frac{0 - {b}^{3}}{\color{blue}{0} + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow2 51.7%

         \[\leadsto \frac{\frac{0 - {b}^{3}}{0 + \left(\color{blue}{{b}^{2}} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 51.7%

      \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{3}}{0 + \left({b}^{2} + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. sub0-neg 51.7%

         \[\leadsto \frac{\frac{\color{blue}{-{b}^{3}}}{0 + \left({b}^{2} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. +-lft-identity 51.7%

         \[\leadsto \frac{\frac{-{b}^{3}}{\color{blue}{{b}^{2} + 0 \cdot b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. mul0-lft 51.7%

         \[\leadsto \frac{\frac{-{b}^{3}}{{b}^{2} + \color{blue}{0}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. +-rgt-identity 51.7%

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

   8. Simplified 51.7%

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

      1. flip-+ 52.0%

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

   10. Applied egg-rr 54.3%

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

       1. associate--r- 99.4%

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

       2. unpow2 99.4%

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

       3. unpow2 99.4%

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

       4. difference-of-squares 99.4%

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

       5. +-commutative 99.4%

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

       6. neg-mul-1 99.4%

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

       7. distribute-rgt1-in 99.4%

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

       8. metadata-eval 99.4%

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

       9. mul0-lft 99.4%

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

       10. *-commutative 99.4%

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

       11. *-commutative 99.4%

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

   12. Simplified 99.4%

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

       1. clear-num 99.3%

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

       2. inv-pow 99.3%

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

       3. +-commutative 99.3%

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

       4. mul0-lft 99.3%

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

       5. div0 99.3%

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

       6. fma-define 99.3%

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

       7. div0 99.3%

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

   14. Applied egg-rr 99.3%

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

       1. unpow-1 99.3%

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

       2. associate-*r* 99.3%

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

       3. *-commutative 99.3%

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

       4. *-commutative 99.3%

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

       5. cancel-sign-sub-inv 99.3%

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

       6. metadata-eval 99.3%

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

       7. +-commutative 99.3%

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

       8. fma-define 99.3%

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

       9. *-commutative 99.3%

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

       10. fma-undefine 99.3%

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

       11. +-rgt-identity 99.3%

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

       12. associate-*r* 99.3%

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

       13. *-commutative 99.3%

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

   16. Simplified 99.3%

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

   17. Taylor expanded in a around 0 91.5%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -2.6:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{-0.5 \cdot \frac{b}{c} + a \cdot \left(0.5 \cdot \frac{c \cdot a}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}{a}}}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{if}\;t\_0 \leq -2.6:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{-0.5 \cdot \frac{b}{c} + a \cdot \left(0.5 \cdot \frac{c \cdot a}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}{a}}}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a))))
   (if (<= t_0 -2.6)
     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 <= -2.6) {
		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 <= (-2.6d0)) 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 <= -2.6) {
		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 <= -2.6:
		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 <= -2.6)
		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 <= -2.6)
		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, -2.6], 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)) < -2.60000000000000009

   1. Initial program 86.3%

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

   if -2.60000000000000009 < (/.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 52.4%

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

      1. *-commutative 52.4%

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

   3. Simplified 52.4%

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

      1. neg-sub0 52.4%

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

      2. flip3-- 51.7%

         \[\leadsto \frac{\color{blue}{\frac{{0}^{3} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. metadata-eval 51.7%

         \[\leadsto \frac{\frac{\color{blue}{0} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. metadata-eval 51.7%

         \[\leadsto \frac{\frac{0 - {b}^{3}}{\color{blue}{0} + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow2 51.7%

         \[\leadsto \frac{\frac{0 - {b}^{3}}{0 + \left(\color{blue}{{b}^{2}} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 51.7%

      \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{3}}{0 + \left({b}^{2} + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. sub0-neg 51.7%

         \[\leadsto \frac{\frac{\color{blue}{-{b}^{3}}}{0 + \left({b}^{2} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. +-lft-identity 51.7%

         \[\leadsto \frac{\frac{-{b}^{3}}{\color{blue}{{b}^{2} + 0 \cdot b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. mul0-lft 51.7%

         \[\leadsto \frac{\frac{-{b}^{3}}{{b}^{2} + \color{blue}{0}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. +-rgt-identity 51.7%

