Cubic critical, narrow range

Percentage Accurate: 55.2% → 99.3%

Time: 12.1s

Alternatives: 9

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: 55.2% 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: 99.3% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/
   (+ (- (pow b 2.0) (pow b 2.0)) (* c (* a (pow 27.0 0.3333333333333333))))
   (- (- b) (sqrt (fma b b (* -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(fma(b, b, (-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(fma(b, b, Float64(-3.0 * Float64(c * a)))))) / 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[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.1%

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

   1. add-cbrt-cube 56.0%

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

   2. pow1/3 55.9%

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

   3. pow3 55.9%

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

   4. associate-*l* 55.9%

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

   5. unpow-prod-down 55.9%

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

   6. metadata-eval 55.9%

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

3. Applied egg-rr 55.9%

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

   1. flip-+ 56.0%

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

   2. pow2 55.9%

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

   3. add-sqr-sqrt 57.2%

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

   4. pow2 57.2%

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

   5. unpow1/3 57.4%

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

   6. *-commutative 57.4%

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

   7. cbrt-prod 57.4%

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

   8. unpow3 57.4%

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

   9. add-cbrt-cube 57.4%

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

5. Applied egg-rr 57.4%

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

   1. associate--r- 98.5%

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

   2. unpow2 98.4%

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

   3. sqr-neg 98.4%

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

   4. unpow2 98.5%

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

   5. *-commutative 98.5%

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

   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%

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

7. Simplified 98.6%

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

   1. pow1/3 99.4%

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

9. Applied egg-rr 99.4%

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

    1. associate-*r* 99.4%

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

    2. *-commutative 99.4%

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

    3. add-cbrt-cube 99.4%

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

    4. cbrt-unprod 99.3%

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

    5. pow3 99.3%

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

    6. metadata-eval 99.3%

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

    7. unpow-prod-down 99.3%

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

    8. *-commutative 99.3%

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

    9. pow3 99.4%

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

    10. add-cbrt-cube 99.4%

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

    11. cancel-sign-sub-inv 99.4%

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

    12. pow2 99.4%

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

    13. fma-def 99.4%

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

    14. metadata-eval 99.4%

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

    15. *-commutative 99.4%

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

11. Applied egg-rr 99.4%

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

12. Final simplification 99.4%

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

Alternative 2: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, {b}^{2}\right)}}{a \cdot \left(c \cdot 3\right)}}}{a \cdot 3} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/ 1.0 (/ (- (- b) (sqrt (fma -3.0 (* c a) (pow b 2.0)))) (* a (* c 3.0))))
  (* a 3.0)))

C

double code(double a, double b, double c) {
	return (1.0 / ((-b - sqrt(fma(-3.0, (c * a), pow(b, 2.0)))) / (a * (c * 3.0)))) / (a * 3.0);
}

Julia

function code(a, b, c)
	return Float64(Float64(1.0 / Float64(Float64(Float64(-b) - sqrt(fma(-3.0, Float64(c * a), (b ^ 2.0)))) / Float64(a * Float64(c * 3.0)))) / Float64(a * 3.0))
end

Wolfram

code[a_, b_, c_] := N[(N[(1.0 / N[(N[((-b) - N[Sqrt[N[(-3.0 * N[(c * a), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.1%

