Cubic critical, medium range

Percentage Accurate: 31.7% → 99.3%

Time: 15.0s

Alternatives: 6

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: 31.7% 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, 0.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double t_0 = c * (a * 3.0);
	return (((pow(b, 2.0) - pow(-b, 2.0)) - t_0) / (b + sqrt((pow(b, 2.0) - t_0)))) / (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
    real(8) :: t_0
    t_0 = c * (a * 3.0d0)
    code = ((((b ** 2.0d0) - (-b ** 2.0d0)) - t_0) / (b + sqrt(((b ** 2.0d0) - t_0)))) / (a * 3.0d0)
end function

Java

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

Python

def code(a, b, c):
	t_0 = c * (a * 3.0)
	return (((math.pow(b, 2.0) - math.pow(-b, 2.0)) - t_0) / (b + math.sqrt((math.pow(b, 2.0) - t_0)))) / (a * 3.0)

Julia

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

MATLAB

function tmp = code(a, b, c)
	t_0 = c * (a * 3.0);
	tmp = ((((b ^ 2.0) - (-b ^ 2.0)) - t_0) / (b + sqrt(((b ^ 2.0) - t_0)))) / (a * 3.0);
end

Wolfram

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

Derivation

1. Initial program 28.7%

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

   1. add-cbrt-cube 28.7%

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

   2. pow3 28.7%

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

4. Applied egg-rr 28.7%

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

   1. flip-+ 28.8%

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

   2. pow2 28.8%

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

   3. add-sqr-sqrt 29.6%

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

   4. pow2 29.6%

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

   5. *-commutative 29.6%

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

   6. *-commutative 29.6%

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

   7. pow2 29.6%

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

   8. *-commutative 29.6%

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

   9. *-commutative 29.6%

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

6. Applied egg-rr 29.6%

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

   1. associate--r- 99.0%

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

8. Simplified 99.0%

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

9. Taylor expanded in a around 0 99.4%

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

    1. *-commutative 99.4%

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

11. Simplified 99.4%

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

12. Final simplification 99.4%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 28.7%

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

   1. add-cbrt-cube 28.7%

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

   2. pow3 28.7%

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

4. Applied egg-rr 28.7%

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

   1. flip-+ 28.8%

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

   2. pow2 28.8%

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

   3. add-sqr-sqrt 29.6%

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

   4. pow2 29.6%

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

   5. *-commutative 29.6%

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

   6. *-commutative 29.6%

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

   7. pow2 29.6%

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

   8. *-commutative 29.6%

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

   9. *-commutative 29.6%

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

6. Applied egg-rr 29.6%

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

   1. associate--r- 99.0%

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

8. Simplified 99.0%

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

9. Taylor expanded in a around 0 99.4%

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

    1. *-commutative 99.4%

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

11. Simplified 99.4%

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

    1. div-inv 99.2%

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

    2. +-commutative 99.2%

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

    3. *-commutative 99.2%

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

    4. fma-define 99.2%

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

    5. *-commutative 99.2%

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

    6. neg-mul-1 99.2%

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

    7. unpow-prod-down 99.2%

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

    8. metadata-eval 99.2%

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

    9. *-un-lft-identity 99.2%

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

13. Applied egg-rr 99.2%

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

    1. associate-*r/ 99.4%

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

    2. *-rgt-identity 99.4%

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

    3. fma-undefine 99.4%

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

    4. +-inverses 99.4%

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

    5. +-rgt-identity 99.4%

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

    6. unpow2 99.4%

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

    7. fmm-def 99.4%

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

    8. distribute-rgt-neg-in 99.4%

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

    9. *-commutative 99.4%

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

    10. distribute-lft-neg-in 99.4%

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

    11. metadata-eval 99.4%

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

    12. *-commutative 99.4%

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

15. Simplified 99.4%

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

Alternative 3: 90.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.7%

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

   1. add-cbrt-cube 28.7%

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

   2. pow3 28.7%

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

4. Applied egg-rr 28.7%

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

5. Taylor expanded in b around inf 91.8%

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

   1. fma-define 91.8%

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

   2. associate-/l* 91.8%

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

   3. unpow2 91.8%

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

   4. unpow2 91.8%

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

   5. times-frac 91.8%

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

   6. unpow1 91.8%

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

   7. pow-plus 91.8%

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

   8. metadata-eval 91.8%

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

7. Simplified 91.8%

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

   1. fma-undefine 91.8%

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

   2. associate-*r* 91.8%

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

   3. clear-num 91.8%

      \[\leadsto \frac{-0.5 \cdot c + \left(-0.375 \cdot a\right) \cdot {\color{blue}{\left(\frac{1}{\frac{b}{c}}\right)}}^{2}}{b} \]

   4. inv-pow 91.8%

      \[\leadsto \frac{-0.5 \cdot c + \left(-0.375 \cdot a\right) \cdot {\color{blue}{\left({\left(\frac{b}{c}\right)}^{-1}\right)}}^{2}}{b} \]

   5. pow-pow 91.8%

      \[\leadsto \frac{-0.5 \cdot c + \left(-0.375 \cdot a\right) \cdot \color{blue}{{\left(\frac{b}{c}\right)}^{\left(-1 \cdot 2\right)}}}{b} \]

   6. metadata-eval 91.8%

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

9. Applied egg-rr 91.8%

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

10. Final simplification 91.8%

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

Alternative 4: 90.5% 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 28.7%

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

   1. add-cbrt-cube 28.7%

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

   2. pow3 28.7%

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

4. Applied egg-rr 28.7%

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

5. Taylor expanded in b around inf 91.8%

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

   1. fma-define 91.8%

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

   2. associate-/l* 91.8%

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

   3. unpow2 91.8%

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

   4. unpow2 91.8%

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

   5. times-frac 91.8%

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

   6. unpow1 91.8%

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

   7. pow-plus 91.8%

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

   8. metadata-eval 91.8%

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

7. Simplified 91.8%

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

8. Taylor expanded in c around 0 91.8%

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

9. Final simplification 91.8%

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

Alternative 5: 90.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return c * ((-0.375 * (a * (c / 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) * (a * (c / (b ** 3.0d0)))) - (0.5d0 / b))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.7%

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

3. Simplified 28.7%

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

5. Taylor expanded in c around 0 91.6%

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

   1. associate-/l* 91.6%

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

   2. associate-*r/ 91.6%

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

   3. metadata-eval 91.6%

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

7. Simplified 91.6%

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

Alternative 6: 81.1% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.7%

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

3. Simplified 28.7%

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

5. Taylor expanded in b around inf 83.0%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024180 
(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)))