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

   8. Simplified 51.7%

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

      1. flip-+ 52.0%

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

   10. Applied egg-rr 54.3%

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

       1. associate--r- 99.4%

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

       2. unpow2 99.4%

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

       3. unpow2 99.4%

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

       4. difference-of-squares 99.4%

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

       5. +-commutative 99.4%

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

       6. neg-mul-1 99.4%

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

       7. distribute-rgt1-in 99.4%

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

       8. metadata-eval 99.4%

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

       9. mul0-lft 99.4%

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

       10. *-commutative 99.4%

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

       11. *-commutative 99.4%

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

   12. Simplified 99.4%

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

       1. clear-num 99.3%

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

       2. inv-pow 99.3%

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

       3. +-commutative 99.3%

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

       4. mul0-lft 99.3%

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

       5. div0 99.3%

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

       6. fma-define 99.3%

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

       7. div0 99.3%

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

   14. Applied egg-rr 99.3%

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

       1. unpow-1 99.3%

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

       2. associate-*r* 99.3%

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

       3. *-commutative 99.3%

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

       4. *-commutative 99.3%

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

       5. cancel-sign-sub-inv 99.3%

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

       6. metadata-eval 99.3%

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

       7. +-commutative 99.3%

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

       8. fma-define 99.3%

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

       9. *-commutative 99.3%

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

       10. fma-undefine 99.3%

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

       11. +-rgt-identity 99.3%

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

       12. associate-*r* 99.3%

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

       13. *-commutative 99.3%

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

   16. Simplified 99.3%

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

   17. Taylor expanded in a around 0 91.5%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -2.6:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{-0.5 \cdot \frac{b}{c} + a \cdot \left(0.5 \cdot \frac{c \cdot a}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}{a}}}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{if}\;t\_0 \leq -2.6:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{-0.5 \cdot \frac{b}{a} + c \cdot \left(0.5 \cdot \frac{c \cdot a}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}{c}}}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a))))
   (if (<= t_0 -2.6)
     t_0
     (/
      (/
       1.0
       (/
        (+
         (* -0.5 (/ b a))
         (* c (+ (* 0.5 (/ (* c a) (pow b 3.0))) (* 0.5 (/ 1.0 b)))))
        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 <= -2.6) {
		tmp = t_0;
	} else {
		tmp = (1.0 / (((-0.5 * (b / a)) + (c * ((0.5 * ((c * a) / pow(b, 3.0))) + (0.5 * (1.0 / b))))) / 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 <= (-2.6d0)) then
        tmp = t_0
    else
        tmp = (1.0d0 / ((((-0.5d0) * (b / a)) + (c * ((0.5d0 * ((c * a) / (b ** 3.0d0))) + (0.5d0 * (1.0d0 / b))))) / 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 <= -2.6) {
		tmp = t_0;
	} else {
		tmp = (1.0 / (((-0.5 * (b / a)) + (c * ((0.5 * ((c * a) / Math.pow(b, 3.0))) + (0.5 * (1.0 / b))))) / 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 <= -2.6:
		tmp = t_0
	else:
		tmp = (1.0 / (((-0.5 * (b / a)) + (c * ((0.5 * ((c * a) / math.pow(b, 3.0))) + (0.5 * (1.0 / b))))) / 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 <= -2.6)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(-0.5 * Float64(b / a)) + Float64(c * Float64(Float64(0.5 * Float64(Float64(c * a) / (b ^ 3.0))) + Float64(0.5 * Float64(1.0 / b))))) / 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 <= -2.6)
		tmp = t_0;
	else
		tmp = (1.0 / (((-0.5 * (b / a)) + (c * ((0.5 * ((c * a) / (b ^ 3.0))) + (0.5 * (1.0 / b))))) / 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, -2.6], t$95$0, N[(N[(1.0 / N[(N[(N[(-0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(c * 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] / 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)) < -2.60000000000000009

   1. Initial program 86.3%

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

   if -2.60000000000000009 < (/.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 52.4%

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

      1. *-commutative 52.4%

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

   3. Simplified 52.4%

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

      1. neg-sub0 52.4%

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

      2. flip3-- 51.7%

         \[\leadsto \frac{\color{blue}{\frac{{0}^{3} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. metadata-eval 51.7%

         \[\leadsto \frac{\frac{\color{blue}{0} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. metadata-eval 51.7%

         \[\leadsto \frac{\frac{0 - {b}^{3}}{\color{blue}{0} + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow2 51.7%

         \[\leadsto \frac{\frac{0 - {b}^{3}}{0 + \left(\color{blue}{{b}^{2}} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 51.7%

      \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{3}}{0 + \left({b}^{2} + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. sub0-neg 51.7%

         \[\leadsto \frac{\frac{\color{blue}{-{b}^{3}}}{0 + \left({b}^{2} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. +-lft-identity 51.7%

         \[\leadsto \frac{\frac{-{b}^{3}}{\color{blue}{{b}^{2} + 0 \cdot b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. mul0-lft 51.7%

         \[\leadsto \frac{\frac{-{b}^{3}}{{b}^{2} + \color{blue}{0}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. +-rgt-identity 51.7%

