Cubic critical, medium range

Percentage Accurate: 31.3% → 99.6%

Time: 14.4s

Alternatives: 6

Speedup: 23.2×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

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

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

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

Initial Program: 31.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

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

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

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

Alternative 1: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{c \cdot a}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -3, {b}^{2}\right)}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (/ (* c a) a) (- (- b) (sqrt (fma (* c a) -3.0 (pow b 2.0))))))

C

double code(double a, double b, double c) {
	return ((c * a) / a) / (-b - sqrt(fma((c * a), -3.0, pow(b, 2.0))));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 31.1%

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

   1. sqr-neg 31.1%

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

   2. sqr-neg 31.1%

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

   3. associate-*l* 31.1%

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

3. Simplified 31.1%

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

   1. expm1-log1p-u 31.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]

   2. expm1-undefine 25.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(e^{\mathsf{log1p}\left(3 \cdot \left(a \cdot c\right)\right)} - 1\right)}}}{3 \cdot a} \]

   3. *-commutative 25.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(\color{blue}{\left(a \cdot c\right) \cdot 3}\right)} - 1\right)}}{3 \cdot a} \]

   4. *-commutative 25.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(\color{blue}{\left(c \cdot a\right)} \cdot 3\right)} - 1\right)}}{3 \cdot a} \]

   5. associate-*l* 25.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(\color{blue}{c \cdot \left(a \cdot 3\right)}\right)} - 1\right)}}{3 \cdot a} \]

   6. *-commutative 25.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \color{blue}{\left(3 \cdot a\right)}\right)} - 1\right)}}{3 \cdot a} \]

6. Applied egg-rr 25.1%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}}{3 \cdot a} \]
7. Step-by-step derivation

   1. flip-+ 25.1%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)} \cdot \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}}}{3 \cdot a} \]

   2. pow2 25.1%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)} \cdot \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}}{3 \cdot a} \]

   3. add-sqr-sqrt 26.0%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}}{3 \cdot a} \]

   4. pow2 26.0%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}}{3 \cdot a} \]

   5. expm1-define 30.5%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}}{3 \cdot a} \]

   6. expm1-log1p-u 30.5%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{c \cdot \left(3 \cdot a\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}}{3 \cdot a} \]

   7. *-commutative 30.5%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \color{blue}{\left(a \cdot 3\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}}{3 \cdot a} \]

   8. pow2 30.5%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 3\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2}} - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}}{3 \cdot a} \]

   9. expm1-define 32.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 3\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)\right)}}}}{3 \cdot a} \]

   10. expm1-log1p-u 32.3%

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

   11. *-commutative 32.3%

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

8. Applied egg-rr 32.3%

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

   1. associate--r- 99.3%

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

   2. associate-*r* 99.1%

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

   3. *-commutative 99.1%

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

   4. associate-*r* 99.1%

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

   5. *-commutative 99.1%

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

10. Simplified 99.1%

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

    1. div-inv 99.1%

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

    2. +-commutative 99.1%

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

    3. associate-*l* 99.1%

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

    4. fma-define 99.1%

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

    5. neg-mul-1 99.1%

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

    6. metadata-eval 99.1%

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

    7. unpow-prod-down 99.1%

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

    8. metadata-eval 99.1%

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

    9. metadata-eval 99.1%

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

    10. *-un-lft-identity 99.1%

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

    11. associate-*l* 99.1%

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

12. Applied egg-rr 99.1%

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

    1. associate-*r/ 99.1%

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

    2. fma-undefine 99.1%

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

    3. +-inverses 99.1%

       \[\leadsto \frac{\frac{\left(a \cdot \left(c \cdot 3\right) + \color{blue}{0}\right) \cdot 1}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}}}{3 \cdot a} \]

