Cubic critical, medium range

Percentage Accurate: 31.9% → 99.4%

Time: 13.4s

Alternatives: 8

Speedup: 23.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 31.9% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.6%

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

   1. add-cbrt-cube 32.7%

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

   2. cbrt-prod 32.4%

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

   3. distribute-rgt-neg-in 32.4%

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

   4. cbrt-prod 32.3%

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

   5. pow2 32.3%

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

4. Applied egg-rr 32.3%

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

   1. distribute-rgt-neg-out 32.3%

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

   2. unpow2 32.3%

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

   3. add-cube-cbrt 32.6%

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

   4. pow1 32.6%

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

   5. metadata-eval 32.6%

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

   6. pow-div 32.6%

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

   7. pow2 32.6%

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

   8. pow1 32.6%

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

   9. distribute-frac-neg 32.6%

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

   10. pow2 32.6%

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

   11. flip-+ 32.4%

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

6. Applied egg-rr 33.3%

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

   1. associate--r- 99.3%

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

   2. associate-*r* 99.2%

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

   3. *-commutative 99.2%

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

   4. associate-*r* 99.1%

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

   5. associate-*r* 99.1%

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

   6. *-commutative 99.1%

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

   7. associate-*r* 99.1%

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

8. Simplified 99.1%

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

9. Taylor expanded in b around 0 99.2%

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

    1. associate-*r* 99.3%

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

    2. *-commutative 99.3%

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

    3. *-commutative 99.3%

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

    4. *-commutative 99.3%

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

11. Simplified 99.3%

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

    1. pow2 99.3%

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

13. Applied egg-rr 99.3%

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

Alternative 2: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.6%

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

   1. add-cbrt-cube 32.7%

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

   2. cbrt-prod 32.4%

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

   3. distribute-rgt-neg-in 32.4%

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

   4. cbrt-prod 32.3%

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

   5. pow2 32.3%

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

4. Applied egg-rr 32.3%

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

   1. distribute-rgt-neg-out 32.3%

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

   2. unpow2 32.3%

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

   3. add-cube-cbrt 32.6%

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

   4. pow1 32.6%

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

   5. metadata-eval 32.6%

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

   6. pow-div 32.6%

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

   7. pow2 32.6%

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

   8. pow1 32.6%

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

   9. distribute-frac-neg 32.6%

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

   10. pow2 32.6%

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

   11. flip-+ 32.4%

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

6. Applied egg-rr 33.3%

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

   1. associate--r- 99.3%

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

   2. associate-*r* 99.2%

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

   3. *-commutative 99.2%

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

   4. associate-*r* 99.1%

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

   5. associate-*r* 99.1%

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

   6. *-commutative 99.1%

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

   7. associate-*r* 99.1%

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

8. Simplified 99.1%

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

9. Taylor expanded in b around 0 99.2%

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

    1. pow2 99.3%

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

11. Applied egg-rr 99.2%

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

12. Final simplification 99.2%

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

Alternative 3: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.00105:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right) + c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.00105)
   (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a))
   (/ (+ (* -0.375 (* a (pow (/ c b) 2.0))) (* c -0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.00105) {
		tmp = (sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	} else {
		tmp = ((-0.375 * (a * pow((c / b), 2.0))) + (c * -0.5)) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 0.00105d0) then
        tmp = (sqrt(((b * b) - (c * (3.0d0 * a)))) - b) / (3.0d0 * a)
    else
        tmp = (((-0.375d0) * (a * ((c / b) ** 2.0d0))) + (c * (-0.5d0))) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.00105) {
		tmp = (Math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	} else {
		tmp = ((-0.375 * (a * Math.pow((c / b), 2.0))) + (c * -0.5)) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.00105:
		tmp = (math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)
	else:
		tmp = ((-0.375 * (a * math.pow((c / b), 2.0))) + (c * -0.5)) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.00105)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(-0.375 * Float64(a * (Float64(c / b) ^ 2.0))) + Float64(c * -0.5)) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.00105)
		tmp = (sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	else
		tmp = ((-0.375 * (a * ((c / b) ^ 2.0))) + (c * -0.5)) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.00105], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.375 * N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.00104999999999999994

   1. Initial program 80.7%

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

   if 0.00104999999999999994 < b

   1. Initial program 29.0%

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

      1. add-cbrt-cube 29.2%

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

      2. cbrt-prod 29.0%

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

      3. distribute-rgt-neg-in 29.0%

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

      4. cbrt-prod 28.9%

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

      5. pow2 28.9%

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

   4. Applied egg-rr 28.9%

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

   5. Taylor expanded in b around inf 92.5%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   6. Step-by-step derivation

      1. +-commutative 92.5%

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

      2. fma-define 92.5%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.5 \cdot c\right)}}{b} \]

      3. associate-/l* 92.5%

         \[\leadsto \frac{\mathsf{fma}\left(-0.375, \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{2}}}, -0.5 \cdot c\right)}{b} \]

      4. unpow2 92.5%

         \[\leadsto \frac{\mathsf{fma}\left(-0.375, a \cdot \frac{\color{blue}{c \cdot c}}{{b}^{2}}, -0.5 \cdot c\right)}{b} \]

      5. unpow2 92.5%

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

      6. times-frac 92.5%

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

      7. unpow1 92.5%

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

      8. pow-plus 92.5%

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

      9. metadata-eval 92.5%

         \[\leadsto \frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{\color{blue}{2}}, -0.5 \cdot c\right)}{b} \]