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

   8. Simplified 51.7%

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

      1. flip-+ 52.0%

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

   10. Applied egg-rr 54.3%

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

       1. associate--r- 99.4%

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

       2. unpow2 99.4%

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

       3. unpow2 99.4%

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

       4. difference-of-squares 99.4%

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

       5. +-commutative 99.4%

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

       6. neg-mul-1 99.4%

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

       7. distribute-rgt1-in 99.4%

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

       8. metadata-eval 99.4%

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

       9. mul0-lft 99.4%

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

       10. *-commutative 99.4%

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

       11. *-commutative 99.4%

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

   12. Simplified 99.4%

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

       1. clear-num 99.3%

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

       2. inv-pow 99.3%

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

       3. +-commutative 99.3%

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

       4. mul0-lft 99.3%

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

       5. div0 99.3%

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

       6. fma-define 99.3%

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

       7. div0 99.3%

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

   14. Applied egg-rr 99.3%

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

       1. unpow-1 99.3%

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

       2. associate-*r* 99.3%

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

       3. *-commutative 99.3%

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

       4. *-commutative 99.3%

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

       5. cancel-sign-sub-inv 99.3%

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

       6. metadata-eval 99.3%

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

       7. +-commutative 99.3%

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

       8. fma-define 99.3%

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

       9. *-commutative 99.3%

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

       10. fma-undefine 99.3%

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

       11. +-rgt-identity 99.3%

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

       12. associate-*r* 99.3%

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

       13. *-commutative 99.3%

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

   16. Simplified 99.3%

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

   17. Taylor expanded in c around 0 91.5%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -2.6:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{-0.5 \cdot \frac{b}{a} + c \cdot \left(0.5 \cdot \frac{c \cdot a}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}{c}}}{2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\left(c \cdot a\right) \cdot 4}{a}}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a))))
   (if (<= t_0 -0.1)
     t_0
     (/ (* 0.5 (/ (* (* c a) 4.0) a)) (- (* 2.0 (/ (* c a) b)) (* 2.0 b))))))

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 <= -0.1) {
		tmp = t_0;
	} else {
		tmp = (0.5 * (((c * a) * 4.0) / a)) / ((2.0 * ((c * a) / b)) - (2.0 * b));
	}
	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 <= (-0.1d0)) then
        tmp = t_0
    else
        tmp = (0.5d0 * (((c * a) * 4.0d0) / a)) / ((2.0d0 * ((c * a) / b)) - (2.0d0 * b))
    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 <= -0.1) {
		tmp = t_0;
	} else {
		tmp = (0.5 * (((c * a) * 4.0) / a)) / ((2.0 * ((c * a) / b)) - (2.0 * b));
	}
	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 <= -0.1:
		tmp = t_0
	else:
		tmp = (0.5 * (((c * a) * 4.0) / a)) / ((2.0 * ((c * a) / b)) - (2.0 * b))
	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 <= -0.1)
		tmp = t_0;
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(Float64(c * a) * 4.0) / a)) / Float64(Float64(2.0 * Float64(Float64(c * a) / b)) - Float64(2.0 * b)));
	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 <= -0.1)
		tmp = t_0;
	else
		tmp = (0.5 * (((c * a) * 4.0) / a)) / ((2.0 * ((c * a) / b)) - (2.0 * b));
	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, -0.1], t$95$0, N[(N[(0.5 * N[(N[(N[(c * a), $MachinePrecision] * 4.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] - N[(2.0 * b), $MachinePrecision]), $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)) < -0.10000000000000001

   1. Initial program 82.6%

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

   if -0.10000000000000001 < (/.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 50.7%

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

      1. *-commutative 50.7%

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

   3. Simplified 50.7%

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

      1. neg-sub0 50.7%

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

      2. flip3-- 50.1%

         \[\leadsto \frac{\color{blue}{\frac{{0}^{3} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. metadata-eval 50.1%

         \[\leadsto \frac{\frac{\color{blue}{0} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. metadata-eval 50.1%

         \[\leadsto \frac{\frac{0 - {b}^{3}}{\color{blue}{0} + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow2 50.1%

         \[\leadsto \frac{\frac{0 - {b}^{3}}{0 + \left(\color{blue}{{b}^{2}} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 50.1%

      \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{3}}{0 + \left({b}^{2} + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. sub0-neg 50.1%

         \[\leadsto \frac{\frac{\color{blue}{-{b}^{3}}}{0 + \left({b}^{2} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. +-lft-identity 50.1%

         \[\leadsto \frac{\frac{-{b}^{3}}{\color{blue}{{b}^{2} + 0 \cdot b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. mul0-lft 50.1%

         \[\leadsto \frac{\frac{-{b}^{3}}{{b}^{2} + \color{blue}{0}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. +-rgt-identity 50.1%