    4. +-rgt-identity 99.1%

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

    5. *-rgt-identity 99.1%

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

    6. *-commutative 99.1%

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

    7. associate-*l* 99.3%

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

    8. *-commutative 99.3%

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

    9. associate-*r* 99.3%

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

    10. *-commutative 99.3%

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

    11. cancel-sign-sub-inv 99.3%

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

    12. metadata-eval 99.3%

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

    13. +-commutative 99.3%

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

    14. fma-define 99.3%

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

    15. *-commutative 99.3%

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

14. Simplified 99.3%

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

    1. div-inv 99.3%

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

    2. associate-/l* 99.2%

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

    3. *-commutative 99.2%

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

16. Applied egg-rr 99.2%

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

    1. *-commutative 99.2%

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

    2. associate-*r/ 99.3%

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

    3. times-frac 99.3%

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

    4. *-lft-identity 99.3%

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

    5. associate-/r* 99.6%

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

    6. associate-*r* 99.2%

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

    7. *-commutative 99.2%

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

    8. *-commutative 99.2%

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

    9. *-commutative 99.2%

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

    10. times-frac 99.6%

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

    11. metadata-eval 99.6%

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

    12. *-commutative 99.6%

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

    13. fma-define 99.6%

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

    14. *-commutative 99.6%

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

    15. *-commutative 99.6%

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

    16. fma-define 99.6%

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

18. Simplified 99.6%

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

19. Final simplification 99.6%

    \[\leadsto \frac{\frac{c \cdot a}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -3, {b}^{2}\right)}} \]
20. Add Preprocessing

Alternative 2: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return ((c * (a * 3.0)) / (-b - sqrt((c * ((a * -3.0) + (pow(b, 2.0) / c)))))) / (a * 3.0);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((c * (a * 3.0d0)) / (-b - sqrt((c * ((a * (-3.0d0)) + ((b ** 2.0d0) / c)))))) / (a * 3.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.1%

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

   1. sqr-neg 31.1%

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

   2. sqr-neg 31.1%

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

   3. associate-*l* 31.1%

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

3. Simplified 31.1%

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

   1. expm1-log1p-u 31.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]

   2. expm1-undefine 25.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(e^{\mathsf{log1p}\left(3 \cdot \left(a \cdot c\right)\right)} - 1\right)}}}{3 \cdot a} \]

   3. *-commutative 25.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(\color{blue}{\left(a \cdot c\right) \cdot 3}\right)} - 1\right)}}{3 \cdot a} \]

   4. *-commutative 25.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(\color{blue}{\left(c \cdot a\right)} \cdot 3\right)} - 1\right)}}{3 \cdot a} \]

   5. associate-*l* 25.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(\color{blue}{c \cdot \left(a \cdot 3\right)}\right)} - 1\right)}}{3 \cdot a} \]

   6. *-commutative 25.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \color{blue}{\left(3 \cdot a\right)}\right)} - 1\right)}}{3 \cdot a} \]

6. Applied egg-rr 25.1%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}}{3 \cdot a} \]
7. Step-by-step derivation

   1. flip-+ 25.1%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)} \cdot \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}}}{3 \cdot a} \]

   2. pow2 25.1%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)} \cdot \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}}{3 \cdot a} \]

   3. add-sqr-sqrt 26.0%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}}{3 \cdot a} \]

   4. pow2 26.0%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}}{3 \cdot a} \]

   5. expm1-define 30.5%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}}{3 \cdot a} \]

   6. expm1-log1p-u 30.5%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{c \cdot \left(3 \cdot a\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}}{3 \cdot a} \]

   7. *-commutative 30.5%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \color{blue}{\left(a \cdot 3\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}}{3 \cdot a} \]

   8. pow2 30.5%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 3\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2}} - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}}{3 \cdot a} \]

   9. expm1-define 32.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 3\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)\right)}}}}{3 \cdot a} \]

   10. expm1-log1p-u 32.3%

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

   11. *-commutative 32.3%

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

8. Applied egg-rr 32.3%

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

   1. associate--r- 99.3%

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

   2. associate-*r* 99.1%

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

   3. *-commutative 99.1%

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

   4. associate-*r* 99.1%

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

   5. *-commutative 99.1%

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

10. Simplified 99.1%

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

    1. div-inv 99.1%

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

    2. +-commutative 99.1%

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

    3. associate-*l* 99.1%

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

    4. fma-define 99.1%

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

    5. neg-mul-1 99.1%

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

    6. metadata-eval 99.1%

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

    7. unpow-prod-down 99.1%

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

    8. metadata-eval 99.1%

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

    9. metadata-eval 99.1%

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

    10. *-un-lft-identity 99.1%

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

    11. associate-*l* 99.1%

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

12. Applied egg-rr 99.1%

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

    1. associate-*r/ 99.1%

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

    2. fma-undefine 99.1%

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

    3. +-inverses 99.1%

       \[\leadsto \frac{\frac{\left(a \cdot \left(c \cdot 3\right) + \color{blue}{0}\right) \cdot 1}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}}}{3 \cdot a} \]

