Cubic critical, narrow range

Percentage Accurate: 55.2% → 99.1%

Time: 15.4s

Alternatives: 7

Speedup: 23.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(c \cdot 3\right)\\ t_1 := \sqrt{t\_0}\\ \frac{\frac{t\_0}{\left(-b\right) - \sqrt{\left(b + t\_1\right) \cdot \left(b - t\_1\right)}}}{a \cdot 3} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* c 3.0))) (t_1 (sqrt t_0)))
   (/ (/ t_0 (- (- b) (sqrt (* (+ b t_1) (- b t_1))))) (* a 3.0))))

C

double code(double a, double b, double c) {
	double t_0 = a * (c * 3.0);
	double t_1 = sqrt(t_0);
	return (t_0 / (-b - sqrt(((b + t_1) * (b - t_1))))) / (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
    real(8) :: t_1
    t_0 = a * (c * 3.0d0)
    t_1 = sqrt(t_0)
    code = (t_0 / (-b - sqrt(((b + t_1) * (b - t_1))))) / (a * 3.0d0)
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = a * (c * 3.0);
	double t_1 = Math.sqrt(t_0);
	return (t_0 / (-b - Math.sqrt(((b + t_1) * (b - t_1))))) / (a * 3.0);
}

Python

def code(a, b, c):
	t_0 = a * (c * 3.0)
	t_1 = math.sqrt(t_0)
	return (t_0 / (-b - math.sqrt(((b + t_1) * (b - t_1))))) / (a * 3.0)

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(c * 3.0))
	t_1 = sqrt(t_0)
	return Float64(Float64(t_0 / Float64(Float64(-b) - sqrt(Float64(Float64(b + t_1) * Float64(b - t_1))))) / Float64(a * 3.0))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = a * (c * 3.0);
	t_1 = sqrt(t_0);
	tmp = (t_0 / (-b - sqrt(((b + t_1) * (b - t_1))))) / (a * 3.0);
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, N[(N[(t$95$0 / N[((-b) - N[Sqrt[N[(N[(b + t$95$1), $MachinePrecision] * N[(b - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 57.3%

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

   1. sqr-neg 57.3%

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

   2. sqr-neg 57.3%

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

   3. associate-*l* 57.3%

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

3. Simplified 57.3%

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

   1. add-sqr-sqrt 57.3%

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

   2. difference-of-squares 57.4%

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

6. Applied egg-rr 57.4%

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

   1. associate-*r* 57.4%

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

   2. *-commutative 57.4%

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

   3. associate-*l* 57.4%

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

   4. associate-*r* 57.4%

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

   5. *-commutative 57.4%

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

   6. associate-*l* 57.4%

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

8. Simplified 57.4%

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

   1. flip-+ 57.1%

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

   2. pow2 57.1%

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

   3. add-sqr-sqrt 58.5%

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

   4. associate-*r* 58.5%

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

   5. associate-*r* 58.5%

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

10. Applied egg-rr 58.5%

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

    1. unpow2 58.5%

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

    2. sqr-neg 58.5%

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

    3. unpow2 58.5%

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

    4. associate-*l* 58.5%

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

    5. *-commutative 58.5%

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

    6. associate-*l* 58.5%

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

    7. *-commutative 58.5%

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

12. Simplified 58.5%

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

13. Taylor expanded in b around 0 98.5%

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

    1. unpow2 98.5%

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

    2. rem-square-sqrt 99.1%

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

15. Simplified 99.1%

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

16. Final simplification 99.1%

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

Alternative 2: 76.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -4e-7)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -4e-7) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -4e-7)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -4e-7], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -3.9999999999999998e-7

   1. Initial program 74.0%

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

      1. /-rgt-identity 74.0%

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

      2. metadata-eval 74.0%

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

   3. Simplified 74.1%

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

   if -3.9999999999999998e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 29.9%

