Cubic critical, medium range

Percentage Accurate: 32.2% → 99.4%

Time: 15.4s

Alternatives: 11

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: 32.2% 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: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot {27}^{0.3333333333333333}\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}}}{a \cdot 3} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/
   (+ (- (pow b 2.0) (pow b 2.0)) (* c (* a (pow 27.0 0.3333333333333333))))
   (- (- b) (sqrt (- (pow b 2.0) (* c (* a (cbrt 27.0)))))))
  (* a 3.0)))

C

double code(double a, double b, double c) {
	return (((pow(b, 2.0) - pow(b, 2.0)) + (c * (a * pow(27.0, 0.3333333333333333)))) / (-b - sqrt((pow(b, 2.0) - (c * (a * cbrt(27.0))))))) / (a * 3.0);
}

Java

public static double code(double a, double b, double c) {
	return (((Math.pow(b, 2.0) - Math.pow(b, 2.0)) + (c * (a * Math.pow(27.0, 0.3333333333333333)))) / (-b - Math.sqrt((Math.pow(b, 2.0) - (c * (a * Math.cbrt(27.0))))))) / (a * 3.0);
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64((b ^ 2.0) - (b ^ 2.0)) + Float64(c * Float64(a * (27.0 ^ 0.3333333333333333)))) / Float64(Float64(-b) - sqrt(Float64((b ^ 2.0) - Float64(c * Float64(a * cbrt(27.0))))))) / Float64(a * 3.0))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(c * N[(a * N[Power[27.0, 0.3333333333333333], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] - N[(c * N[(a * N[Power[27.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.5%

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

   1. add-cbrt-cube 32.5%

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

   2. pow1/3 32.4%

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

   3. pow3 32.4%

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

   4. associate-*l* 32.4%

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

   5. unpow-prod-down 32.4%

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

   6. metadata-eval 32.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\color{blue}{27} \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}{3 \cdot a} \]

4. Applied egg-rr 32.4%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]
5. Step-by-step derivation

   1. flip-+ 32.3%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}} \cdot \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}}{3 \cdot a} \]

   2. pow2 32.3%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}} \cdot \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   3. add-sqr-sqrt 33.0%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}\right)}}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   4. pow2 33.0%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   5. unpow1/3 33.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\sqrt[3]{27 \cdot {\left(a \cdot c\right)}^{3}}}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   6. *-commutative 33.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \sqrt[3]{\color{blue}{{\left(a \cdot c\right)}^{3} \cdot 27}}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   7. cbrt-prod 33.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\sqrt[3]{{\left(a \cdot c\right)}^{3}} \cdot \sqrt[3]{27}}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   8. unpow3 33.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \sqrt[3]{\color{blue}{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right) \cdot \left(a \cdot c\right)}} \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   9. add-cbrt-cube 33.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\left(a \cdot c\right)} \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

6. Applied egg-rr 33.1%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}}{3 \cdot a} \]
7. Step-by-step derivation

   1. associate--r- 98.6%

      \[\leadsto \frac{\frac{\color{blue}{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \left(a \cdot c\right) \cdot \sqrt[3]{27}}}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   2. unpow2 98.6%

      \[\leadsto \frac{\frac{\left(\color{blue}{\left(-b\right) \cdot \left(-b\right)} - {b}^{2}\right) + \left(a \cdot c\right) \cdot \sqrt[3]{27}}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   3. sqr-neg 98.6%

      \[\leadsto \frac{\frac{\left(\color{blue}{b \cdot b} - {b}^{2}\right) + \left(a \cdot c\right) \cdot \sqrt[3]{27}}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   4. unpow2 98.6%

      \[\leadsto \frac{\frac{\left(\color{blue}{{b}^{2}} - {b}^{2}\right) + \left(a \cdot c\right) \cdot \sqrt[3]{27}}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   5. *-commutative 98.6%

      \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + \color{blue}{\left(c \cdot a\right)} \cdot \sqrt[3]{27}}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   6. associate-*l* 98.6%