    4. +-rgt-identity 99.1%

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

    5. *-rgt-identity 99.1%

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

    6. *-commutative 99.1%

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

    7. associate-*l* 99.3%

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

    8. *-commutative 99.3%

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

    9. associate-*r* 99.3%

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

    10. *-commutative 99.3%

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

    11. cancel-sign-sub-inv 99.3%

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

    12. metadata-eval 99.3%

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

    13. +-commutative 99.3%

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

    14. fma-define 99.3%

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

    15. *-commutative 99.3%

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

14. Simplified 99.3%

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

15. Taylor expanded in c around inf 99.3%

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

16. Final simplification 99.3%

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

Alternative 3: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return ((c * (a * 3.0)) / (-b - sqrt((pow(b, 2.0) + ((c * a) * -3.0))))) / (a * 3.0);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((c * (a * 3.0d0)) / (-b - sqrt(((b ** 2.0d0) + ((c * a) * (-3.0d0)))))) / (a * 3.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.1%

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

   1. sqr-neg 31.1%

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

   2. sqr-neg 31.1%

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

   3. associate-*l* 31.1%

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

3. Simplified 31.1%

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

   1. expm1-log1p-u 31.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]

   2. expm1-undefine 25.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(e^{\mathsf{log1p}\left(3 \cdot \left(a \cdot c\right)\right)} - 1\right)}}}{3 \cdot a} \]

   3. *-commutative 25.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(\color{blue}{\left(a \cdot c\right) \cdot 3}\right)} - 1\right)}}{3 \cdot a} \]

   4. *-commutative 25.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(\color{blue}{\left(c \cdot a\right)} \cdot 3\right)} - 1\right)}}{3 \cdot a} \]

   5. associate-*l* 25.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(\color{blue}{c \cdot \left(a \cdot 3\right)}\right)} - 1\right)}}{3 \cdot a} \]

   6. *-commutative 25.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \color{blue}{\left(3 \cdot a\right)}\right)} - 1\right)}}{3 \cdot a} \]

6. Applied egg-rr 25.1%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}}{3 \cdot a} \]
7. Step-by-step derivation

   1. flip-+ 25.1%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)} \cdot \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}}}{3 \cdot a} \]

   2. pow2 25.1%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)} \cdot \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}}{3 \cdot a} \]

   3. add-sqr-sqrt 26.0%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}}{3 \cdot a} \]

   4. pow2 26.0%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}}{3 \cdot a} \]

   5. expm1-define 30.5%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}}{3 \cdot a} \]

   6. expm1-log1p-u 30.5%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{c \cdot \left(3 \cdot a\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}}{3 \cdot a} \]

   7. *-commutative 30.5%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \color{blue}{\left(a \cdot 3\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}}{3 \cdot a} \]

   8. pow2 30.5%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 3\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2}} - \left(e^{\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)} - 1\right)}}}{3 \cdot a} \]

   9. expm1-define 32.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 3\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot \left(3 \cdot a\right)\right)\right)}}}}{3 \cdot a} \]