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

   8. Simplified 50.1%

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

      1. flip-+ 50.3%

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

   10. Applied egg-rr 52.7%

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

       1. associate--r- 99.4%

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

       2. unpow2 99.4%

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

       3. unpow2 99.4%

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

       4. difference-of-squares 99.4%

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

       5. +-commutative 99.4%

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

       6. neg-mul-1 99.4%

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

       7. distribute-rgt1-in 99.4%

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

       8. metadata-eval 99.4%

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

       9. mul0-lft 99.4%

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

       10. *-commutative 99.4%

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

       11. *-commutative 99.4%

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

   12. Simplified 99.4%

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

       1. div-inv 99.2%

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

       2. +-commutative 99.2%

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

       3. mul0-lft 99.2%

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

       4. div0 99.2%

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

       5. fma-define 99.2%

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

       6. div0 99.2%

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

   14. Applied egg-rr 99.2%

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

       1. *-commutative 99.2%

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

       2. times-frac 99.3%

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

       3. *-lft-identity 99.3%

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

       4. associate-/r* 99.6%

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

       5. *-lft-identity 99.6%

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

       6. *-commutative 99.6%

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

       7. times-frac 99.6%

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

       8. metadata-eval 99.6%

          \[\leadsto \frac{\color{blue}{0.5} \cdot \frac{\mathsf{fma}\left(c, a \cdot 4, 0\right)}{a}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}} \]

       9. fma-undefine 99.6%

          \[\leadsto \frac{0.5 \cdot \frac{\color{blue}{c \cdot \left(a \cdot 4\right) + 0}}{a}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}} \]

       10. +-rgt-identity 99.6%

           \[\leadsto \frac{0.5 \cdot \frac{\color{blue}{c \cdot \left(a \cdot 4\right)}}{a}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}} \]

       11. associate-*r* 99.6%

           \[\leadsto \frac{0.5 \cdot \frac{\color{blue}{\left(c \cdot a\right) \cdot 4}}{a}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}} \]

       12. *-commutative 99.6%

           \[\leadsto \frac{0.5 \cdot \frac{\color{blue}{4 \cdot \left(c \cdot a\right)}}{a}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}} \]

       13. associate-*r* 99.6%

           \[\leadsto \frac{0.5 \cdot \frac{4 \cdot \left(c \cdot a\right)}{a}}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(c \cdot a\right) \cdot 4}}} \]

       14. *-commutative 99.6%

           \[\leadsto \frac{0.5 \cdot \frac{4 \cdot \left(c \cdot a\right)}{a}}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right)} \cdot 4}} \]

       15. *-commutative 99.6%

           \[\leadsto \frac{0.5 \cdot \frac{4 \cdot \left(c \cdot a\right)}{a}}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}} \]

       16. cancel-sign-sub-inv 99.6%

           \[\leadsto \frac{0.5 \cdot \frac{4 \cdot \left(c \cdot a\right)}{a}}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2} + \left(-4\right) \cdot \left(a \cdot c\right)}}} \]

       17. metadata-eval 99.6%

           \[\leadsto \frac{0.5 \cdot \frac{4 \cdot \left(c \cdot a\right)}{a}}{\left(-b\right) - \sqrt{{b}^{2} + \color{blue}{-4} \cdot \left(a \cdot c\right)}} \]

   16. Simplified 99.6%

       \[\leadsto \color{blue}{\frac{0.5 \cdot \frac{4 \cdot \left(c \cdot a\right)}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}}} \]

   17. Taylor expanded in c around 0 86.8%

       \[\leadsto \frac{0.5 \cdot \frac{4 \cdot \left(c \cdot a\right)}{a}}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.1:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \frac{\left(c \cdot a\right) \cdot 4}{a}}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \frac{4 \cdot \left(c \cdot \left(-a\right)\right)}{a}}{b + \sqrt{{b}^{2} + -4 \cdot \left(c \cdot a\right)}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (* 0.5 (/ (* 4.0 (* c (- a))) a))
  (+ b (sqrt (+ (pow b 2.0) (* -4.0 (* c a)))))))

C

double code(double a, double b, double c) {
	return (0.5 * ((4.0 * (c * -a)) / a)) / (b + sqrt((pow(b, 2.0) + (-4.0 * (c * 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 = (0.5d0 * ((4.0d0 * (c * -a)) / a)) / (b + sqrt(((b ** 2.0d0) + ((-4.0d0) * (c * a)))))
end function

Java

public static double code(double a, double b, double c) {
	return (0.5 * ((4.0 * (c * -a)) / a)) / (b + Math.sqrt((Math.pow(b, 2.0) + (-4.0 * (c * a)))));
}

Python

def code(a, b, c):
	return (0.5 * ((4.0 * (c * -a)) / a)) / (b + math.sqrt((math.pow(b, 2.0) + (-4.0 * (c * a)))))

Julia

function code(a, b, c)
	return Float64(Float64(0.5 * Float64(Float64(4.0 * Float64(c * Float64(-a))) / a)) / Float64(b + sqrt(Float64((b ^ 2.0) + Float64(-4.0 * Float64(c * a))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (0.5 * ((4.0 * (c * -a)) / a)) / (b + sqrt(((b ^ 2.0) + (-4.0 * (c * a)))));
end