      \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + \color{blue}{c \cdot \left(a \cdot \sqrt[3]{27}\right)}}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   7. *-commutative 98.6%

      \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(c \cdot a\right)} \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   8. associate-*l* 98.6%

      \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{c \cdot \left(a \cdot \sqrt[3]{27}\right)}}}}{3 \cdot a} \]

8. Simplified 98.6%

   \[\leadsto \frac{\color{blue}{\frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}}}}{3 \cdot a} \]
9. Step-by-step derivation

   1. pow1/3 99.4%

      \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot \color{blue}{{27}^{0.3333333333333333}}\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}}}{3 \cdot a} \]

10. Applied egg-rr 99.4%

    \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot \color{blue}{{27}^{0.3333333333333333}}\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}}}{3 \cdot a} \]

11. Final simplification 99.4%

    \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot {27}^{0.3333333333333333}\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}}}{a \cdot 3} \]
12. Add Preprocessing

Alternative 2: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left({b}^{2} - {b}^{2}\right) + 3 \cdot \left(c \cdot a\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}}}{a \cdot 3} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/
   (+ (- (pow b 2.0) (pow b 2.0)) (* 3.0 (* c a)))
   (- (- b) (sqrt (- (pow b 2.0) (* c (* a (cbrt 27.0)))))))
  (* a 3.0)))

C

double code(double a, double b, double c) {
	return (((pow(b, 2.0) - pow(b, 2.0)) + (3.0 * (c * a))) / (-b - sqrt((pow(b, 2.0) - (c * (a * cbrt(27.0))))))) / (a * 3.0);
}

Java

public static double code(double a, double b, double c) {
	return (((Math.pow(b, 2.0) - Math.pow(b, 2.0)) + (3.0 * (c * a))) / (-b - Math.sqrt((Math.pow(b, 2.0) - (c * (a * Math.cbrt(27.0))))))) / (a * 3.0);
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64((b ^ 2.0) - (b ^ 2.0)) + Float64(3.0 * Float64(c * a))) / Float64(Float64(-b) - sqrt(Float64((b ^ 2.0) - Float64(c * Float64(a * cbrt(27.0))))))) / Float64(a * 3.0))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] - N[(c * N[(a * N[Power[27.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.5%

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

   1. add-cbrt-cube 32.5%

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

   2. pow1/3 32.4%

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

   3. pow3 32.4%

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

   4. associate-*l* 32.4%

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

   5. unpow-prod-down 32.4%

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

   6. metadata-eval 32.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\color{blue}{27} \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}{3 \cdot a} \]

4. Applied egg-rr 32.4%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]
5. Step-by-step derivation

   1. flip-+ 32.3%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}} \cdot \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}}{3 \cdot a} \]

   2. pow2 32.3%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}} \cdot \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   3. add-sqr-sqrt 33.0%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}\right)}}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   4. pow2 33.0%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   5. unpow1/3 33.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\sqrt[3]{27 \cdot {\left(a \cdot c\right)}^{3}}}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   6. *-commutative 33.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \sqrt[3]{\color{blue}{{\left(a \cdot c\right)}^{3} \cdot 27}}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   7. cbrt-prod 33.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\sqrt[3]{{\left(a \cdot c\right)}^{3}} \cdot \sqrt[3]{27}}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   8. unpow3 33.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \sqrt[3]{\color{blue}{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right) \cdot \left(a \cdot c\right)}} \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   9. add-cbrt-cube 33.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\left(a \cdot c\right)} \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

6. Applied egg-rr 33.1%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}}{3 \cdot a} \]
7. Step-by-step derivation

   1. associate--r- 98.6%

      \[\leadsto \frac{\frac{\color{blue}{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \left(a \cdot c\right) \cdot \sqrt[3]{27}}}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   2. unpow2 98.6%

      \[\leadsto \frac{\frac{\left(\color{blue}{\left(-b\right) \cdot \left(-b\right)} - {b}^{2}\right) + \left(a \cdot c\right) \cdot \sqrt[3]{27}}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   3. sqr-neg 98.6%

      \[\leadsto \frac{\frac{\left(\color{blue}{b \cdot b} - {b}^{2}\right) + \left(a \cdot c\right) \cdot \sqrt[3]{27}}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   4. unpow2 98.6%

      \[\leadsto \frac{\frac{\left(\color{blue}{{b}^{2}} - {b}^{2}\right) + \left(a \cdot c\right) \cdot \sqrt[3]{27}}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   5. *-commutative 98.6%