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

      1. sqr-neg 29.9%

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

      2. sqr-neg 29.9%

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

      3. associate-*l* 29.9%

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

   3. Simplified 29.9%

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

   5. Taylor expanded in b around inf 84.6%

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

      1. *-commutative 84.6%

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

      2. associate-*l/ 84.6%

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

   7. Simplified 84.6%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -4e-7) t_0 (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -4e-7) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    if (t_0 <= (-4d-7)) then
        tmp = t_0
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -4e-7) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	tmp = 0
	if t_0 <= -4e-7:
		tmp = t_0
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (t_0 <= -4e-7)
		tmp = t_0;
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	tmp = 0.0;
	if (t_0 <= -4e-7)
		tmp = t_0;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-7], t$95$0, N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -3.9999999999999998e-7

   1. Initial program 74.0%

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

   if -3.9999999999999998e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 29.9%

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

      1. sqr-neg 29.9%

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

      2. sqr-neg 29.9%

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

      3. associate-*l* 29.9%

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

   3. Simplified 29.9%

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

   5. Taylor expanded in b around inf 84.6%

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

      1. *-commutative 84.6%

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

      2. associate-*l/ 84.6%

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

   7. Simplified 84.6%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.8)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.8) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.8)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.8], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.80000000000000004

   1. Initial program 84.7%

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

      1. /-rgt-identity 84.7%

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

      2. metadata-eval 84.7%

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

   3. Simplified 84.9%

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

   if 1.80000000000000004 < b

   1. Initial program 49.8%

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

      1. sqr-neg 49.8%

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

      2. sqr-neg 49.8%

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

      3. associate-*l* 49.8%

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

   3. Simplified 49.8%

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

   5. Taylor expanded in b around inf 86.3%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 820:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 820.0)
   (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* a 3.0))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 820.0) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 820.0d0) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 820.0) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 820.0:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 820.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 820.0)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 820.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 820

   1. Initial program 77.3%

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

      1. sqr-neg 77.3%

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

      2. sqr-neg 77.3%

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

      3. associate-*l* 77.3%

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

   3. Simplified 77.3%

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

   if 820 < b

   1. Initial program 41.5%

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

      1. sqr-neg 41.5%

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

      2. sqr-neg 41.5%

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

      3. associate-*l* 41.5%

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

   3. Simplified 41.5%

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

   5. Taylor expanded in b around inf 75.9%

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

      1. *-commutative 75.9%

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

      2. associate-*l/ 75.9%

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

   7. Simplified 75.9%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 820:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.6% 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 57.3%

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

   1. sqr-neg 57.3%

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

   2. sqr-neg 57.3%

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

   3. associate-*l* 57.3%

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

3. Simplified 57.3%

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

5. Taylor expanded in b around inf 62.4%

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

   1. *-commutative 62.4%

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

   2. associate-*l/ 62.4%

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

7. Simplified 62.4%

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

8. Final simplification 62.4%

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

Alternative 7: 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 57.3%

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

   1. sqr-neg 57.3%

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

   2. sqr-neg 57.3%

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

   3. associate-*l* 57.3%

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

3. Simplified 57.3%

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

   1. add-sqr-sqrt 57.3%

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

   2. difference-of-squares 57.4%

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

6. Applied egg-rr 57.4%

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

   1. associate-*r* 57.4%

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

   2. *-commutative 57.4%

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

   3. associate-*l* 57.4%

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

   4. associate-*r* 57.4%

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

   5. *-commutative 57.4%

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

   6. associate-*l* 57.4%

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

8. Simplified 57.4%

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

9. Taylor expanded in b around inf 3.2%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{-1 \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right) + \sqrt{a \cdot c} \cdot \sqrt{3}}{a}} \]
10. Step-by-step derivation

    1. associate-*r/ 3.2%

       \[\leadsto \color{blue}{\frac{0.16666666666666666 \cdot \left(-1 \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right) + \sqrt{a \cdot c} \cdot \sqrt{3}\right)}{a}} \]

    2. distribute-lft1-in 3.2%

       \[\leadsto \frac{0.16666666666666666 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right)\right)}}{a} \]

    3. metadata-eval 3.2%

       \[\leadsto \frac{0.16666666666666666 \cdot \left(\color{blue}{0} \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right)\right)}{a} \]

    4. mul0-lft 3.2%

       \[\leadsto \frac{0.16666666666666666 \cdot \color{blue}{0}}{a} \]

    5. metadata-eval 3.2%

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

11. Simplified 3.2%

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

12. Final simplification 3.2%

    \[\leadsto \frac{0}{a} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))