Wolfram

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

Derivation

1. Initial program 55.8%

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

   1. *-commutative 55.8%

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

3. Simplified 55.8%

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

   1. neg-sub0 55.8%

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

   2. flip3-- 55.1%

      \[\leadsto \frac{\color{blue}{\frac{{0}^{3} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. metadata-eval 55.1%

      \[\leadsto \frac{\frac{\color{blue}{0} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. metadata-eval 55.1%

      \[\leadsto \frac{\frac{0 - {b}^{3}}{\color{blue}{0} + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. pow2 55.1%

      \[\leadsto \frac{\frac{0 - {b}^{3}}{0 + \left(\color{blue}{{b}^{2}} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 55.1%

   \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{3}}{0 + \left({b}^{2} + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. sub0-neg 55.1%

      \[\leadsto \frac{\frac{\color{blue}{-{b}^{3}}}{0 + \left({b}^{2} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   2. +-lft-identity 55.1%

      \[\leadsto \frac{\frac{-{b}^{3}}{\color{blue}{{b}^{2} + 0 \cdot b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. mul0-lft 55.1%

      \[\leadsto \frac{\frac{-{b}^{3}}{{b}^{2} + \color{blue}{0}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. +-rgt-identity 55.1%

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

8. Simplified 55.1%

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

   1. flip-+ 55.4%

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

10. Applied egg-rr 57.8%

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

    1. associate--r- 99.4%

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

    2. unpow2 99.4%

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

    3. unpow2 99.4%

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

    4. difference-of-squares 99.4%

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

    5. +-commutative 99.4%

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

    6. neg-mul-1 99.4%

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

    7. distribute-rgt1-in 99.4%

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

    8. metadata-eval 99.4%

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

    9. mul0-lft 99.4%

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

    10. *-commutative 99.4%

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

    11. *-commutative 99.4%

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

12. Simplified 99.4%

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

    1. div-inv 99.2%

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

    2. +-commutative 99.2%

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

    3. mul0-lft 99.2%

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

    4. div0 99.2%

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

    5. fma-define 99.2%

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

    6. div0 99.2%

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

14. Applied egg-rr 99.2%

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

    1. *-commutative 99.2%

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

    2. times-frac 99.3%

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

    3. *-lft-identity 99.3%

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

    4. associate-/r* 99.6%

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

    5. *-lft-identity 99.6%

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

    6. *-commutative 99.6%

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

    7. times-frac 99.6%

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

    8. metadata-eval 99.6%

       \[\leadsto \frac{\color{blue}{0.5} \cdot \frac{\mathsf{fma}\left(c, a \cdot 4, 0\right)}{a}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}} \]

    9. fma-undefine 99.6%

       \[\leadsto \frac{0.5 \cdot \frac{\color{blue}{c \cdot \left(a \cdot 4\right) + 0}}{a}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}} \]

    10. +-rgt-identity 99.6%

        \[\leadsto \frac{0.5 \cdot \frac{\color{blue}{c \cdot \left(a \cdot 4\right)}}{a}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}} \]

    11. associate-*r* 99.6%

        \[\leadsto \frac{0.5 \cdot \frac{\color{blue}{\left(c \cdot a\right) \cdot 4}}{a}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}} \]

    12. *-commutative 99.6%

        \[\leadsto \frac{0.5 \cdot \frac{\color{blue}{4 \cdot \left(c \cdot a\right)}}{a}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}} \]

    13. associate-*r* 99.6%

        \[\leadsto \frac{0.5 \cdot \frac{4 \cdot \left(c \cdot a\right)}{a}}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(c \cdot a\right) \cdot 4}}} \]

    14. *-commutative 99.6%

        \[\leadsto \frac{0.5 \cdot \frac{4 \cdot \left(c \cdot a\right)}{a}}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right)} \cdot 4}} \]

    15. *-commutative 99.6%

        \[\leadsto \frac{0.5 \cdot \frac{4 \cdot \left(c \cdot a\right)}{a}}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}} \]

    16. cancel-sign-sub-inv 99.6%

        \[\leadsto \frac{0.5 \cdot \frac{4 \cdot \left(c \cdot a\right)}{a}}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2} + \left(-4\right) \cdot \left(a \cdot c\right)}}} \]

    17. metadata-eval 99.6%

        \[\leadsto \frac{0.5 \cdot \frac{4 \cdot \left(c \cdot a\right)}{a}}{\left(-b\right) - \sqrt{{b}^{2} + \color{blue}{-4} \cdot \left(a \cdot c\right)}} \]

16. Simplified 99.6%

    \[\leadsto \color{blue}{\frac{0.5 \cdot \frac{4 \cdot \left(c \cdot a\right)}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}}} \]
17. Step-by-step derivation

    1. fma-undefine 99.6%

       \[\leadsto \frac{0.5 \cdot \frac{4 \cdot \left(c \cdot a\right)}{a}}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(c \cdot a\right) + {b}^{2}}}} \]