      \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + \color{blue}{\left(c \cdot a\right)} \cdot \sqrt[3]{27}}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   6. associate-*l* 98.6%

      \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + \color{blue}{c \cdot \left(a \cdot \sqrt[3]{27}\right)}}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   7. *-commutative 98.6%

      \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(c \cdot a\right)} \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   8. associate-*l* 98.6%

      \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{c \cdot \left(a \cdot \sqrt[3]{27}\right)}}}}{3 \cdot a} \]

8. Simplified 98.6%

   \[\leadsto \frac{\color{blue}{\frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}}}}{3 \cdot a} \]

9. Taylor expanded in c around 0 99.2%

   \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + \color{blue}{3 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}}}{3 \cdot a} \]
10. Step-by-step derivation

    1. *-commutative 99.2%

       \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + 3 \cdot \color{blue}{\left(c \cdot a\right)}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}}}{3 \cdot a} \]

11. Simplified 99.2%

    \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + \color{blue}{3 \cdot \left(c \cdot a\right)}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}}}{3 \cdot a} \]

12. Final simplification 99.2%

    \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + 3 \cdot \left(c \cdot a\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}}}{a \cdot 3} \]
13. Add Preprocessing

Alternative 3: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot \sqrt[3]{27}\right)\\ \frac{t\_0}{\left(\left(-b\right) - \sqrt{{b}^{2} - t\_0}\right) \cdot \left(a \cdot 3\right)} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a (cbrt 27.0)))))
   (/ t_0 (* (- (- b) (sqrt (- (pow b 2.0) t_0))) (* a 3.0)))))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * cbrt(27.0));
	return t_0 / ((-b - sqrt((pow(b, 2.0) - t_0))) * (a * 3.0));
}

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (a * Math.cbrt(27.0));
	return t_0 / ((-b - Math.sqrt((Math.pow(b, 2.0) - t_0))) * (a * 3.0));
}

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * cbrt(27.0)))
	return Float64(t_0 / Float64(Float64(Float64(-b) - sqrt(Float64((b ^ 2.0) - t_0))) * Float64(a * 3.0)))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * N[Power[27.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 / N[(N[((-b) - N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 32.5%

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

   1. add-cbrt-cube 32.5%

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

   2. pow1/3 32.4%

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

   3. pow3 32.4%

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

   4. associate-*l* 32.4%

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

   5. unpow-prod-down 32.4%

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

   6. metadata-eval 32.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\color{blue}{27} \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}{3 \cdot a} \]

4. Applied egg-rr 32.4%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]
5. Step-by-step derivation

   1. flip-+ 32.3%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}} \cdot \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}}{3 \cdot a} \]

   2. pow2 32.3%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}} \cdot \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   3. add-sqr-sqrt 33.0%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}\right)}}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   4. pow2 33.0%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   5. unpow1/3 33.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\sqrt[3]{27 \cdot {\left(a \cdot c\right)}^{3}}}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   6. *-commutative 33.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \sqrt[3]{\color{blue}{{\left(a \cdot c\right)}^{3} \cdot 27}}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   7. cbrt-prod 33.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\sqrt[3]{{\left(a \cdot c\right)}^{3}} \cdot \sqrt[3]{27}}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   8. unpow3 33.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \sqrt[3]{\color{blue}{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right) \cdot \left(a \cdot c\right)}} \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   9. add-cbrt-cube 33.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\left(a \cdot c\right)} \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

