Cubic critical, narrow range

Percentage Accurate: 55.3% → 99.3%

Time: 14.4s

Alternatives: 10

Speedup: 23.2×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

Math

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

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: 55.3% 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: 99.3% 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 53.7%

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

   1. add-cbrt-cube 53.6%

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

   2. pow1/3 53.5%

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

   3. pow3 53.5%

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

   4. associate-*l* 53.6%

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

   5. unpow-prod-down 53.6%

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

   6. metadata-eval 53.6%

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

3. Applied egg-rr 53.6%

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

   1. unpow1/3 53.7%

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

   2. cube-prod 53.6%

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

   3. *-commutative 53.6%

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

   4. cube-prod 53.7%

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

5. Simplified 53.7%

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

   1. flip-+ 53.8%

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

   2. pow2 53.8%

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

   3. add-sqr-sqrt 55.2%

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

   4. pow2 55.2%

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

   5. cbrt-prod 55.2%

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

   6. unpow3 55.2%

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

   7. add-cbrt-cube 55.2%

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

   8. pow2 55.2%

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

   9. cbrt-prod 55.2%

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

7. Applied egg-rr 55.2%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}}{3 \cdot a} \]
8. 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%

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

9. Simplified 98.6%

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

    1. pow1/3 99.3%

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

11. Applied egg-rr 99.3%

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

12. Final simplification 99.3%

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

Alternative 2: 99.3% accurate, 0.2× speedup

Math

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

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) (* -3.0 (* c a))))))
  (* 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) + (-3.0 * (c * a)))))) / (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 = ((((b ** 2.0d0) - (b ** 2.0d0)) + (c * (a * (27.0d0 ** 0.3333333333333333d0)))) / (-b - sqrt(((b ** 2.0d0) + ((-3.0d0) * (c * a)))))) / (a * 3.0d0)
end function

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) + (-3.0 * (c * a)))))) / (a * 3.0);
}

Python

def code(a, b, 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) + (-3.0 * (c * a)))))) / (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(-3.0 * Float64(c * a)))))) / Float64(a * 3.0))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((((b ^ 2.0) - (b ^ 2.0)) + (c * (a * (27.0 ^ 0.3333333333333333)))) / (-b - sqrt(((b ^ 2.0) + (-3.0 * (c * a)))))) / (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[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.7%

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

   1. add-cbrt-cube 53.6%

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

   2. pow1/3 53.5%

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

   3. pow3 53.5%

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

   4. associate-*l* 53.6%

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

   5. unpow-prod-down 53.6%

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

   6. metadata-eval 53.6%

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

3. Applied egg-rr 53.6%

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

   1. unpow1/3 53.7%

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

   2. cube-prod 53.6%

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

   3. *-commutative 53.6%

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

   4. cube-prod 53.7%

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

5. Simplified 53.7%

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

   1. flip-+ 53.8%

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

   2. pow2 53.8%

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

   3. add-sqr-sqrt 55.2%

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

   4. pow2 55.2%

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

   5. cbrt-prod 55.2%

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

   6. unpow3 55.2%

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

   7. add-cbrt-cube 55.2%

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

   8. pow2 55.2%

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

   9. cbrt-prod 55.2%

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

7. Applied egg-rr 55.2%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}}{3 \cdot a} \]
8. 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%

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

9. Simplified 98.6%

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

    1. pow1/3 99.3%

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

11. Applied egg-rr 99.3%

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

12. Taylor expanded in b around 0 99.3%

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

13. Final simplification 99.3%

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

Alternative 3: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot \sqrt[3]{27}\right)\\ \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{{b}^{2} - t_0}}{t_0}}}{a \cdot 3} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a (cbrt 27.0)))))
   (/ (/ 1.0 (/ (- (- b) (sqrt (- (pow b 2.0) t_0))) t_0)) (* a 3.0))))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * cbrt(27.0));
	return (1.0 / ((-b - sqrt((pow(b, 2.0) - t_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 (1.0 / ((-b - Math.sqrt((Math.pow(b, 2.0) - t_0))) / t_0)) / (a * 3.0);
}

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * cbrt(27.0)))
	return Float64(Float64(1.0 / Float64(Float64(Float64(-b) - sqrt(Float64((b ^ 2.0) - t_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[(N[(1.0 / N[(N[((-b) - N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 53.7%

