Cubic critical, wide range

Percentage Accurate: 17.8% → 99.4%

Time: 9.3s

Alternatives: 5

Speedup: 23.2×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

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: 17.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((c * (a * (27.0d0 ** 0.3333333333333333d0))) + ((b ** 2.0d0) - (b ** 2.0d0))) / -(b + sqrt(((b ** 2.0d0) - (3.0d0 * (c * a)))))) / (a * 3.0d0)
end function

Java

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

Python

def code(a, b, c):
	return (((c * (a * math.pow(27.0, 0.3333333333333333))) + (math.pow(b, 2.0) - math.pow(b, 2.0))) / -(b + math.sqrt((math.pow(b, 2.0) - (3.0 * (c * a)))))) / (a * 3.0)

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (((c * (a * (27.0 ^ 0.3333333333333333))) + ((b ^ 2.0) - (b ^ 2.0))) / -(b + sqrt(((b ^ 2.0) - (3.0 * (c * a)))))) / (a * 3.0);
end

Wolfram

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

Derivation

1. Initial program 16.5%

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

   1. add-cbrt-cube 16.5%

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

   2. pow1/3 16.4%

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

   3. pow3 16.4%

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

   4. associate-*l* 16.4%

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

   5. unpow-prod-down 16.4%

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

   6. metadata-eval 16.4%

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

3. Applied egg-rr 16.4%

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

   1. flip-+ 16.5%

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

   2. pow2 16.5%

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

   3. add-sqr-sqrt 17.0%

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

   4. pow2 17.0%

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

   5. unpow1/3 17.0%

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

   6. *-commutative 17.0%

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

   7. cbrt-prod 17.1%

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

   8. unpow3 17.0%

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

   9. add-cbrt-cube 17.0%

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

5. Applied egg-rr 17.1%

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

   1. associate--r- 98.6%

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

   2. +-commutative 98.6%

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

   3. *-commutative 98.6%

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

   4. associate-*l* 98.6%

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

   5. unpow2 98.6%

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

   6. sqr-neg 98.6%

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

   7. unpow2 98.6%

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

   8. *-commutative 98.6%

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

   9. associate-*l* 98.6%

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

7. Simplified 98.6%

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

   1. pow1/3 99.3%

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

9. Applied egg-rr 99.3%

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

10. Taylor expanded in c around 0 99.3%

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

    1. *-commutative 99.3%

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

12. Simplified 99.3%

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

13. Final simplification 99.3%

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

Alternative 2: 99.2% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.5%

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

   1. add-cbrt-cube 16.5%

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

   2. pow1/3 16.4%

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

   3. pow3 16.4%

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

   4. associate-*l* 16.4%

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

   5. unpow-prod-down 16.4%

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

   6. metadata-eval 16.4%

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

3. Applied egg-rr 16.4%

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

   1. flip-+ 16.5%

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

   2. pow2 16.5%

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

   3. add-sqr-sqrt 17.0%

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

   4. pow2 17.0%

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

   5. unpow1/3 17.0%

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

   6. *-commutative 17.0%

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

   7. cbrt-prod 17.1%

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

   8. unpow3 17.0%

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

   9. add-cbrt-cube 17.0%

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

5. Applied egg-rr 17.1%

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

   1. associate--r- 98.6%

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

   2. +-commutative 98.6%

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

   3. *-commutative 98.6%

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

   4. associate-*l* 98.6%

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

   5. unpow2 98.6%

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

   6. sqr-neg 98.6%

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

   7. unpow2 98.6%

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

   8. *-commutative 98.6%

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

   9. associate-*l* 98.6%

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

7. Simplified 98.6%

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

8. Taylor expanded in c around 0 99.2%

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

   1. *-commutative 99.3%

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

10. Simplified 99.2%

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

11. Final simplification 99.2%

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

Alternative 3: 98.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.5%

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

   1. add-cbrt-cube 16.5%

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

   2. pow1/3 16.4%

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

   3. pow3 16.4%

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

   4. associate-*l* 16.4%

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

   5. unpow-prod-down 16.4%

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

   6. metadata-eval 16.4%

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

3. Applied egg-rr 16.4%

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

   1. flip-+ 16.5%

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

   2. pow2 16.5%

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

   3. add-sqr-sqrt 17.0%

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

   4. pow2 17.0%

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

   5. unpow1/3 17.0%

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

   6. *-commutative 17.0%

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

   7. cbrt-prod 17.1%

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

   8. unpow3 17.0%

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

   9. add-cbrt-cube 17.0%

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

5. Applied egg-rr 17.1%

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

   1. associate--r- 98.6%

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

   2. +-commutative 98.6%

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

   3. *-commutative 98.6%

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

   4. associate-*l* 98.6%

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

   5. unpow2 98.6%

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

   6. sqr-neg 98.6%

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

   7. unpow2 98.6%

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

   8. *-commutative 98.6%

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

   9. associate-*l* 98.6%

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

7. Simplified 98.6%

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

   1. expm1-log1p-u 83.9%

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

   2. expm1-udef 19.9%

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

9. Applied egg-rr 19.9%

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

    1. expm1-def 83.9%

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

    2. expm1-log1p 98.5%

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

    3. associate-/r* 98.6%

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

    4. *-rgt-identity 98.6%

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

    5. associate-*r/ 98.6%

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

    6. associate-/l* 98.6%

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

    7. fma-udef 98.6%

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

    8. +-rgt-identity 98.6%

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

    9. *-commutative 98.6%

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

11. Simplified 99.0%

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

12. Final simplification 99.0%

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

Alternative 4: 95.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.5%

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

2. Taylor expanded in b around inf 96.3%

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

3. Final simplification 96.3%

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

Alternative 5: 90.4% 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 16.5%

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

2. Taylor expanded in b around inf 91.5%

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

   1. *-commutative 91.5%

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

   2. associate-*l/ 91.5%

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

4. Simplified 91.5%

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

5. Final simplification 91.5%

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

Reproduce

herbie shell --seed 2023301 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))