6. Applied egg-rr 33.1%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}}{3 \cdot a} \]
7. Step-by-step derivation

   1. associate--r- 98.6%

      \[\leadsto \frac{\frac{\color{blue}{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \left(a \cdot c\right) \cdot \sqrt[3]{27}}}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   2. unpow2 98.6%

      \[\leadsto \frac{\frac{\left(\color{blue}{\left(-b\right) \cdot \left(-b\right)} - {b}^{2}\right) + \left(a \cdot c\right) \cdot \sqrt[3]{27}}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   3. sqr-neg 98.6%

      \[\leadsto \frac{\frac{\left(\color{blue}{b \cdot b} - {b}^{2}\right) + \left(a \cdot c\right) \cdot \sqrt[3]{27}}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   4. unpow2 98.6%

      \[\leadsto \frac{\frac{\left(\color{blue}{{b}^{2}} - {b}^{2}\right) + \left(a \cdot c\right) \cdot \sqrt[3]{27}}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   5. *-commutative 98.6%

      \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + \color{blue}{\left(c \cdot a\right)} \cdot \sqrt[3]{27}}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   6. associate-*l* 98.6%

      \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + \color{blue}{c \cdot \left(a \cdot \sqrt[3]{27}\right)}}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   7. *-commutative 98.6%

      \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(c \cdot a\right)} \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   8. associate-*l* 98.6%

      \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{c \cdot \left(a \cdot \sqrt[3]{27}\right)}}}}{3 \cdot a} \]

8. Simplified 98.6%

   \[\leadsto \frac{\color{blue}{\frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}}}}{3 \cdot a} \]
9. Step-by-step derivation

   1. pow1/3 99.4%

      \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot \color{blue}{{27}^{0.3333333333333333}}\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}}}{3 \cdot a} \]

10. Applied egg-rr 99.4%

    \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot \color{blue}{{27}^{0.3333333333333333}}\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}}}{3 \cdot a} \]
11. Step-by-step derivation

    1. expm1-log1p-u 85.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot {27}^{0.3333333333333333}\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}}}{3 \cdot a}\right)\right)} \]

    2. expm1-udef 36.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot {27}^{0.3333333333333333}\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}}}{3 \cdot a}\right)} - 1} \]

12. Applied egg-rr 36.6%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(c, a \cdot \sqrt[3]{27}, 0\right)}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}\right)}\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 84.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(c, a \cdot \sqrt[3]{27}, 0\right)}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}\right)}\right)\right)} \]

    2. expm1-log1p 98.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c, a \cdot \sqrt[3]{27}, 0\right)}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}\right)}} \]

    3. fma-udef 98.6%

       \[\leadsto \frac{\color{blue}{c \cdot \left(a \cdot \sqrt[3]{27}\right) + 0}}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}\right)} \]

    4. +-rgt-identity 98.6%

       \[\leadsto \frac{\color{blue}{c \cdot \left(a \cdot \sqrt[3]{27}\right)}}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}\right)} \]

14. Simplified 98.6%

    \[\leadsto \color{blue}{\frac{c \cdot \left(a \cdot \sqrt[3]{27}\right)}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}\right)}} \]

15. Final simplification 98.6%

    \[\leadsto \frac{c \cdot \left(a \cdot \sqrt[3]{27}\right)}{\left(\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}\right) \cdot \left(a \cdot 3\right)} \]
16. Add Preprocessing

Alternative 4: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot \sqrt[3]{27}\right)\\ \frac{t\_0 \cdot \frac{0.3333333333333333}{a}}{\left(-b\right) - \sqrt{{b}^{2} - t\_0}} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a (cbrt 27.0)))))
   (/ (* t_0 (/ 0.3333333333333333 a)) (- (- b) (sqrt (- (pow b 2.0) t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * cbrt(27.0));
	return (t_0 * (0.3333333333333333 / a)) / (-b - sqrt((pow(b, 2.0) - t_0)));
}

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (a * Math.cbrt(27.0));
	return (t_0 * (0.3333333333333333 / a)) / (-b - Math.sqrt((Math.pow(b, 2.0) - t_0)));
}