      10. *-commutative 92.5%

          \[\leadsto \frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, \color{blue}{c \cdot -0.5}\right)}{b} \]

   7. Simplified 92.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}} \]
   8. Step-by-step derivation

      1. fma-undefine 92.5%

         \[\leadsto \frac{\color{blue}{-0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right) + c \cdot -0.5}}{b} \]

   9. Applied egg-rr 92.5%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.00105:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right) + c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return ((-0.375 * (a * pow((c / b), 2.0))) + (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 = (((-0.375d0) * (a * ((c / b) ** 2.0d0))) + (c * (-0.5d0))) / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.6%

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

   1. add-cbrt-cube 32.7%

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

   2. cbrt-prod 32.4%

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

   3. distribute-rgt-neg-in 32.4%

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

   4. cbrt-prod 32.3%

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

   5. pow2 32.3%

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

4. Applied egg-rr 32.3%

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

5. Taylor expanded in b around inf 90.0%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
6. Step-by-step derivation

   1. +-commutative 90.0%

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

   2. fma-define 90.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.5 \cdot c\right)}}{b} \]

   3. associate-/l* 90.0%

      \[\leadsto \frac{\mathsf{fma}\left(-0.375, \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{2}}}, -0.5 \cdot c\right)}{b} \]

   4. unpow2 90.0%

      \[\leadsto \frac{\mathsf{fma}\left(-0.375, a \cdot \frac{\color{blue}{c \cdot c}}{{b}^{2}}, -0.5 \cdot c\right)}{b} \]

   5. unpow2 90.0%

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

   6. times-frac 90.0%

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

   7. unpow1 90.0%

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

   8. pow-plus 90.0%

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

   9. metadata-eval 90.0%

      \[\leadsto \frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{\color{blue}{2}}, -0.5 \cdot c\right)}{b} \]

   10. *-commutative 90.0%

       \[\leadsto \frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, \color{blue}{c \cdot -0.5}\right)}{b} \]

7. Simplified 90.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}} \]
8. Step-by-step derivation

   1. fma-undefine 90.0%

      \[\leadsto \frac{\color{blue}{-0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right) + c \cdot -0.5}}{b} \]

9. Applied egg-rr 90.0%

   \[\leadsto \frac{\color{blue}{-0.375 \cdot \left(a \cdot {\left(\frac{c}{b}\right)}^{2}\right) + c \cdot -0.5}}{b} \]
10. Add Preprocessing

Alternative 5: 90.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(-0.375 \cdot \frac{c \cdot a}{{b}^{2}} - 0.5\right)}{b} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.6%

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

   1. add-cbrt-cube 32.7%

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

   2. cbrt-prod 32.4%

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

   3. distribute-rgt-neg-in 32.4%

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

   4. cbrt-prod 32.3%

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

   5. pow2 32.3%

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

4. Applied egg-rr 32.3%

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

5. Taylor expanded in b around inf 90.0%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
6. Step-by-step derivation

   1. +-commutative 90.0%

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

   2. fma-define 90.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.5 \cdot c\right)}}{b} \]

   3. associate-/l* 90.0%

      \[\leadsto \frac{\mathsf{fma}\left(-0.375, \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{2}}}, -0.5 \cdot c\right)}{b} \]

   4. unpow2 90.0%

      \[\leadsto \frac{\mathsf{fma}\left(-0.375, a \cdot \frac{\color{blue}{c \cdot c}}{{b}^{2}}, -0.5 \cdot c\right)}{b} \]

   5. unpow2 90.0%

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

   6. times-frac 90.0%

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

   7. unpow1 90.0%

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

   8. pow-plus 90.0%

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

   9. metadata-eval 90.0%

      \[\leadsto \frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{\color{blue}{2}}, -0.5 \cdot c\right)}{b} \]

   10. *-commutative 90.0%

       \[\leadsto \frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, \color{blue}{c \cdot -0.5}\right)}{b} \]

7. Simplified 90.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}} \]

8. Taylor expanded in c around 0 90.0%

   \[\leadsto \frac{\color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5\right)}}{b} \]

9. Final simplification 90.0%

   \[\leadsto \frac{c \cdot \left(-0.375 \cdot \frac{c \cdot a}{{b}^{2}} - 0.5\right)}{b} \]
10. Add Preprocessing

Alternative 6: 90.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.6%

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

3. Simplified 32.8%

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

5. Taylor expanded in c around 0 89.7%

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

   1. associate-*r/ 89.7%

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

   2. metadata-eval 89.7%

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

7. Simplified 89.7%

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

8. Final simplification 89.7%

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

Alternative 7: 81.0% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

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 / b) * (-0.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.6%

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

3. Simplified 32.8%

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

5. Taylor expanded in b around inf 80.2%

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

6. Final simplification 80.2%

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

Alternative 8: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0 / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.6%

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

   1. add-cbrt-cube 32.7%

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

   2. cbrt-prod 32.4%

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

   3. distribute-rgt-neg-in 32.4%

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

   4. cbrt-prod 32.3%

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

   5. pow2 32.3%

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

4. Applied egg-rr 32.3%

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

5. Taylor expanded in a around 0 3.2%

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

   1. associate-*r/ 3.2%

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

   2. distribute-rgt1-in 3.2%

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

   3. metadata-eval 3.2%

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

   4. mul0-lft 3.2%

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

   5. metadata-eval 3.2%

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

7. Simplified 3.2%

   \[\leadsto \color{blue}{\frac{0}{a}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024170 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))