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

   1. add-cbrt-cube 53.6%

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

   2. pow1/3 53.5%

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

   3. pow3 53.5%

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

   4. associate-*l* 53.6%

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

   5. unpow-prod-down 53.6%

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

   6. metadata-eval 53.6%

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

3. Applied egg-rr 53.6%

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

   1. unpow1/3 53.7%

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

   2. cube-prod 53.6%

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

   3. *-commutative 53.6%

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

   4. cube-prod 53.7%

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

5. Simplified 53.7%

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

   1. flip-+ 53.8%

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

   2. pow2 53.8%

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

   3. add-sqr-sqrt 55.2%

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

   4. pow2 55.2%

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

   5. cbrt-prod 55.2%

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

   6. unpow3 55.2%

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

   7. add-cbrt-cube 55.2%

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

   8. pow2 55.2%

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

   9. cbrt-prod 55.2%

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

7. Applied egg-rr 55.2%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}}{3 \cdot a} \]
8. 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%

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

9. Simplified 98.6%

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

    1. pow1/3 99.3%

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

11. Applied egg-rr 99.3%

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

    1. clear-num 99.3%

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

    2. inv-pow 99.3%

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

    3. +-commutative 99.3%

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

    4. pow1/3 98.7%

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

    5. fma-def 98.7%

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

    6. +-inverses 98.7%

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

13. Applied egg-rr 98.7%

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

    1. unpow-1 98.7%

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

    2. fma-udef 98.7%

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

    3. +-rgt-identity 98.7%

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

15. Simplified 98.7%

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

16. Final simplification 98.7%

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

Alternative 4: 98.6% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a (cbrt 27.0)))))
   (/ (/ t_0 (* a 3.0)) (- (- 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 / (a * 3.0)) / (-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 / (a * 3.0)) / (-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(a * 3.0)) / 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[(a * 3.0), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 53.7%

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

   1. add-cbrt-cube 53.6%

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

   2. pow1/3 53.5%

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

   3. pow3 53.5%

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

   4. associate-*l* 53.6%

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

   5. unpow-prod-down 53.6%

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

   6. metadata-eval 53.6%

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

3. Applied egg-rr 53.6%

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

   1. unpow1/3 53.7%

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

   2. cube-prod 53.6%

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

   3. *-commutative 53.6%

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

   4. cube-prod 53.7%

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

5. Simplified 53.7%

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

   1. flip-+ 53.8%

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

   2. pow2 53.8%

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

   3. add-sqr-sqrt 55.2%

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

   4. pow2 55.2%

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

   5. cbrt-prod 55.2%

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

   6. unpow3 55.2%

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

   7. add-cbrt-cube 55.2%

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

   8. pow2 55.2%

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

   9. cbrt-prod 55.2%

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

7. Applied egg-rr 55.2%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}}{3 \cdot a} \]
8. 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%

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

9. Simplified 98.6%

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

    1. pow1/3 99.3%

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

11. Applied egg-rr 99.3%

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

    1. expm1-log1p-u 86.9%

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

    2. expm1-udef 62.3%

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

13. Applied egg-rr 62.3%

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

    1. expm1-def 86.3%

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

    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. associate-/r* 98.6%

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

    4. fma-udef 98.6%

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

    5. +-rgt-identity 98.6%

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

15. Simplified 98.6%

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

16. Final simplification 98.6%

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

Alternative 5: 98.6% accurate, 0.3× speedup

Math

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

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(Float64(t_0 / 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[(N[(t$95$0 / N[((-b) - N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 53.7%

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

   1. add-cbrt-cube 53.6%

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

   2. pow1/3 53.5%

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

   3. pow3 53.5%

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

   4. associate-*l* 53.6%

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

   5. unpow-prod-down 53.6%

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

   6. metadata-eval 53.6%

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

3. Applied egg-rr 53.6%

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

   1. unpow1/3 53.7%

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

   2. cube-prod 53.6%

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

   3. *-commutative 53.6%

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

   4. cube-prod 53.7%

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

5. Simplified 53.7%

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

   1. flip-+ 53.8%

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

   2. pow2 53.8%

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

   3. add-sqr-sqrt 55.2%

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

   4. pow2 55.2%

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

   5. cbrt-prod 55.2%

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

   6. unpow3 55.2%

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

   7. add-cbrt-cube 55.2%

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

   8. pow2 55.2%

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

   9. cbrt-prod 55.2%

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

7. Applied egg-rr 55.2%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}}{3 \cdot a} \]
8. 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%

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

9. Simplified 98.6%

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

    1. pow1/3 99.3%

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

11. Applied egg-rr 99.3%

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

    1. expm1-log1p-u 88.5%

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

    2. expm1-udef 58.3%

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

    3. +-commutative 58.3%

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

    4. pow1/3 58.3%

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

    5. fma-def 58.3%

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

    6. +-inverses 58.3%

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

13. Applied egg-rr 58.3%

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

    1. expm1-def 87.9%

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

    2. expm1-log1p 98.6%

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

    3. fma-udef 98.6%

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

    4. +-rgt-identity 98.6%

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

15. Simplified 98.6%

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

16. Final simplification 98.6%

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

Alternative 6: 76.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -5.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -5.5e-7)
   (/ (- (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)) <= -5.5e-7) {
		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)) <= -5.5e-7)
		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], -5.5e-7], 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)) < -5.5000000000000003e-7