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * cbrt(27.0)))
	return Float64(Float64(t_0 * Float64(0.3333333333333333 / a)) / Float64(Float64(-b) - sqrt(Float64((b ^ 2.0) - t_0))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * N[Power[27.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$0 * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 32.5%

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

   1. add-cbrt-cube 32.5%

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

   2. pow1/3 32.4%

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

   3. pow3 32.4%

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

   4. associate-*l* 32.4%

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

   5. unpow-prod-down 32.4%

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

   6. metadata-eval 32.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\color{blue}{27} \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}{3 \cdot a} \]

4. Applied egg-rr 32.4%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]
5. Step-by-step derivation

   1. flip-+ 32.3%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}} \cdot \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}}{3 \cdot a} \]

   2. pow2 32.3%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}} \cdot \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   3. add-sqr-sqrt 33.0%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}\right)}}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   4. pow2 33.0%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   5. unpow1/3 33.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\sqrt[3]{27 \cdot {\left(a \cdot c\right)}^{3}}}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   6. *-commutative 33.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \sqrt[3]{\color{blue}{{\left(a \cdot c\right)}^{3} \cdot 27}}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   7. cbrt-prod 33.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\sqrt[3]{{\left(a \cdot c\right)}^{3}} \cdot \sqrt[3]{27}}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   8. unpow3 33.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \sqrt[3]{\color{blue}{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right) \cdot \left(a \cdot c\right)}} \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   9. add-cbrt-cube 33.1%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\left(a \cdot c\right)} \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

6. Applied egg-rr 33.1%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}}{3 \cdot a} \]
7. Step-by-step derivation

   1. associate--r- 98.6%

      \[\leadsto \frac{\frac{\color{blue}{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \left(a \cdot c\right) \cdot \sqrt[3]{27}}}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   2. unpow2 98.6%

      \[\leadsto \frac{\frac{\left(\color{blue}{\left(-b\right) \cdot \left(-b\right)} - {b}^{2}\right) + \left(a \cdot c\right) \cdot \sqrt[3]{27}}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   3. sqr-neg 98.6%

      \[\leadsto \frac{\frac{\left(\color{blue}{b \cdot b} - {b}^{2}\right) + \left(a \cdot c\right) \cdot \sqrt[3]{27}}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   4. unpow2 98.6%

      \[\leadsto \frac{\frac{\left(\color{blue}{{b}^{2}} - {b}^{2}\right) + \left(a \cdot c\right) \cdot \sqrt[3]{27}}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   5. *-commutative 98.6%

      \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + \color{blue}{\left(c \cdot a\right)} \cdot \sqrt[3]{27}}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   6. associate-*l* 98.6%

      \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + \color{blue}{c \cdot \left(a \cdot \sqrt[3]{27}\right)}}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   7. *-commutative 98.6%

      \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(c \cdot a\right)} \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   8. associate-*l* 98.6%

      \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{c \cdot \left(a \cdot \sqrt[3]{27}\right)}}}}{3 \cdot a} \]

8. Simplified 98.6%

   \[\leadsto \frac{\color{blue}{\frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}}}}{3 \cdot a} \]
9. Step-by-step derivation

   1. pow1/3 99.4%

      \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot \color{blue}{{27}^{0.3333333333333333}}\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}}}{3 \cdot a} \]

10. Applied egg-rr 99.4%

    \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot \color{blue}{{27}^{0.3333333333333333}}\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}}}{3 \cdot a} \]
11. Step-by-step derivation

    1. expm1-log1p-u 85.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot {27}^{0.3333333333333333}\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}}}{3 \cdot a}\right)\right)} \]

    2. expm1-udef 36.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\left({b}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot {27}^{0.3333333333333333}\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}}}{3 \cdot a}\right)} - 1} \]

12. Applied egg-rr 36.6%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(c, a \cdot \sqrt[3]{27}, 0\right)}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}\right)}\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 84.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(c, a \cdot \sqrt[3]{27}, 0\right)}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}\right)}\right)\right)} \]

    2. expm1-log1p 98.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c, a \cdot \sqrt[3]{27}, 0\right)}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}\right)}} \]