18. Applied egg-rr 99.6%

    \[\leadsto \frac{0.5 \cdot \frac{4 \cdot \left(c \cdot a\right)}{a}}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(c \cdot a\right) + {b}^{2}}}} \]

19. Final simplification 99.6%

    \[\leadsto \frac{0.5 \cdot \frac{4 \cdot \left(c \cdot \left(-a\right)\right)}{a}}{b + \sqrt{{b}^{2} + -4 \cdot \left(c \cdot a\right)}} \]
20. Add Preprocessing

Alternative 7: 82.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \frac{\left(c \cdot a\right) \cdot 4}{a}}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* 0.5 (/ (* (* c a) 4.0) a)) (- (* 2.0 (/ (* c a) b)) (* 2.0 b))))

C

double code(double a, double b, double c) {
	return (0.5 * (((c * a) * 4.0) / a)) / ((2.0 * ((c * a) / b)) - (2.0 * b));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (0.5d0 * (((c * a) * 4.0d0) / a)) / ((2.0d0 * ((c * a) / b)) - (2.0d0 * b))
end function

Java

public static double code(double a, double b, double c) {
	return (0.5 * (((c * a) * 4.0) / a)) / ((2.0 * ((c * a) / b)) - (2.0 * b));
}

Python

def code(a, b, c):
	return (0.5 * (((c * a) * 4.0) / a)) / ((2.0 * ((c * a) / b)) - (2.0 * b))

Julia

function code(a, b, c)
	return Float64(Float64(0.5 * Float64(Float64(Float64(c * a) * 4.0) / a)) / Float64(Float64(2.0 * Float64(Float64(c * a) / b)) - Float64(2.0 * b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (0.5 * (((c * a) * 4.0) / a)) / ((2.0 * ((c * a) / b)) - (2.0 * b));
end

Wolfram

code[a_, b_, c_] := N[(N[(0.5 * N[(N[(N[(c * a), $MachinePrecision] * 4.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] - N[(2.0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.8%

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

   1. *-commutative 55.8%

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

3. Simplified 55.8%

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

   1. neg-sub0 55.8%

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

   2. flip3-- 55.1%

      \[\leadsto \frac{\color{blue}{\frac{{0}^{3} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. metadata-eval 55.1%

      \[\leadsto \frac{\frac{\color{blue}{0} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. metadata-eval 55.1%

      \[\leadsto \frac{\frac{0 - {b}^{3}}{\color{blue}{0} + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. pow2 55.1%

      \[\leadsto \frac{\frac{0 - {b}^{3}}{0 + \left(\color{blue}{{b}^{2}} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 55.1%

   \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{3}}{0 + \left({b}^{2} + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. sub0-neg 55.1%

      \[\leadsto \frac{\frac{\color{blue}{-{b}^{3}}}{0 + \left({b}^{2} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   2. +-lft-identity 55.1%

      \[\leadsto \frac{\frac{-{b}^{3}}{\color{blue}{{b}^{2} + 0 \cdot b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. mul0-lft 55.1%

      \[\leadsto \frac{\frac{-{b}^{3}}{{b}^{2} + \color{blue}{0}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. +-rgt-identity 55.1%

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

8. Simplified 55.1%

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

   1. flip-+ 55.4%

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

10. Applied egg-rr 57.8%

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

    1. associate--r- 99.4%

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

    2. unpow2 99.4%

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

    3. unpow2 99.4%

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

    4. difference-of-squares 99.4%

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

    5. +-commutative 99.4%

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

    6. neg-mul-1 99.4%

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

    7. distribute-rgt1-in 99.4%

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

    8. metadata-eval 99.4%

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

    9. mul0-lft 99.4%

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

    10. *-commutative 99.4%

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

    11. *-commutative 99.4%

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

12. Simplified 99.4%

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

    1. div-inv 99.2%

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

    2. +-commutative 99.2%

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

    3. mul0-lft 99.2%

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

    4. div0 99.2%

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

    5. fma-define 99.2%

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

    6. div0 99.2%

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

14. Applied egg-rr 99.2%

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

    1. *-commutative 99.2%

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

    2. times-frac 99.3%

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

    3. *-lft-identity 99.3%

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

    4. associate-/r* 99.6%

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

    5. *-lft-identity 99.6%

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

    6. *-commutative 99.6%

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

    7. times-frac 99.6%

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

    8. metadata-eval 99.6%

       \[\leadsto \frac{\color{blue}{0.5} \cdot \frac{\mathsf{fma}\left(c, a \cdot 4, 0\right)}{a}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}} \]

    9. fma-undefine 99.6%

       \[\leadsto \frac{0.5 \cdot \frac{\color{blue}{c \cdot \left(a \cdot 4\right) + 0}}{a}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}} \]

    10. +-rgt-identity 99.6%

        \[\leadsto \frac{0.5 \cdot \frac{\color{blue}{c \cdot \left(a \cdot 4\right)}}{a}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}} \]

