Quadratic roots, narrow range

Percentage Accurate: 55.6% → 99.6%

Time: 21.9s

Alternatives: 9

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: 55.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \sqrt[3]{64}\\ \frac{t\_0 \cdot \left(-0.5\right)}{b + \sqrt{{b}^{2} - t\_0 \cdot a}} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (cbrt 64.0))))
   (/ (* t_0 (- 0.5)) (+ b (sqrt (- (pow b 2.0) (* t_0 a)))))))

C

double code(double a, double b, double c) {
	double t_0 = c * cbrt(64.0);
	return (t_0 * -0.5) / (b + sqrt((pow(b, 2.0) - (t_0 * a))));
}

Java

public static double code(double a, double b, double c) {
	double t_0 = c * Math.cbrt(64.0);
	return (t_0 * -0.5) / (b + Math.sqrt((Math.pow(b, 2.0) - (t_0 * a))));
}

Julia

function code(a, b, c)
	t_0 = Float64(c * cbrt(64.0))
	return Float64(Float64(t_0 * Float64(-0.5)) / Float64(b + sqrt(Float64((b ^ 2.0) - Float64(t_0 * a)))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[Power[64.0, 1/3], $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$0 * (-0.5)), $MachinePrecision] / N[(b + N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] - N[(t$95$0 * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 57.3%

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

   1. *-commutative 57.3%

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

3. Simplified 57.3%

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

   1. add-cbrt-cube 57.2%

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

   2. pow1/3 57.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}^{0.3333333333333333}}}}{a \cdot 2} \]

   3. pow3 57.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\color{blue}{\left({\left(\left(4 \cdot a\right) \cdot c\right)}^{3}\right)}}^{0.3333333333333333}}}{a \cdot 2} \]

   4. associate-*l* 57.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({\color{blue}{\left(4 \cdot \left(a \cdot c\right)\right)}}^{3}\right)}^{0.3333333333333333}}}{a \cdot 2} \]

   5. unpow-prod-down 57.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\color{blue}{\left({4}^{3} \cdot {\left(a \cdot c\right)}^{3}\right)}}^{0.3333333333333333}}}{a \cdot 2} \]

   6. metadata-eval 57.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\color{blue}{64} \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}{a \cdot 2} \]

6. Applied egg-rr 57.1%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{a \cdot 2} \]
7. Step-by-step derivation

   1. flip-+ 57.0%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}} \cdot \sqrt{b \cdot b - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}{\left(-b\right) - \sqrt{b \cdot b - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}}{a \cdot 2} \]

   2. pow2 57.1%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}} \cdot \sqrt{b \cdot b - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}{\left(-b\right) - \sqrt{b \cdot b - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{a \cdot 2} \]

   3. add-sqr-sqrt 58.6%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}\right)}}{\left(-b\right) - \sqrt{b \cdot b - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{a \cdot 2} \]

   4. pow2 58.5%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{a \cdot 2} \]

   5. unpow1/3 58.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\sqrt[3]{64 \cdot {\left(a \cdot c\right)}^{3}}}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{a \cdot 2} \]

   6. *-commutative 58.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \sqrt[3]{\color{blue}{{\left(a \cdot c\right)}^{3} \cdot 64}}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{a \cdot 2} \]

   7. cbrt-prod 58.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\sqrt[3]{{\left(a \cdot c\right)}^{3}} \cdot \sqrt[3]{64}}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{a \cdot 2} \]

   8. unpow3 58.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \sqrt[3]{\color{blue}{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right) \cdot \left(a \cdot c\right)}} \cdot \sqrt[3]{64}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{a \cdot 2} \]

   9. add-cbrt-cube 58.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\left(a \cdot c\right)} \cdot \sqrt[3]{64}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{a \cdot 2} \]

   10. *-commutative 58.7%

       \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\left(c \cdot a\right)} \cdot \sqrt[3]{64}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{a \cdot 2} \]