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

   1. add-cbrt-cube 56.0%

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

   2. pow1/3 55.9%

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

   3. pow3 55.9%

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

   4. associate-*l* 55.9%

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

   5. unpow-prod-down 55.9%

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

   6. metadata-eval 55.9%

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

3. Applied egg-rr 55.9%

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

   1. flip-+ 56.0%

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

   2. pow2 55.9%

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

   3. add-sqr-sqrt 57.2%

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

   4. pow2 57.2%

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

   5. unpow1/3 57.4%

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

   6. *-commutative 57.4%

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

   7. cbrt-prod 57.4%

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

   8. unpow3 57.4%

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

   9. add-cbrt-cube 57.4%

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

5. Applied egg-rr 57.4%

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

   1. associate--r- 98.5%

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

   2. unpow2 98.4%

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

   3. sqr-neg 98.4%

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

   4. unpow2 98.5%

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

   5. *-commutative 98.5%

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

   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%

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

7. 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. Applied egg-rr 98.6%

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

    1. unpow-1 98.6%

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

    2. fma-udef 98.6%

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

    3. unpow2 98.6%

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

    4. *-commutative 98.6%

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

    5. +-commutative 98.6%

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

    6. fma-def 98.6%

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

    7. fma-udef 98.6%

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

    8. +-rgt-identity 98.6%

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

    9. *-commutative 98.6%

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

11. Simplified 98.6%

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

12. Taylor expanded in a around 0 99.1%

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

    1. associate-*r* 99.2%

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

    2. *-commutative 99.2%

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

    3. associate-*l* 99.1%

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

14. Simplified 99.1%

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

15. Final simplification 99.1%

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

Alternative 3: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, {b}^{2}\right)}}{3 \cdot \left(c \cdot a\right)}}}{a \cdot 3} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/ 1.0 (/ (- (- b) (sqrt (fma -3.0 (* c a) (pow b 2.0)))) (* 3.0 (* c a))))
  (* a 3.0)))

C

double code(double a, double b, double c) {
	return (1.0 / ((-b - sqrt(fma(-3.0, (c * a), pow(b, 2.0)))) / (3.0 * (c * a)))) / (a * 3.0);
}

Julia

function code(a, b, c)
	return Float64(Float64(1.0 / Float64(Float64(Float64(-b) - sqrt(fma(-3.0, Float64(c * a), (b ^ 2.0)))) / Float64(3.0 * Float64(c * a)))) / Float64(a * 3.0))
end

Wolfram

code[a_, b_, c_] := N[(N[(1.0 / N[(N[((-b) - N[Sqrt[N[(-3.0 * N[(c * a), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.1%

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

   1. add-cbrt-cube 56.0%

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

   2. pow1/3 55.9%

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

   3. pow3 55.9%

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

   4. associate-*l* 55.9%

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

   5. unpow-prod-down 55.9%

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

   6. metadata-eval 55.9%

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

3. Applied egg-rr 55.9%

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

   1. flip-+ 56.0%

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

   2. pow2 55.9%

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

   3. add-sqr-sqrt 57.2%

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

   4. pow2 57.2%

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

   5. unpow1/3 57.4%

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

   6. *-commutative 57.4%

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

   7. cbrt-prod 57.4%

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

   8. unpow3 57.4%

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

   9. add-cbrt-cube 57.4%

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

5. Applied egg-rr 57.4%

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

   1. associate--r- 98.5%

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

   2. unpow2 98.4%

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

   3. sqr-neg 98.4%

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

   4. unpow2 98.5%

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

   5. *-commutative 98.5%

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

   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%

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

7. 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. Applied egg-rr 98.6%

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

    1. unpow-1 98.6%

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

    2. fma-udef 98.6%

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

    3. unpow2 98.6%

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

    4. *-commutative 98.6%

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

    5. +-commutative 98.6%

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

    6. fma-def 98.6%

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

    7. fma-udef 98.6%

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

    8. +-rgt-identity 98.6%

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

    9. *-commutative 98.6%

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

11. Simplified 98.6%

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

12. Taylor expanded in a around 0 99.1%

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

    1. *-commutative 99.1%

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

14. Simplified 99.1%

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

15. Final simplification 99.1%

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

Alternative 4: 89.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{-0.6666666666666666 \cdot \frac{b}{c \cdot a} + \left(0.375 \cdot \frac{c \cdot a}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.0)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* a 3.0))
   (/
    (/
     1.0
     (+
      (* -0.6666666666666666 (/ b (* c a)))
      (+ (* 0.375 (/ (* c a) (pow b 3.0))) (* 0.5 (/ 1.0 b)))))
    (* a 3.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (1.0 / ((-0.6666666666666666 * (b / (c * a))) + ((0.375 * ((c * a) / pow(b, 3.0))) + (0.5 * (1.0 / b))))) / (a * 3.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(-0.6666666666666666 * Float64(b / Float64(c * a))) + Float64(Float64(0.375 * Float64(Float64(c * a) / (b ^ 3.0))) + Float64(0.5 * Float64(1.0 / b))))) / Float64(a * 3.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(-0.6666666666666666 * N[(b / N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.375 * N[(N[(c * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2