   1. Initial program 71.8%

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

      1. +-commutative 71.8%

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

      2. sqr-neg 71.8%

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

      3. unsub-neg 71.8%

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

      4. div-sub 71.1%

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

      5. --rgt-identity 71.1%

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

      6. div-sub 71.8%

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

   3. Simplified 71.9%

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

   if -5.5000000000000003e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 27.1%

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

   2. Taylor expanded in b around inf 86.8%

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

      1. *-commutative 86.8%

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

      2. associate-*l/ 86.8%

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

   4. Simplified 86.8%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -5.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 7: 76.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -5.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -5.5e-7)
   (/ (- (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)) <= -5.5e-7) {
		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)) <= (-5.5d-7)) 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)) <= -5.5e-7) {
		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)) <= -5.5e-7:
		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)) <= -5.5e-7)
		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)) <= -5.5e-7)
		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], -5.5e-7], 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)) < -5.5000000000000003e-7

   1. Initial program 71.8%

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

      1. add-cbrt-cube 71.8%

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

      2. pow1/3 71.6%

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

      3. pow3 71.6%

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

      4. associate-*l* 71.7%

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

      5. unpow-prod-down 71.7%

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

      6. metadata-eval 71.7%

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

   3. Applied egg-rr 71.7%

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

      1. unpow1/3 71.8%

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

      2. cube-prod 71.8%

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

      3. *-commutative 71.8%

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

      4. cube-prod 71.8%

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

   5. Simplified 71.8%

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

   6. Taylor expanded in a around 0 71.8%

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

      1. associate-*r* 71.8%

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

      2. *-commutative 71.8%

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

      3. associate-*l* 71.8%

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

   8. Simplified 71.8%

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

   if -5.5000000000000003e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 27.1%

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

   2. Taylor expanded in b around inf 86.8%

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

      1. *-commutative 86.8%

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

      2. associate-*l/ 86.8%

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

   4. Simplified 86.8%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -5.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 8: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.4:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6.4)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.4) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.4)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.4], 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[(-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. Split input into 2 regimes

2. if b < 6.4000000000000004

   1. Initial program 81.7%

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

      1. +-commutative 81.7%

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

      2. sqr-neg 81.7%

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

      3. unsub-neg 81.7%

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

      4. div-sub 81.0%

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

      5. --rgt-identity 81.0%

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

      6. div-sub 81.7%

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

   3. Simplified 81.7%

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

   if 6.4000000000000004 < b

   1. Initial program 43.9%

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

   2. Taylor expanded in b around inf 88.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.4:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]

Alternative 9: 64.5% 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 53.7%

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

2. Taylor expanded in b around inf 65.4%

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

   1. *-commutative 65.4%

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

   2. associate-*l/ 65.4%

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

4. Simplified 65.4%

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

5. Final simplification 65.4%

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

Alternative 10: 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 53.7%

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

   1. add-cbrt-cube 53.6%

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

   2. pow1/3 53.5%

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

   3. pow3 53.5%

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

   4. associate-*l* 53.6%

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

   5. unpow-prod-down 53.6%

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

   6. metadata-eval 53.6%

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

3. Applied egg-rr 53.6%

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

   1. unpow1/3 53.7%

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

   2. cube-prod 53.6%

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

   3. *-commutative 53.6%

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

   4. cube-prod 53.7%

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

5. Simplified 53.7%

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

   1. clear-num 53.7%

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

   2. inv-pow 53.7%

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

   3. *-commutative 53.7%

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

   4. neg-mul-1 53.7%

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

   5. fma-def 53.7%

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

   6. pow2 53.7%

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

   7. cbrt-prod 53.6%

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

   8. unpow3 53.6%

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

   9. add-cbrt-cube 53.6%

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

7. Applied egg-rr 53.6%

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

   1. unpow-1 53.6%

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

   2. associate-/l* 53.7%

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

   3. fma-udef 53.7%

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

   4. *-commutative 53.7%

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

   5. fma-def 53.7%

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

   6. *-commutative 53.7%

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

   7. associate-*l* 53.7%

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

9. Simplified 53.7%

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

10. Taylor expanded in a around 0 3.2%

    \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + -1 \cdot b}{a}} \]
11. 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%

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

12. Simplified 3.2%

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

13. Final simplification 3.2%

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

Reproduce

herbie shell --seed 2023322 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))