8. Applied egg-rr 58.7%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(c \cdot a\right) \cdot \sqrt[3]{64}\right)}{\left(-b\right) - \sqrt{{b}^{2} - \left(c \cdot a\right) \cdot \sqrt[3]{64}}}}}{a \cdot 2} \]
9. Step-by-step derivation

   1. *-un-lft-identity 58.7%

      \[\leadsto \color{blue}{1 \cdot \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(c \cdot a\right) \cdot \sqrt[3]{64}\right)}{\left(-b\right) - \sqrt{{b}^{2} - \left(c \cdot a\right) \cdot \sqrt[3]{64}}}}{a \cdot 2}} \]

   2. associate-/l/ 58.7%

      \[\leadsto 1 \cdot \color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(c \cdot a\right) \cdot \sqrt[3]{64}\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \left(c \cdot a\right) \cdot \sqrt[3]{64}}\right)}} \]

   3. associate--r- 99.2%

      \[\leadsto 1 \cdot \frac{\color{blue}{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \left(c \cdot a\right) \cdot \sqrt[3]{64}}}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \left(c \cdot a\right) \cdot \sqrt[3]{64}}\right)} \]

   4. neg-mul-1 99.2%

      \[\leadsto 1 \cdot \frac{\left({\color{blue}{\left(-1 \cdot b\right)}}^{2} - {b}^{2}\right) + \left(c \cdot a\right) \cdot \sqrt[3]{64}}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \left(c \cdot a\right) \cdot \sqrt[3]{64}}\right)} \]

   5. unpow-prod-down 99.2%

      \[\leadsto 1 \cdot \frac{\left(\color{blue}{{-1}^{2} \cdot {b}^{2}} - {b}^{2}\right) + \left(c \cdot a\right) \cdot \sqrt[3]{64}}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \left(c \cdot a\right) \cdot \sqrt[3]{64}}\right)} \]

   6. metadata-eval 99.2%

      \[\leadsto 1 \cdot \frac{\left(\color{blue}{1} \cdot {b}^{2} - {b}^{2}\right) + \left(c \cdot a\right) \cdot \sqrt[3]{64}}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \left(c \cdot a\right) \cdot \sqrt[3]{64}}\right)} \]

   7. *-un-lft-identity 99.2%

      \[\leadsto 1 \cdot \frac{\left(\color{blue}{{b}^{2}} - {b}^{2}\right) + \left(c \cdot a\right) \cdot \sqrt[3]{64}}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \left(c \cdot a\right) \cdot \sqrt[3]{64}}\right)} \]

   8. associate-*l* 99.2%

      \[\leadsto 1 \cdot \frac{\left({b}^{2} - {b}^{2}\right) + \color{blue}{c \cdot \left(a \cdot \sqrt[3]{64}\right)}}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \left(c \cdot a\right) \cdot \sqrt[3]{64}}\right)} \]

   9. associate-*l* 99.2%

      \[\leadsto 1 \cdot \frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot \sqrt[3]{64}\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{c \cdot \left(a \cdot \sqrt[3]{64}\right)}}\right)} \]

10. Applied egg-rr 99.2%

    \[\leadsto \color{blue}{1 \cdot \frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot \sqrt[3]{64}\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{64}\right)}\right)}} \]
11. Step-by-step derivation

    1. *-lft-identity 99.2%

       \[\leadsto \color{blue}{\frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot \sqrt[3]{64}\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{64}\right)}\right)}} \]

    2. associate-/r* 99.4%

       \[\leadsto \color{blue}{\frac{\frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot \sqrt[3]{64}\right)}{a \cdot 2}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{64}\right)}}} \]

    3. +-inverses 99.4%

       \[\leadsto \frac{\frac{\color{blue}{0} + c \cdot \left(a \cdot \sqrt[3]{64}\right)}{a \cdot 2}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{64}\right)}} \]

    4. +-lft-identity 99.4%

       \[\leadsto \frac{\frac{\color{blue}{c \cdot \left(a \cdot \sqrt[3]{64}\right)}}{a \cdot 2}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{64}\right)}} \]