    11. associate-*r* 99.6%

        \[\leadsto \frac{0.5 \cdot \frac{\color{blue}{\left(c \cdot a\right) \cdot 4}}{a}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}} \]

    12. *-commutative 99.6%

        \[\leadsto \frac{0.5 \cdot \frac{\color{blue}{4 \cdot \left(c \cdot a\right)}}{a}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}} \]

    13. associate-*r* 99.6%

        \[\leadsto \frac{0.5 \cdot \frac{4 \cdot \left(c \cdot a\right)}{a}}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(c \cdot a\right) \cdot 4}}} \]

    14. *-commutative 99.6%

        \[\leadsto \frac{0.5 \cdot \frac{4 \cdot \left(c \cdot a\right)}{a}}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right)} \cdot 4}} \]

    15. *-commutative 99.6%

        \[\leadsto \frac{0.5 \cdot \frac{4 \cdot \left(c \cdot a\right)}{a}}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}} \]

    16. cancel-sign-sub-inv 99.6%

        \[\leadsto \frac{0.5 \cdot \frac{4 \cdot \left(c \cdot a\right)}{a}}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2} + \left(-4\right) \cdot \left(a \cdot c\right)}}} \]

    17. metadata-eval 99.6%

        \[\leadsto \frac{0.5 \cdot \frac{4 \cdot \left(c \cdot a\right)}{a}}{\left(-b\right) - \sqrt{{b}^{2} + \color{blue}{-4} \cdot \left(a \cdot c\right)}} \]

16. Simplified 99.6%

    \[\leadsto \color{blue}{\frac{0.5 \cdot \frac{4 \cdot \left(c \cdot a\right)}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}}} \]

17. Taylor expanded in c around 0 82.5%

    \[\leadsto \frac{0.5 \cdot \frac{4 \cdot \left(c \cdot a\right)}{a}}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}} \]

18. Final simplification 82.5%

    \[\leadsto \frac{0.5 \cdot \frac{\left(c \cdot a\right) \cdot 4}{a}}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b} \]
19. Add Preprocessing

Alternative 8: 82.3% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

   1. *-commutative 55.8%

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

3. Simplified 55.8%

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

   1. neg-sub0 55.8%

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

   2. flip3-- 55.1%

      \[\leadsto \frac{\color{blue}{\frac{{0}^{3} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. metadata-eval 55.1%

      \[\leadsto \frac{\frac{\color{blue}{0} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. metadata-eval 55.1%

      \[\leadsto \frac{\frac{0 - {b}^{3}}{\color{blue}{0} + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. pow2 55.1%

      \[\leadsto \frac{\frac{0 - {b}^{3}}{0 + \left(\color{blue}{{b}^{2}} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 55.1%

   \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{3}}{0 + \left({b}^{2} + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. sub0-neg 55.1%

      \[\leadsto \frac{\frac{\color{blue}{-{b}^{3}}}{0 + \left({b}^{2} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   2. +-lft-identity 55.1%

      \[\leadsto \frac{\frac{-{b}^{3}}{\color{blue}{{b}^{2} + 0 \cdot b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. mul0-lft 55.1%

      \[\leadsto \frac{\frac{-{b}^{3}}{{b}^{2} + \color{blue}{0}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. +-rgt-identity 55.1%

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

8. Simplified 55.1%

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

   1. flip-+ 55.4%

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

10. Applied egg-rr 57.8%

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

    1. associate--r- 99.4%

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

    2. unpow2 99.4%

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

    3. unpow2 99.4%

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

    4. difference-of-squares 99.4%

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

    5. +-commutative 99.4%

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

    6. neg-mul-1 99.4%

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

    7. distribute-rgt1-in 99.4%

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

    8. metadata-eval 99.4%

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

    9. mul0-lft 99.4%

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

    10. *-commutative 99.4%

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

    11. *-commutative 99.4%

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

12. Simplified 99.4%

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

13. Taylor expanded in c around 0 82.4%

    \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + c \cdot \left(a \cdot 4\right)}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}}{a \cdot 2} \]
14. Step-by-step derivation

    1. distribute-lft-out-- 82.4%

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

    2. *-commutative 82.4%

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

15. Simplified 82.4%

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

16. Taylor expanded in b around 0 82.4%

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

    1. *-commutative 82.4%

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

18. Simplified 82.4%

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

19. Final simplification 82.4%

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

Alternative 9: 82.3% 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 55.8%

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

   1. *-commutative 55.8%

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

3. Simplified 55.8%

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

   1. neg-sub0 55.8%

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

   2. flip3-- 55.1%

      \[\leadsto \frac{\color{blue}{\frac{{0}^{3} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. metadata-eval 55.1%

      \[\leadsto \frac{\frac{\color{blue}{0} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. metadata-eval 55.1%

      \[\leadsto \frac{\frac{0 - {b}^{3}}{\color{blue}{0} + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. pow2 55.1%

      \[\leadsto \frac{\frac{0 - {b}^{3}}{0 + \left(\color{blue}{{b}^{2}} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 55.1%