    3. *-lft-identity 98.6%

       \[\leadsto \color{blue}{1 \cdot \frac{\mathsf{fma}\left(c, a \cdot \sqrt[3]{27}, 0\right)}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}\right)}} \]

    4. associate-*r/ 98.6%

       \[\leadsto \color{blue}{\frac{1 \cdot \mathsf{fma}\left(c, a \cdot \sqrt[3]{27}, 0\right)}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}\right)}} \]

    5. times-frac 98.6%

       \[\leadsto \color{blue}{\frac{1}{a \cdot 3} \cdot \frac{\mathsf{fma}\left(c, a \cdot \sqrt[3]{27}, 0\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}}} \]

    6. *-commutative 98.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c, a \cdot \sqrt[3]{27}, 0\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}} \cdot \frac{1}{a \cdot 3}} \]

    7. associate-*l/ 98.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c, a \cdot \sqrt[3]{27}, 0\right) \cdot \frac{1}{a \cdot 3}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}}} \]

    8. fma-udef 98.6%

       \[\leadsto \frac{\color{blue}{\left(c \cdot \left(a \cdot \sqrt[3]{27}\right) + 0\right)} \cdot \frac{1}{a \cdot 3}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}} \]

    9. +-rgt-identity 98.6%

       \[\leadsto \frac{\color{blue}{\left(c \cdot \left(a \cdot \sqrt[3]{27}\right)\right)} \cdot \frac{1}{a \cdot 3}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}} \]

    10. *-commutative 98.6%

        \[\leadsto \frac{\left(c \cdot \left(a \cdot \sqrt[3]{27}\right)\right) \cdot \frac{1}{\color{blue}{3 \cdot a}}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}} \]

    11. associate-/r* 99.0%

        \[\leadsto \frac{\left(c \cdot \left(a \cdot \sqrt[3]{27}\right)\right) \cdot \color{blue}{\frac{\frac{1}{3}}{a}}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}} \]

    12. metadata-eval 99.0%

        \[\leadsto \frac{\left(c \cdot \left(a \cdot \sqrt[3]{27}\right)\right) \cdot \frac{\color{blue}{0.3333333333333333}}{a}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}} \]

14. Simplified 99.0%

    \[\leadsto \color{blue}{\frac{\left(c \cdot \left(a \cdot \sqrt[3]{27}\right)\right) \cdot \frac{0.3333333333333333}{a}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}}} \]

15. Final simplification 99.0%

    \[\leadsto \frac{\left(c \cdot \left(a \cdot \sqrt[3]{27}\right)\right) \cdot \frac{0.3333333333333333}{a}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}} \]
16. Add Preprocessing

Alternative 5: 84.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -2e-6)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -2e-6) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -2e-6)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / 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 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -2e-6], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $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 3 a) c)))) (*.f64 3 a)) < -1.99999999999999991e-6

   1. Initial program 69.2%

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

      1. +-commutative 69.2%

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

      2. sqr-neg 69.2%

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

      3. unsub-neg 69.2%

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

      4. div-sub 68.0%

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

      5. --rgt-identity 68.0%

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

      6. div-sub 69.2%

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

   3. Simplified 69.3%

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

   if -1.99999999999999991e-6 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 18.4%

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

   3. Taylor expanded in b around inf 90.6%

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

      1. associate-*r/ 90.6%

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

   5. Simplified 90.6%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -2e-6)
   (/ (- (sqrt (- (* b b) (* a (* c 3.0)))) b) (* a 3.0))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -2e-6) {
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / 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) :: tmp
    if (((sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)) <= (-2d-6)) then
        tmp = (sqrt(((b * b) - (a * (c * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -2e-6) {
		tmp = (Math.sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -2e-6:
		tmp = (math.sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -2e-6)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(a * Float64(c * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -2e-6)
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -2e-6], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $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 3 a) c)))) (*.f64 3 a)) < -1.99999999999999991e-6