    5. associate-*r* 99.4%

       \[\leadsto \frac{\frac{\color{blue}{\left(c \cdot a\right) \cdot \sqrt[3]{64}}}{a \cdot 2}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{64}\right)}} \]

    6. *-commutative 99.4%

       \[\leadsto \frac{\frac{\color{blue}{\left(a \cdot c\right)} \cdot \sqrt[3]{64}}{a \cdot 2}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{64}\right)}} \]

    7. associate-*r* 99.4%

       \[\leadsto \frac{\frac{\color{blue}{a \cdot \left(c \cdot \sqrt[3]{64}\right)}}{a \cdot 2}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{64}\right)}} \]

    8. associate-*r* 99.4%

       \[\leadsto \frac{\frac{a \cdot \left(c \cdot \sqrt[3]{64}\right)}{a \cdot 2}}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(c \cdot a\right) \cdot \sqrt[3]{64}}}} \]

    9. *-commutative 99.4%

       \[\leadsto \frac{\frac{a \cdot \left(c \cdot \sqrt[3]{64}\right)}{a \cdot 2}}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right)} \cdot \sqrt[3]{64}}} \]

    10. associate-*r* 99.4%

        \[\leadsto \frac{\frac{a \cdot \left(c \cdot \sqrt[3]{64}\right)}{a \cdot 2}}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{a \cdot \left(c \cdot \sqrt[3]{64}\right)}}} \]

12. Simplified 99.4%

    \[\leadsto \color{blue}{\frac{\frac{a \cdot \left(c \cdot \sqrt[3]{64}\right)}{a \cdot 2}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot \sqrt[3]{64}\right)}}} \]

13. Taylor expanded in a around 0 99.5%

    \[\leadsto \frac{\color{blue}{0.5 \cdot \left(c \cdot \sqrt[3]{64}\right)}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot \sqrt[3]{64}\right)}} \]
14. Step-by-step derivation

    1. *-commutative 99.5%

       \[\leadsto \frac{\color{blue}{\left(c \cdot \sqrt[3]{64}\right) \cdot 0.5}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot \sqrt[3]{64}\right)}} \]

15. Simplified 99.5%

    \[\leadsto \frac{\color{blue}{\left(c \cdot \sqrt[3]{64}\right) \cdot 0.5}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot \sqrt[3]{64}\right)}} \]

16. Final simplification 99.5%

    \[\leadsto \frac{\left(c \cdot \sqrt[3]{64}\right) \cdot \left(-0.5\right)}{b + \sqrt{{b}^{2} - \left(c \cdot \sqrt[3]{64}\right) \cdot a}} \]
17. Add Preprocessing

Alternative 2: 89.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.4:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left({c}^{3} \cdot \left(-2 \cdot \frac{a}{{b}^{5}} + \frac{-1}{c \cdot {b}^{3}}\right)\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a)) -0.4)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* 2.0 a))
   (-
    (*
     a
     (* (pow c 3.0) (+ (* -2.0 (/ a (pow b 5.0))) (/ -1.0 (* c (pow b 3.0))))))
    (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)) <= -0.4) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (2.0 * a);
	} else {
		tmp = (a * (pow(c, 3.0) * ((-2.0 * (a / pow(b, 5.0))) + (-1.0 / (c * pow(b, 3.0)))))) - (c / b);
	}
	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)) <= -0.4)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(a * Float64((c ^ 3.0) * Float64(Float64(-2.0 * Float64(a / (b ^ 5.0))) + Float64(-1.0 / Float64(c * (b ^ 3.0)))))) - Float64(c / b));
	end
	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], -0.4], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[Power[c, 3.0], $MachinePrecision] * N[(N[(-2.0 * N[(a / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(c * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. 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)) < -0.40000000000000002