   1. Initial program 82.7%

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

   3. Simplified 82.9%

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

   if 2 < b

   1. Initial program 49.4%

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

      1. add-cbrt-cube 49.4%

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

      2. pow1/3 49.3%

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

      3. pow3 49.3%

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

      4. associate-*l* 49.3%

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

      5. unpow-prod-down 49.3%

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

      6. metadata-eval 49.3%

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

   3. Applied egg-rr 49.3%

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

      1. flip-+ 49.3%

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

      2. pow2 49.3%

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

      3. add-sqr-sqrt 50.6%

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

      4. pow2 50.7%

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

      5. unpow1/3 50.8%

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

      6. *-commutative 50.8%

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

      7. cbrt-prod 50.8%

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

      8. unpow3 50.8%

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

      9. add-cbrt-cube 50.8%

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

   5. Applied egg-rr 50.8%

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

      1. associate--r- 98.5%

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

      2. unpow2 98.4%

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

      3. sqr-neg 98.4%

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

      4. unpow2 98.5%

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

      5. *-commutative 98.5%

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

      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%

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

   7. 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. Applied egg-rr 98.6%

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

       1. unpow-1 98.6%

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

       2. fma-udef 98.5%

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

       3. unpow2 98.5%

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

       4. *-commutative 98.5%

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

       5. +-commutative 98.5%

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

       6. fma-def 98.5%

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

       7. fma-udef 98.5%

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

       8. +-rgt-identity 98.5%

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

       9. *-commutative 98.5%

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

   11. Simplified 98.5%

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

   12. Taylor expanded in b around inf 92.5%

       \[\leadsto \frac{\frac{1}{\color{blue}{-0.6666666666666666 \cdot \frac{b}{a \cdot c} + \left(0.375 \cdot \frac{a \cdot c}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}}}{3 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{-0.6666666666666666 \cdot \frac{b}{c \cdot a} + \left(0.375 \cdot \frac{c \cdot a}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}}{a \cdot 3}\\ \end{array} \]

Alternative 5: 89.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{-0.6666666666666666 \cdot \frac{b}{c \cdot a} + \left(0.375 \cdot \frac{c \cdot a}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.0)
   (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
   (/
    (/
     1.0
     (+
      (* -0.6666666666666666 (/ b (* c a)))
      (+ (* 0.375 (/ (* c a) (pow b 3.0))) (* 0.5 (/ 1.0 b)))))
    (* a 3.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.0) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (1.0 / ((-0.6666666666666666 * (b / (c * a))) + ((0.375 * ((c * a) / pow(b, 3.0))) + (0.5 * (1.0 / b))))) / (a * 3.0);
	}
	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 (b <= 2.0d0) then
        tmp = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (1.0d0 / (((-0.6666666666666666d0) * (b / (c * a))) + ((0.375d0 * ((c * a) / (b ** 3.0d0))) + (0.5d0 * (1.0d0 / b))))) / (a * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.0) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (1.0 / ((-0.6666666666666666 * (b / (c * a))) + ((0.375 * ((c * a) / Math.pow(b, 3.0))) + (0.5 * (1.0 / b))))) / (a * 3.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.0:
		tmp = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = (1.0 / ((-0.6666666666666666 * (b / (c * a))) + ((0.375 * ((c * a) / math.pow(b, 3.0))) + (0.5 * (1.0 / b))))) / (a * 3.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(-0.6666666666666666 * Float64(b / Float64(c * a))) + Float64(Float64(0.375 * Float64(Float64(c * a) / (b ^ 3.0))) + Float64(0.5 * Float64(1.0 / b))))) / Float64(a * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.0)
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	else
		tmp = (1.0 / ((-0.6666666666666666 * (b / (c * a))) + ((0.375 * ((c * a) / (b ^ 3.0))) + (0.5 * (1.0 / b))))) / (a * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.0], 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], N[(N[(1.0 / N[(N[(-0.6666666666666666 * N[(b / N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.375 * N[(N[(c * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2