   \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{3}}{0 + \left({b}^{2} + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. sub0-neg 55.1%

      \[\leadsto \frac{\frac{\color{blue}{-{b}^{3}}}{0 + \left({b}^{2} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   2. +-lft-identity 55.1%

      \[\leadsto \frac{\frac{-{b}^{3}}{\color{blue}{{b}^{2} + 0 \cdot b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. mul0-lft 55.1%

      \[\leadsto \frac{\frac{-{b}^{3}}{{b}^{2} + \color{blue}{0}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. +-rgt-identity 55.1%

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

8. Simplified 55.1%

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

   1. flip-+ 55.4%

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

10. Applied egg-rr 57.8%

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

    1. associate--r- 99.4%

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

    2. unpow2 99.4%

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

    3. unpow2 99.4%

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

    4. difference-of-squares 99.4%

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

    5. +-commutative 99.4%

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

    6. neg-mul-1 99.4%

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

    7. distribute-rgt1-in 99.4%

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

    8. metadata-eval 99.4%

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

    9. mul0-lft 99.4%

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

    10. *-commutative 99.4%

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

    11. *-commutative 99.4%

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

12. Simplified 99.4%

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

    1. clear-num 99.2%

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

    2. inv-pow 99.2%

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

    3. +-commutative 99.2%

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

    4. mul0-lft 99.2%

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

    5. div0 99.2%

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

    6. fma-define 99.2%

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

    7. div0 99.2%

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

14. Applied egg-rr 99.2%

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

    1. unpow-1 99.2%

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

    2. associate-*r* 99.2%

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

    3. *-commutative 99.2%

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

    4. *-commutative 99.2%

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

    5. cancel-sign-sub-inv 99.2%

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

    6. metadata-eval 99.2%

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

    7. +-commutative 99.2%

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

    8. fma-define 99.2%

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

    9. *-commutative 99.2%

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

    10. fma-undefine 99.2%

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

    11. +-rgt-identity 99.2%

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

    12. associate-*r* 99.2%

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

    13. *-commutative 99.2%

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

16. Simplified 99.2%

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

17. Taylor expanded in c around 0 82.3%

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

18. Final simplification 82.3%

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

Alternative 10: 64.6% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

   1. *-commutative 55.8%

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

3. Simplified 55.8%

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

5. Taylor expanded in b around inf 64.4%

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

   1. associate-*r/ 64.4%

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

   2. mul-1-neg 64.4%

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

7. Simplified 64.4%

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

8. Final simplification 64.4%

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

Alternative 11: 3.2% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 55.8%

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

   1. *-commutative 55.8%

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

3. Simplified 55.8%

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

   1. neg-sub0 55.8%

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

   2. flip3-- 55.1%

      \[\leadsto \frac{\color{blue}{\frac{{0}^{3} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. metadata-eval 55.1%

      \[\leadsto \frac{\frac{\color{blue}{0} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. metadata-eval 55.1%

      \[\leadsto \frac{\frac{0 - {b}^{3}}{\color{blue}{0} + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. pow2 55.1%

      \[\leadsto \frac{\frac{0 - {b}^{3}}{0 + \left(\color{blue}{{b}^{2}} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 55.1%

   \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{3}}{0 + \left({b}^{2} + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. sub0-neg 55.1%

      \[\leadsto \frac{\frac{\color{blue}{-{b}^{3}}}{0 + \left({b}^{2} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   2. +-lft-identity 55.1%

      \[\leadsto \frac{\frac{-{b}^{3}}{\color{blue}{{b}^{2} + 0 \cdot b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. mul0-lft 55.1%

      \[\leadsto \frac{\frac{-{b}^{3}}{{b}^{2} + \color{blue}{0}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. +-rgt-identity 55.1%

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

8. Simplified 55.1%

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

9. Taylor expanded in a around 0 3.2%

   \[\leadsto \color{blue}{0.5 \cdot \frac{b + -1 \cdot b}{a}} \]
10. Step-by-step derivation

    1. associate-*r/ 3.2%

       \[\leadsto \color{blue}{\frac{0.5 \cdot \left(b + -1 \cdot b\right)}{a}} \]

    2. distribute-rgt1-in 3.2%

       \[\leadsto \frac{0.5 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot b\right)}}{a} \]

    3. metadata-eval 3.2%

       \[\leadsto \frac{0.5 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

    4. mul0-lft 3.2%

       \[\leadsto \frac{0.5 \cdot \color{blue}{0}}{a} \]

    5. metadata-eval 3.2%

       \[\leadsto \frac{\color{blue}{0}}{a} \]

11. Simplified 3.2%

    \[\leadsto \color{blue}{\frac{0}{a}} \]

12. Taylor expanded in a around 0 3.2%

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

Reproduce

herbie shell --seed 2024123 
(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)))