   1. Initial program 86.0%

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

      1. *-commutative 86.0%

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

      2. +-commutative 86.0%

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

      3. sqr-neg 86.0%

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

      4. unsub-neg 86.0%

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

      5. sqr-neg 86.0%

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

      6. fma-neg 86.4%

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

      7. distribute-lft-neg-in 86.4%

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

      8. *-commutative 86.4%

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

      9. *-commutative 86.4%

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

      10. distribute-rgt-neg-in 86.4%

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

      11. metadata-eval 86.4%

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

   3. Simplified 86.4%

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

   if -0.40000000000000002 < (/.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 50.6%

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

      1. *-commutative 50.6%

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

   3. Simplified 50.6%

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

   5. Taylor expanded in a around 0 93.7%

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

   6. Taylor expanded in c around 0 93.7%

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

   7. Taylor expanded in a around 0 91.4%

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

      1. associate-*r/ 91.4%

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

      2. associate-*r* 91.4%

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

   9. Simplified 91.4%

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

   10. Taylor expanded in c around inf 91.4%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.4:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left({c}^{3} \cdot \left(-2 \cdot \frac{a}{{b}^{5}} + \frac{-1}{c \cdot {b}^{3}}\right)\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.4:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a)) -0.4)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* 2.0 a))
   (*
    c
    (+
     (* c (- (* -2.0 (/ (* c (pow a 2.0)) (pow b 5.0))) (/ a (pow b 3.0))))
     (/ -1.0 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)) <= -0.4) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (2.0 * a);
	} else {
		tmp = c * ((c * ((-2.0 * ((c * pow(a, 2.0)) / pow(b, 5.0))) - (a / pow(b, 3.0)))) + (-1.0 / b));
	}
	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)) <= -0.4)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(c * Float64(Float64(c * Float64(Float64(-2.0 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 5.0))) - Float64(a / (b ^ 3.0)))) + Float64(-1.0 / b)));
	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], -0.4], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(N[(-2.0 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / 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)) < -0.40000000000000002

   1. Initial program 86.0%

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

      1. *-commutative 86.0%

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

      2. +-commutative 86.0%

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

      3. sqr-neg 86.0%

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

      4. unsub-neg 86.0%

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

      5. sqr-neg 86.0%

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

      6. fma-neg 86.4%

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

      7. distribute-lft-neg-in 86.4%

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

      8. *-commutative 86.4%

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

      9. *-commutative 86.4%

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

      10. distribute-rgt-neg-in 86.4%

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

      11. metadata-eval 86.4%

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

   3. Simplified 86.4%

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

   if -0.40000000000000002 < (/.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 50.6%

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

      1. *-commutative 50.6%

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

   3. Simplified 50.6%

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

   5. Taylor expanded in c around 0 91.2%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.4:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-2 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} - \frac{a}{{b}^{3}}\right) + \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot \sqrt[3]{64}\right) \cdot a\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.002648:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{2 \cdot a}}{0.5 \cdot \frac{t\_0}{b} - b \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* c (cbrt 64.0)) a)))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a)) -0.002648)
     (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* 2.0 a))
     (/ (/ t_0 (* 2.0 a)) (- (* 0.5 (/ t_0 b)) (* b 2.0))))))

C

double code(double a, double b, double c) {
	double t_0 = (c * cbrt(64.0)) * a;
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)) <= -0.002648) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (2.0 * a);
	} else {
		tmp = (t_0 / (2.0 * a)) / ((0.5 * (t_0 / b)) - (b * 2.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * cbrt(64.0)) * a)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a)) <= -0.002648)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(t_0 / Float64(2.0 * a)) / Float64(Float64(0.5 * Float64(t_0 / b)) - Float64(b * 2.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * N[Power[64.0, 1/3], $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, 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], -0.002648], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[(t$95$0 / b), $MachinePrecision]), $MachinePrecision] - N[(b * 2.0), $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)) < -0.0026480000000000002