   1. Initial program 82.7%

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

   if 2 < b

   1. Initial program 49.4%

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

      1. add-cbrt-cube 49.4%

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

      2. pow1/3 49.3%

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

      3. pow3 49.3%

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

      4. associate-*l* 49.3%

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

      5. unpow-prod-down 49.3%

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

      6. metadata-eval 49.3%

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

   3. Applied egg-rr 49.3%

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

      1. flip-+ 49.3%

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

      2. pow2 49.3%

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

      3. add-sqr-sqrt 50.6%

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

      4. pow2 50.7%

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

      5. unpow1/3 50.8%

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

      6. *-commutative 50.8%

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

      7. cbrt-prod 50.8%

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

      8. unpow3 50.8%

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

      9. add-cbrt-cube 50.8%

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

   5. Applied egg-rr 50.8%

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

      1. associate--r- 98.5%

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

      2. unpow2 98.4%

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

      3. sqr-neg 98.4%

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

      4. unpow2 98.5%

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

      5. *-commutative 98.5%

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

      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%

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

   7. 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. Applied egg-rr 98.6%

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

       1. unpow-1 98.6%

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

       2. fma-udef 98.5%

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

       3. unpow2 98.5%

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

       4. *-commutative 98.5%

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

       5. +-commutative 98.5%

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

       6. fma-def 98.5%

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

       7. fma-udef 98.5%

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

       8. +-rgt-identity 98.5%

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

       9. *-commutative 98.5%

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

   11. Simplified 98.5%

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

   12. Taylor expanded in b around inf 92.5%

       \[\leadsto \frac{\frac{1}{\color{blue}{-0.6666666666666666 \cdot \frac{b}{a \cdot c} + \left(0.375 \cdot \frac{a \cdot c}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}}}{3 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{-0.6666666666666666 \cdot \frac{b}{c \cdot a} + \left(0.375 \cdot \frac{c \cdot a}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}}{a \cdot 3}\\ \end{array} \]

Alternative 6: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.05:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{-0.6666666666666666 \cdot \frac{b}{c \cdot a} + 0.5 \cdot \frac{1}{b}}}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.05)
   (/ (- (sqrt (- (* b b) (* 3.0 (* c a)))) b) (* a 3.0))
   (/
    (/ 1.0 (+ (* -0.6666666666666666 (/ b (* c a))) (* 0.5 (/ 1.0 b))))
    (* a 3.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.05) {
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = (1.0 / ((-0.6666666666666666 * (b / (c * a))) + (0.5 * (1.0 / b)))) / (a * 3.0);
	}
	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 (b <= 2.05d0) then
        tmp = (sqrt(((b * b) - (3.0d0 * (c * a)))) - b) / (a * 3.0d0)
    else
        tmp = (1.0d0 / (((-0.6666666666666666d0) * (b / (c * a))) + (0.5d0 * (1.0d0 / b)))) / (a * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.05) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = (1.0 / ((-0.6666666666666666 * (b / (c * a))) + (0.5 * (1.0 / b)))) / (a * 3.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.05:
		tmp = (math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0)
	else:
		tmp = (1.0 / ((-0.6666666666666666 * (b / (c * a))) + (0.5 * (1.0 / b)))) / (a * 3.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.05)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(c * a)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(-0.6666666666666666 * Float64(b / Float64(c * a))) + Float64(0.5 * Float64(1.0 / b)))) / Float64(a * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.05)
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	else
		tmp = (1.0 / ((-0.6666666666666666 * (b / (c * a))) + (0.5 * (1.0 / b)))) / (a * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.05], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(-0.6666666666666666 * N[(b / N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.0499999999999998