   1. Initial program 69.2%

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

   3. Taylor expanded in a around 0 69.2%

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

      1. associate-*r* 69.2%

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

      2. *-commutative 69.2%

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

      3. associate-*l* 69.3%

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

   5. Simplified 69.3%

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

   if -1.99999999999999991e-6 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 18.4%

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

   3. Taylor expanded in b around inf 90.6%

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

      1. associate-*r/ 90.6%

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

   5. Simplified 90.6%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.3% 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 32.5%

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

3. Taylor expanded in b around inf 90.3%

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

4. Final simplification 90.3%

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

Alternative 8: 80.5% accurate, 23.2× speedup

Math

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

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(c * Float64(-0.5 / b))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.5%

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

3. Taylor expanded in b around inf 80.4%

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

   1. associate-*r/ 80.4%

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

   2. associate-/l* 80.2%

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

5. Simplified 80.2%

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

   1. associate-/r/ 80.2%

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

7. Applied egg-rr 80.2%

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

8. Final simplification 80.2%

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

Alternative 9: 80.5% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

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) / (b / c)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.5%

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

3. Taylor expanded in b around inf 80.4%

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

   1. associate-*r/ 80.4%

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

   2. associate-/l* 80.2%

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

5. Simplified 80.2%

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

6. Final simplification 80.2%

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

Alternative 10: 80.7% 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 32.5%

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

3. Taylor expanded in b around inf 80.4%

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

   1. associate-*r/ 80.4%

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

5. Simplified 80.4%

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

6. Final simplification 80.4%

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

Alternative 11: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 32.5%

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

   1. add-cbrt-cube 32.5%

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

   2. pow1/3 32.4%

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

   3. pow3 32.4%

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

   4. associate-*l* 32.4%

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

   5. unpow-prod-down 32.4%

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

   6. metadata-eval 32.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\color{blue}{27} \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}{3 \cdot a} \]

4. Applied egg-rr 32.4%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]
5. Step-by-step derivation

   1. log1p-expm1-u 24.6%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-b\right) + \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}\right)\right)}}{3 \cdot a} \]

   2. neg-mul-1 24.6%

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{-1 \cdot b} + \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}\right)\right)}{3 \cdot a} \]

   3. fma-def 24.6%

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}\right)}\right)\right)}{3 \cdot a} \]

   4. pow2 24.6%

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{2}} - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}\right)\right)\right)}{3 \cdot a} \]

   5. unpow1/3 24.7%

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{\sqrt[3]{27 \cdot {\left(a \cdot c\right)}^{3}}}}\right)\right)\right)}{3 \cdot a} \]

   6. *-commutative 24.7%

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \sqrt[3]{\color{blue}{{\left(a \cdot c\right)}^{3} \cdot 27}}}\right)\right)\right)}{3 \cdot a} \]

   7. cbrt-prod 24.7%

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{\sqrt[3]{{\left(a \cdot c\right)}^{3}} \cdot \sqrt[3]{27}}}\right)\right)\right)}{3 \cdot a} \]

   8. unpow3 24.7%

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \sqrt[3]{\color{blue}{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right) \cdot \left(a \cdot c\right)}} \cdot \sqrt[3]{27}}\right)\right)\right)}{3 \cdot a} \]

   9. add-cbrt-cube 24.7%

      \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right)} \cdot \sqrt[3]{27}}\right)\right)\right)}{3 \cdot a} \]

6. Applied egg-rr 24.7%

   \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}\right)\right)\right)}}{3 \cdot a} \]

7. Taylor expanded in a around 0 3.2%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + -1 \cdot b}{a}} \]
8. Step-by-step derivation

   1. associate-*r/ 3.2%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(b + -1 \cdot b\right)}{a}} \]

   2. distribute-rgt1-in 3.2%

      \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot b\right)}}{a} \]

   3. metadata-eval 3.2%

      \[\leadsto \frac{0.3333333333333333 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

   4. mul0-lft 3.2%

      \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{0}}{a} \]

   5. metadata-eval 3.2%

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

9. Simplified 3.2%

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

10. Final simplification 3.2%

    \[\leadsto \frac{0}{a} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024027 
(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)))