   1. Initial program 80.6%

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

      1. *-commutative 80.6%

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

      2. +-commutative 80.6%

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

      3. sqr-neg 80.6%

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

      4. unsub-neg 80.6%

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

      5. sqr-neg 80.6%

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

      6. fma-neg 80.9%

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

      7. distribute-lft-neg-in 80.9%

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

      8. *-commutative 80.9%

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

      9. *-commutative 80.9%

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

      10. distribute-rgt-neg-in 80.9%

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

      11. metadata-eval 80.9%

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

   3. Simplified 80.9%

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

   if -0.0026480000000000002 < (/.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 45.9%

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

      1. *-commutative 45.9%

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

   3. Simplified 45.9%

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

      1. add-cbrt-cube 45.8%

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

      2. pow1/3 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}^{0.3333333333333333}}}}{a \cdot 2} \]

      3. pow3 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\color{blue}{\left({\left(\left(4 \cdot a\right) \cdot c\right)}^{3}\right)}}^{0.3333333333333333}}}{a \cdot 2} \]

      4. associate-*l* 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({\color{blue}{\left(4 \cdot \left(a \cdot c\right)\right)}}^{3}\right)}^{0.3333333333333333}}}{a \cdot 2} \]

      5. unpow-prod-down 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\color{blue}{\left({4}^{3} \cdot {\left(a \cdot c\right)}^{3}\right)}}^{0.3333333333333333}}}{a \cdot 2} \]

      6. metadata-eval 45.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\color{blue}{64} \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}{a \cdot 2} \]

   6. Applied egg-rr 45.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. flip-+ 45.7%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}} \cdot \sqrt{b \cdot b - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}{\left(-b\right) - \sqrt{b \cdot b - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}}{a \cdot 2} \]

      2. pow2 45.8%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}} \cdot \sqrt{b \cdot b - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}{\left(-b\right) - \sqrt{b \cdot b - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{a \cdot 2} \]

      3. add-sqr-sqrt 47.4%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}\right)}}{\left(-b\right) - \sqrt{b \cdot b - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{a \cdot 2} \]

      4. pow2 47.3%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{a \cdot 2} \]

      5. unpow1/3 47.3%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\sqrt[3]{64 \cdot {\left(a \cdot c\right)}^{3}}}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{a \cdot 2} \]

      6. *-commutative 47.3%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \sqrt[3]{\color{blue}{{\left(a \cdot c\right)}^{3} \cdot 64}}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{a \cdot 2} \]

      7. cbrt-prod 47.3%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\sqrt[3]{{\left(a \cdot c\right)}^{3}} \cdot \sqrt[3]{64}}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{a \cdot 2} \]

      8. unpow3 47.3%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \sqrt[3]{\color{blue}{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right) \cdot \left(a \cdot c\right)}} \cdot \sqrt[3]{64}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{a \cdot 2} \]

      9. add-cbrt-cube 47.3%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\left(a \cdot c\right)} \cdot \sqrt[3]{64}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{a \cdot 2} \]

      10. *-commutative 47.3%

          \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\left(c \cdot a\right)} \cdot \sqrt[3]{64}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(64 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{a \cdot 2} \]

   8. Applied egg-rr 47.4%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(c \cdot a\right) \cdot \sqrt[3]{64}\right)}{\left(-b\right) - \sqrt{{b}^{2} - \left(c \cdot a\right) \cdot \sqrt[3]{64}}}}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 47.4%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(c \cdot a\right) \cdot \sqrt[3]{64}\right)}{\left(-b\right) - \sqrt{{b}^{2} - \left(c \cdot a\right) \cdot \sqrt[3]{64}}}}{a \cdot 2}} \]

      2. associate-/l/ 47.3%

         \[\leadsto 1 \cdot \color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(c \cdot a\right) \cdot \sqrt[3]{64}\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \left(c \cdot a\right) \cdot \sqrt[3]{64}}\right)}} \]

      3. associate--r- 99.3%

         \[\leadsto 1 \cdot \frac{\color{blue}{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \left(c \cdot a\right) \cdot \sqrt[3]{64}}}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \left(c \cdot a\right) \cdot \sqrt[3]{64}}\right)} \]