   1. Initial program 82.7%

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

   2. Taylor expanded in a around 0 82.6%

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

   if 2.0499999999999998 < b

   1. Initial program 49.4%

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

      1. add-cbrt-cube 49.4%

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

      2. pow1/3 49.3%

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

      3. pow3 49.3%

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

      4. associate-*l* 49.3%

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

      5. unpow-prod-down 49.3%

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

      6. metadata-eval 49.3%

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

   3. Applied egg-rr 49.3%

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

      1. flip-+ 49.3%

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

      2. pow2 49.3%

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

      3. add-sqr-sqrt 50.6%

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

      4. pow2 50.7%

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

      5. unpow1/3 50.8%

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

      6. *-commutative 50.8%

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

      7. cbrt-prod 50.8%

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

      8. unpow3 50.8%

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

      9. add-cbrt-cube 50.8%

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

   5. Applied egg-rr 50.8%

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

      1. associate--r- 98.5%

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

      2. unpow2 98.4%

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

      3. sqr-neg 98.4%

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

      4. unpow2 98.5%

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

      5. *-commutative 98.5%

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

      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%

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

   7. 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. Applied egg-rr 98.6%

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

       1. unpow-1 98.6%

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

       2. fma-udef 98.5%

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

       3. unpow2 98.5%

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

       4. *-commutative 98.5%

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

       5. +-commutative 98.5%

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

       6. fma-def 98.5%

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

       7. fma-udef 98.5%

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

       8. +-rgt-identity 98.5%

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

       9. *-commutative 98.5%

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

   11. Simplified 98.5%

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

   12. Taylor expanded in b around inf 88.2%

       \[\leadsto \frac{\frac{1}{\color{blue}{-0.6666666666666666 \cdot \frac{b}{a \cdot c} + 0.5 \cdot \frac{1}{b}}}}{3 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.05:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{-0.6666666666666666 \cdot \frac{b}{c \cdot a} + 0.5 \cdot \frac{1}{b}}}{a \cdot 3}\\ \end{array} \]

Alternative 7: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{-0.6666666666666666 \cdot \frac{b}{c \cdot a} + 0.5 \cdot \frac{1}{b}}}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.1)
   (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
   (/
    (/ 1.0 (+ (* -0.6666666666666666 (/ b (* c a))) (* 0.5 (/ 1.0 b))))
    (* a 3.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.1) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (1.0 / ((-0.6666666666666666 * (b / (c * a))) + (0.5 * (1.0 / b)))) / (a * 3.0);
	}
	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 (b <= 2.1d0) then
        tmp = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (1.0d0 / (((-0.6666666666666666d0) * (b / (c * a))) + (0.5d0 * (1.0d0 / b)))) / (a * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.1) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (1.0 / ((-0.6666666666666666 * (b / (c * a))) + (0.5 * (1.0 / b)))) / (a * 3.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.1:
		tmp = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = (1.0 / ((-0.6666666666666666 * (b / (c * a))) + (0.5 * (1.0 / b)))) / (a * 3.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.1)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(-0.6666666666666666 * Float64(b / Float64(c * a))) + Float64(0.5 * Float64(1.0 / b)))) / Float64(a * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.1)
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	else
		tmp = (1.0 / ((-0.6666666666666666 * (b / (c * a))) + (0.5 * (1.0 / b)))) / (a * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.1], 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], N[(N[(1.0 / N[(N[(-0.6666666666666666 * N[(b / N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.10000000000000009

   1. Initial program 82.7%

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

   if 2.10000000000000009 < b

   1. Initial program 49.4%

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

      1. add-cbrt-cube 49.4%

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

      2. pow1/3 49.3%

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

      3. pow3 49.3%

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

      4. associate-*l* 49.3%

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

      5. unpow-prod-down 49.3%

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

      6. metadata-eval 49.3%

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

   3. Applied egg-rr 49.3%

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

      1. flip-+ 49.3%

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

      2. pow2 49.3%

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

      3. add-sqr-sqrt 50.6%

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

      4. pow2 50.7%

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

      5. unpow1/3 50.8%

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

      6. *-commutative 50.8%

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

      7. cbrt-prod 50.8%

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

      8. unpow3 50.8%

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

      9. add-cbrt-cube 50.8%

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

   5. Applied egg-rr 50.8%

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

      1. associate--r- 98.5%

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

      2. unpow2 98.4%

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

      3. sqr-neg 98.4%

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

      4. unpow2 98.5%

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

      5. *-commutative 98.5%

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

      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%

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

   7. 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. Applied egg-rr 98.6%

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

       1. unpow-1 98.6%

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

       2. fma-udef 98.5%

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

       3. unpow2 98.5%

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

       4. *-commutative 98.5%

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

       5. +-commutative 98.5%

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

       6. fma-def 98.5%

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

       7. fma-udef 98.5%

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

       8. +-rgt-identity 98.5%

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

       9. *-commutative 98.5%

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

   11. Simplified 98.5%

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

   12. Taylor expanded in b around inf 88.2%

       \[\leadsto \frac{\frac{1}{\color{blue}{-0.6666666666666666 \cdot \frac{b}{a \cdot c} + 0.5 \cdot \frac{1}{b}}}}{3 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.1:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{-0.6666666666666666 \cdot \frac{b}{c \cdot a} + 0.5 \cdot \frac{1}{b}}}{a \cdot 3}\\ \end{array} \]