   10. expm1-log1p-u 32.3%

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

   11. *-commutative 32.3%

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

8. Applied egg-rr 32.3%

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

   1. associate--r- 99.3%

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

   2. associate-*r* 99.1%

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

   3. *-commutative 99.1%

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

   4. associate-*r* 99.1%

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

   5. *-commutative 99.1%

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

10. Simplified 99.1%

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

    1. div-inv 99.1%

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

    2. +-commutative 99.1%

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

    3. associate-*l* 99.1%

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

    4. fma-define 99.1%

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

    5. neg-mul-1 99.1%

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

    6. metadata-eval 99.1%

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

    7. unpow-prod-down 99.1%

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

    8. metadata-eval 99.1%

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

    9. metadata-eval 99.1%

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

    10. *-un-lft-identity 99.1%

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

    11. associate-*l* 99.1%

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

12. Applied egg-rr 99.1%

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

    1. associate-*r/ 99.1%

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

    2. fma-undefine 99.1%

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

    3. +-inverses 99.1%

       \[\leadsto \frac{\frac{\left(a \cdot \left(c \cdot 3\right) + \color{blue}{0}\right) \cdot 1}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}}}{3 \cdot a} \]

    4. +-rgt-identity 99.1%

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

    5. *-rgt-identity 99.1%

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

    6. *-commutative 99.1%

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

    7. associate-*l* 99.3%

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

    8. *-commutative 99.3%

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

    9. associate-*r* 99.3%

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

    10. *-commutative 99.3%

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

    11. cancel-sign-sub-inv 99.3%

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

    12. metadata-eval 99.3%

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

    13. +-commutative 99.3%

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

    14. fma-define 99.3%

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

    15. *-commutative 99.3%

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

14. Simplified 99.3%

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

    1. fma-undefine 99.3%

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

16. Applied egg-rr 99.3%

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

17. Final simplification 99.3%

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

Alternative 4: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0)));
}

Python

def code(a, b, c):
	return (-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0)))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0)));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.1%

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

   1. sqr-neg 31.1%

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

   2. sqr-neg 31.1%

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

   3. associate-*l* 31.1%

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

3. Simplified 31.1%

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

5. Taylor expanded in a around 0 91.9%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

6. Final simplification 91.9%

   \[\leadsto -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} \]
7. Add Preprocessing

Alternative 5: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(-0.375 \cdot \frac{c \cdot a}{{b}^{3}} - \frac{0.5}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (- (* -0.375 (/ (* c a) (pow b 3.0))) (/ 0.5 b))))

C

double code(double a, double b, double c) {
	return c * ((-0.375 * ((c * a) / pow(b, 3.0))) - (0.5 / 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 * (((-0.375d0) * ((c * a) / (b ** 3.0d0))) - (0.5d0 / b))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((-0.375 * ((c * a) / Math.pow(b, 3.0))) - (0.5 / b));
}

Python

def code(a, b, c):
	return c * ((-0.375 * ((c * a) / math.pow(b, 3.0))) - (0.5 / b))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(-0.375 * Float64(Float64(c * a) / (b ^ 3.0))) - Float64(0.5 / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((-0.375 * ((c * a) / (b ^ 3.0))) - (0.5 / b));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(-0.375 * N[(N[(c * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.1%

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

   1. sqr-neg 31.1%

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

   2. sqr-neg 31.1%

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

   3. associate-*l* 31.1%

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

3. Simplified 31.1%

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

5. Taylor expanded in a around 0 91.9%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

6. Taylor expanded in c around 0 91.7%

   \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]
7. Step-by-step derivation

   1. associate-*r/ 91.7%

      \[\leadsto c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - \color{blue}{\frac{0.5 \cdot 1}{b}}\right) \]

   2. metadata-eval 91.7%

      \[\leadsto c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{\color{blue}{0.5}}{b}\right) \]

8. Simplified 91.7%

   \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{0.5}{b}\right)} \]

9. Final simplification 91.7%

   \[\leadsto c \cdot \left(-0.375 \cdot \frac{c \cdot a}{{b}^{3}} - \frac{0.5}{b}\right) \]
10. Add Preprocessing

Alternative 6: 81.3% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot -0.5}{b} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.1%

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

   1. sqr-neg 31.1%

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

   2. sqr-neg 31.1%

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

   3. associate-*l* 31.1%

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

3. Simplified 31.1%

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

5. Taylor expanded in b around inf 81.8%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
6. Step-by-step derivation

   1. associate-*r/ 81.8%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   2. *-commutative 81.8%

      \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b} \]

7. Simplified 81.8%

   \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]

8. Final simplification 81.8%

   \[\leadsto \frac{c \cdot -0.5}{b} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024080 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))