      4. neg-mul-1 99.3%

         \[\leadsto 1 \cdot \frac{\left({\color{blue}{\left(-1 \cdot b\right)}}^{2} - {b}^{2}\right) + \left(c \cdot a\right) \cdot \sqrt[3]{64}}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \left(c \cdot a\right) \cdot \sqrt[3]{64}}\right)} \]

      5. unpow-prod-down 99.3%

         \[\leadsto 1 \cdot \frac{\left(\color{blue}{{-1}^{2} \cdot {b}^{2}} - {b}^{2}\right) + \left(c \cdot a\right) \cdot \sqrt[3]{64}}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \left(c \cdot a\right) \cdot \sqrt[3]{64}}\right)} \]

      6. metadata-eval 99.3%

         \[\leadsto 1 \cdot \frac{\left(\color{blue}{1} \cdot {b}^{2} - {b}^{2}\right) + \left(c \cdot a\right) \cdot \sqrt[3]{64}}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \left(c \cdot a\right) \cdot \sqrt[3]{64}}\right)} \]

      7. *-un-lft-identity 99.3%

         \[\leadsto 1 \cdot \frac{\left(\color{blue}{{b}^{2}} - {b}^{2}\right) + \left(c \cdot a\right) \cdot \sqrt[3]{64}}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \left(c \cdot a\right) \cdot \sqrt[3]{64}}\right)} \]

      8. associate-*l* 99.3%

         \[\leadsto 1 \cdot \frac{\left({b}^{2} - {b}^{2}\right) + \color{blue}{c \cdot \left(a \cdot \sqrt[3]{64}\right)}}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \left(c \cdot a\right) \cdot \sqrt[3]{64}}\right)} \]

      9. associate-*l* 99.3%

         \[\leadsto 1 \cdot \frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot \sqrt[3]{64}\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{c \cdot \left(a \cdot \sqrt[3]{64}\right)}}\right)} \]

   10. Applied egg-rr 99.3%

       \[\leadsto \color{blue}{1 \cdot \frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot \sqrt[3]{64}\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{64}\right)}\right)}} \]
   11. Step-by-step derivation

       1. *-lft-identity 99.3%

          \[\leadsto \color{blue}{\frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot \sqrt[3]{64}\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{64}\right)}\right)}} \]

       2. associate-/r* 99.5%

          \[\leadsto \color{blue}{\frac{\frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot \sqrt[3]{64}\right)}{a \cdot 2}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{64}\right)}}} \]

       3. +-inverses 99.5%

          \[\leadsto \frac{\frac{\color{blue}{0} + c \cdot \left(a \cdot \sqrt[3]{64}\right)}{a \cdot 2}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{64}\right)}} \]

       4. +-lft-identity 99.5%

          \[\leadsto \frac{\frac{\color{blue}{c \cdot \left(a \cdot \sqrt[3]{64}\right)}}{a \cdot 2}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{64}\right)}} \]

       5. associate-*r* 99.5%

          \[\leadsto \frac{\frac{\color{blue}{\left(c \cdot a\right) \cdot \sqrt[3]{64}}}{a \cdot 2}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{64}\right)}} \]

       6. *-commutative 99.5%

          \[\leadsto \frac{\frac{\color{blue}{\left(a \cdot c\right)} \cdot \sqrt[3]{64}}{a \cdot 2}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{64}\right)}} \]

       7. associate-*r* 99.5%

          \[\leadsto \frac{\frac{\color{blue}{a \cdot \left(c \cdot \sqrt[3]{64}\right)}}{a \cdot 2}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{64}\right)}} \]

       8. associate-*r* 99.5%

          \[\leadsto \frac{\frac{a \cdot \left(c \cdot \sqrt[3]{64}\right)}{a \cdot 2}}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(c \cdot a\right) \cdot \sqrt[3]{64}}}} \]