Alternative 8: 81.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{-0.6666666666666666 \cdot \frac{b}{c \cdot a} + 0.5 \cdot \frac{1}{b}}}{a \cdot 3} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/ 1.0 (+ (* -0.6666666666666666 (/ b (* c a))) (* 0.5 (/ 1.0 b))))
  (* a 3.0)))

C

double code(double a, double b, double c) {
	return (1.0 / ((-0.6666666666666666 * (b / (c * a))) + (0.5 * (1.0 / b)))) / (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 = (1.0d0 / (((-0.6666666666666666d0) * (b / (c * a))) + (0.5d0 * (1.0d0 / b)))) / (a * 3.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return (1.0 / ((-0.6666666666666666 * (b / (c * a))) + (0.5 * (1.0 / b)))) / (a * 3.0);
}

Python

def code(a, b, c):
	return (1.0 / ((-0.6666666666666666 * (b / (c * a))) + (0.5 * (1.0 / b)))) / (a * 3.0)

Julia

function code(a, b, c)
	return Float64(Float64(1.0 / Float64(Float64(-0.6666666666666666 * Float64(b / Float64(c * a))) + Float64(0.5 * Float64(1.0 / b)))) / Float64(a * 3.0))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (1.0 / ((-0.6666666666666666 * (b / (c * a))) + (0.5 * (1.0 / b)))) / (a * 3.0);
end

Wolfram

code[a_, b_, c_] := N[(N[(1.0 / N[(N[(-0.6666666666666666 * N[(b / N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.1%

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

   1. add-cbrt-cube 56.0%

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

   2. pow1/3 55.9%

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

   3. pow3 55.9%

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

   4. associate-*l* 55.9%

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

   5. unpow-prod-down 55.9%

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

   6. metadata-eval 55.9%

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

3. Applied egg-rr 55.9%

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

   1. flip-+ 56.0%

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

   2. pow2 55.9%

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

   3. add-sqr-sqrt 57.2%

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

   4. pow2 57.2%

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

   5. unpow1/3 57.4%

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

   6. *-commutative 57.4%

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

   7. cbrt-prod 57.4%

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

   8. unpow3 57.4%

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

   9. add-cbrt-cube 57.4%

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

5. Applied egg-rr 57.4%

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

   1. associate--r- 98.5%

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

   2. unpow2 98.4%

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

   3. sqr-neg 98.4%

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

   4. unpow2 98.5%

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

   5. *-commutative 98.5%

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

   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%

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

7. 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. Applied egg-rr 98.6%

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

    1. unpow-1 98.6%

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

    2. fma-udef 98.6%

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

    3. unpow2 98.6%

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

    4. *-commutative 98.6%

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

    5. +-commutative 98.6%

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

    6. fma-def 98.6%

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

    7. fma-udef 98.6%

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

    8. +-rgt-identity 98.6%

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

    9. *-commutative 98.6%

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

11. Simplified 98.6%

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

12. Taylor expanded in b around inf 82.6%

    \[\leadsto \frac{\frac{1}{\color{blue}{-0.6666666666666666 \cdot \frac{b}{a \cdot c} + 0.5 \cdot \frac{1}{b}}}}{3 \cdot a} \]

13. Final simplification 82.6%

    \[\leadsto \frac{\frac{1}{-0.6666666666666666 \cdot \frac{b}{c \cdot a} + 0.5 \cdot \frac{1}{b}}}{a \cdot 3} \]

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 56.1%

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

2. Taylor expanded in b around inf 64.0%

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

   1. *-commutative 64.0%

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

   2. associate-*l/ 64.0%

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

4. Simplified 64.0%

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

5. Final simplification 64.0%

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

Reproduce

herbie shell --seed 2023301 
(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)))