       9. *-commutative 99.5%

          \[\leadsto \frac{\frac{a \cdot \left(c \cdot \sqrt[3]{64}\right)}{a \cdot 2}}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right)} \cdot \sqrt[3]{64}}} \]

       10. associate-*r* 99.5%

           \[\leadsto \frac{\frac{a \cdot \left(c \cdot \sqrt[3]{64}\right)}{a \cdot 2}}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{a \cdot \left(c \cdot \sqrt[3]{64}\right)}}} \]

   12. Simplified 99.5%

       \[\leadsto \color{blue}{\frac{\frac{a \cdot \left(c \cdot \sqrt[3]{64}\right)}{a \cdot 2}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot \sqrt[3]{64}\right)}}} \]

   13. Taylor expanded in a around 0 89.8%

       \[\leadsto \frac{\frac{a \cdot \left(c \cdot \sqrt[3]{64}\right)}{a \cdot 2}}{\color{blue}{0.5 \cdot \frac{a \cdot \left(c \cdot \sqrt[3]{64}\right)}{b} - 2 \cdot b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.002648:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(c \cdot \sqrt[3]{64}\right) \cdot a}{2 \cdot a}}{0.5 \cdot \frac{\left(c \cdot \sqrt[3]{64}\right) \cdot a}{b} - b \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.002648:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(-{\left(\frac{c}{b}\right)}^{2}\right) - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a)) -0.002648)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* 2.0 a))
   (/ (- (* a (- (pow (/ c b) 2.0))) c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)) <= -0.002648) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (2.0 * a);
	} else {
		tmp = ((a * -pow((c / b), 2.0)) - c) / b;
	}
	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)) <= -0.002648)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(a * Float64(-(Float64(c / b) ^ 2.0))) - c) / b);
	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], -0.002648], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * (-N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision] - c), $MachinePrecision] / b), $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)) < -0.0026480000000000002

   1. Initial program 80.6%

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

      1. *-commutative 80.6%

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

      2. +-commutative 80.6%

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

      3. sqr-neg 80.6%

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

      4. unsub-neg 80.6%

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

      5. sqr-neg 80.6%

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

      6. fma-neg 80.9%

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

      7. distribute-lft-neg-in 80.9%

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

      8. *-commutative 80.9%

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

      9. *-commutative 80.9%

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

      10. distribute-rgt-neg-in 80.9%

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

      11. metadata-eval 80.9%

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

   3. Simplified 80.9%

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

   if -0.0026480000000000002 < (/.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 45.9%

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

      1. *-commutative 45.9%

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

   3. Simplified 45.9%

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

   5. Taylor expanded in a around 0 89.4%

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

      1. distribute-lft-out 89.4%

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

      2. associate-/l* 89.4%

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

   7. Simplified 89.4%

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

   8. Taylor expanded in b around inf 89.5%

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

      1. neg-mul-1 89.5%

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

      2. +-commutative 89.5%

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

      3. unsub-neg 89.5%

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

      4. mul-1-neg 89.5%

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

      5. associate-/l* 89.5%

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

      6. distribute-lft-neg-in 89.5%

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

      7. unpow2 89.5%

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

      8. unpow2 89.5%

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

      9. times-frac 89.5%

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

      10. unpow1 89.5%

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

      11. pow-plus 89.5%

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

      12. metadata-eval 89.5%

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

   10. Simplified 89.5%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.002648:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(-{\left(\frac{c}{b}\right)}^{2}\right) - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{if}\;t\_0 \leq -0.002648:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(-{\left(\frac{c}{b}\right)}^{2}\right) - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a))))
   (if (<= t_0 -0.002648) t_0 (/ (- (* a (- (pow (/ c b) 2.0))) c) 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 <= -0.002648) {
		tmp = t_0;
	} else {
		tmp = ((a * -pow((c / b), 2.0)) - c) / 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 <= (-0.002648d0)) then
        tmp = t_0
    else
        tmp = ((a * -((c / b) ** 2.0d0)) - c) / 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 <= -0.002648) {
		tmp = t_0;
	} else {
		tmp = ((a * -Math.pow((c / b), 2.0)) - c) / 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 <= -0.002648:
		tmp = t_0
	else:
		tmp = ((a * -math.pow((c / b), 2.0)) - c) / 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 <= -0.002648)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(a * Float64(-(Float64(c / b) ^ 2.0))) - c) / 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 <= -0.002648)
		tmp = t_0;
	else
		tmp = ((a * -((c / b) ^ 2.0)) - c) / 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, -0.002648], t$95$0, N[(N[(N[(a * (-N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision] - c), $MachinePrecision] / b), $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)) < -0.0026480000000000002

   1. Initial program 80.6%

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

   if -0.0026480000000000002 < (/.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 45.9%

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

      1. *-commutative 45.9%

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

   3. Simplified 45.9%

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

   5. Taylor expanded in a around 0 89.4%

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

      1. distribute-lft-out 89.4%

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

      2. associate-/l* 89.4%

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

   7. Simplified 89.4%

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

   8. Taylor expanded in b around inf 89.5%

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

      1. neg-mul-1 89.5%

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

      2. +-commutative 89.5%

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

      3. unsub-neg 89.5%

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

      4. mul-1-neg 89.5%

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

      5. associate-/l* 89.5%

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

      6. distribute-lft-neg-in 89.5%

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

      7. unpow2 89.5%

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

      8. unpow2 89.5%

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

      9. times-frac 89.5%

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

      10. unpow1 89.5%

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

      11. pow-plus 89.5%

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

      12. metadata-eval 89.5%

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

   10. Simplified 89.5%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.002648:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(-{\left(\frac{c}{b}\right)}^{2}\right) - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.3%

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

   1. *-commutative 57.3%

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

3. Simplified 57.3%

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

5. Taylor expanded in a around 0 79.9%

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

   1. distribute-lft-out 79.9%

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

   2. associate-/l* 79.9%

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

7. Simplified 79.9%

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

8. Taylor expanded in b around inf 80.0%

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

   1. neg-mul-1 80.0%

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

   2. +-commutative 80.0%

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

   3. unsub-neg 80.0%

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

   4. mul-1-neg 80.0%

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

   5. associate-/l* 80.0%

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

   6. distribute-lft-neg-in 80.0%

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

   7. unpow2 80.0%

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

   8. unpow2 80.0%

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

   9. times-frac 80.0%

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

   10. unpow1 80.0%

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

   11. pow-plus 80.0%

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

   12. metadata-eval 80.0%

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

10. Simplified 80.0%

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

11. Final simplification 80.0%

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

Alternative 8: 81.3% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.3%

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

   1. *-commutative 57.3%

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

3. Simplified 57.3%

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

5. Taylor expanded in a around 0 79.9%

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

   1. distribute-lft-out 79.9%

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

   2. associate-/l* 79.9%

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

7. Simplified 79.9%

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

8. Taylor expanded in b around inf 79.9%

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

   1. distribute-lft-out 79.9%

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

   2. associate-/l* 79.9%

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

   3. unpow2 79.9%

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

   4. unpow2 79.9%

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

   5. times-frac 79.9%

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

   6. unpow1 79.9%

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

   7. pow-plus 79.9%

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

   8. metadata-eval 79.9%

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

10. Simplified 79.9%

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

    1. unpow2 79.9%

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

12. Applied egg-rr 79.9%

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

13. Final simplification 79.9%

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

Alternative 9: 64.2% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(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 57.3%

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

   1. *-commutative 57.3%

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

3. Simplified 57.3%

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

5. Taylor expanded in b around inf 62.2%

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

   1. associate-*r/ 62.2%

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

   2. mul-1-neg 62.2%

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

7. Simplified 62.2%

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

8. Final simplification 62.2%

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

Reproduce

herbie shell --seed 2024110 
(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)))