Cubic critical, narrow range

Percentage Accurate: 55.2% → 92.3%

Time: 18.4s

Alternatives: 14

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: 92.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot {a}^{2}\\ t_1 := -0.75 \cdot t_0 + t_0 \cdot 0.375\\ t_2 := \left(a \cdot c\right) \cdot 3\\ \mathbf{if}\;b \leq 0.009:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(t_2 - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - t_2}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;{\left(-3 \cdot \frac{-0.75 \cdot \left(a \cdot \left(c \cdot t_1\right)\right) + \left(-0.2222222222222222 \cdot \frac{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {c}^{2}} + 0.5625 \cdot \left({c}^{2} \cdot {a}^{3}\right)\right)}{{b}^{5}} + \left(-3 \cdot \frac{t_1}{{b}^{3}} + \left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (pow a 2.0)))
        (t_1 (+ (* -0.75 t_0) (* t_0 0.375)))
        (t_2 (* (* a c) 3.0)))
   (if (<= b 0.009)
     (/
      (/
       (+ (pow (- b) 2.0) (- t_2 (pow b 2.0)))
       (- (- b) (sqrt (- (pow b 2.0) t_2))))
      (* a 3.0))
     (pow
      (+
       (*
        -3.0
        (/
         (+
          (* -0.75 (* a (* c t_1)))
          (+
           (*
            -0.2222222222222222
            (/
             (+
              (* 5.0625 (* (pow a 4.0) (pow c 4.0)))
              (pow (* -1.125 (* (pow a 2.0) (pow c 2.0))) 2.0))
             (* a (pow c 2.0))))
           (* 0.5625 (* (pow c 2.0) (pow a 3.0)))))
         (pow b 5.0)))
       (+ (* -3.0 (/ t_1 (pow b 3.0))) (+ (* -2.0 (/ b c)) (* 1.5 (/ a b)))))
      -1.0))))

C

double code(double a, double b, double c) {
	double t_0 = c * pow(a, 2.0);
	double t_1 = (-0.75 * t_0) + (t_0 * 0.375);
	double t_2 = (a * c) * 3.0;
	double tmp;
	if (b <= 0.009) {
		tmp = ((pow(-b, 2.0) + (t_2 - pow(b, 2.0))) / (-b - sqrt((pow(b, 2.0) - t_2)))) / (a * 3.0);
	} else {
		tmp = pow(((-3.0 * (((-0.75 * (a * (c * t_1))) + ((-0.2222222222222222 * (((5.0625 * (pow(a, 4.0) * pow(c, 4.0))) + pow((-1.125 * (pow(a, 2.0) * pow(c, 2.0))), 2.0)) / (a * pow(c, 2.0)))) + (0.5625 * (pow(c, 2.0) * pow(a, 3.0))))) / pow(b, 5.0))) + ((-3.0 * (t_1 / pow(b, 3.0))) + ((-2.0 * (b / c)) + (1.5 * (a / b))))), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = c * (a ** 2.0d0)
    t_1 = ((-0.75d0) * t_0) + (t_0 * 0.375d0)
    t_2 = (a * c) * 3.0d0
    if (b <= 0.009d0) then
        tmp = (((-b ** 2.0d0) + (t_2 - (b ** 2.0d0))) / (-b - sqrt(((b ** 2.0d0) - t_2)))) / (a * 3.0d0)
    else
        tmp = (((-3.0d0) * ((((-0.75d0) * (a * (c * t_1))) + (((-0.2222222222222222d0) * (((5.0625d0 * ((a ** 4.0d0) * (c ** 4.0d0))) + (((-1.125d0) * ((a ** 2.0d0) * (c ** 2.0d0))) ** 2.0d0)) / (a * (c ** 2.0d0)))) + (0.5625d0 * ((c ** 2.0d0) * (a ** 3.0d0))))) / (b ** 5.0d0))) + (((-3.0d0) * (t_1 / (b ** 3.0d0))) + (((-2.0d0) * (b / c)) + (1.5d0 * (a / b))))) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * Math.pow(a, 2.0);
	double t_1 = (-0.75 * t_0) + (t_0 * 0.375);
	double t_2 = (a * c) * 3.0;
	double tmp;
	if (b <= 0.009) {
		tmp = ((Math.pow(-b, 2.0) + (t_2 - Math.pow(b, 2.0))) / (-b - Math.sqrt((Math.pow(b, 2.0) - t_2)))) / (a * 3.0);
	} else {
		tmp = Math.pow(((-3.0 * (((-0.75 * (a * (c * t_1))) + ((-0.2222222222222222 * (((5.0625 * (Math.pow(a, 4.0) * Math.pow(c, 4.0))) + Math.pow((-1.125 * (Math.pow(a, 2.0) * Math.pow(c, 2.0))), 2.0)) / (a * Math.pow(c, 2.0)))) + (0.5625 * (Math.pow(c, 2.0) * Math.pow(a, 3.0))))) / Math.pow(b, 5.0))) + ((-3.0 * (t_1 / Math.pow(b, 3.0))) + ((-2.0 * (b / c)) + (1.5 * (a / b))))), -1.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c * math.pow(a, 2.0)
	t_1 = (-0.75 * t_0) + (t_0 * 0.375)
	t_2 = (a * c) * 3.0
	tmp = 0
	if b <= 0.009:
		tmp = ((math.pow(-b, 2.0) + (t_2 - math.pow(b, 2.0))) / (-b - math.sqrt((math.pow(b, 2.0) - t_2)))) / (a * 3.0)
	else:
		tmp = math.pow(((-3.0 * (((-0.75 * (a * (c * t_1))) + ((-0.2222222222222222 * (((5.0625 * (math.pow(a, 4.0) * math.pow(c, 4.0))) + math.pow((-1.125 * (math.pow(a, 2.0) * math.pow(c, 2.0))), 2.0)) / (a * math.pow(c, 2.0)))) + (0.5625 * (math.pow(c, 2.0) * math.pow(a, 3.0))))) / math.pow(b, 5.0))) + ((-3.0 * (t_1 / math.pow(b, 3.0))) + ((-2.0 * (b / c)) + (1.5 * (a / b))))), -1.0)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c * (a ^ 2.0))
	t_1 = Float64(Float64(-0.75 * t_0) + Float64(t_0 * 0.375))
	t_2 = Float64(Float64(a * c) * 3.0)
	tmp = 0.0
	if (b <= 0.009)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) + Float64(t_2 - (b ^ 2.0))) / Float64(Float64(-b) - sqrt(Float64((b ^ 2.0) - t_2)))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-3.0 * Float64(Float64(Float64(-0.75 * Float64(a * Float64(c * t_1))) + Float64(Float64(-0.2222222222222222 * Float64(Float64(Float64(5.0625 * Float64((a ^ 4.0) * (c ^ 4.0))) + (Float64(-1.125 * Float64((a ^ 2.0) * (c ^ 2.0))) ^ 2.0)) / Float64(a * (c ^ 2.0)))) + Float64(0.5625 * Float64((c ^ 2.0) * (a ^ 3.0))))) / (b ^ 5.0))) + Float64(Float64(-3.0 * Float64(t_1 / (b ^ 3.0))) + Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))))) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = c * (a ^ 2.0);
	t_1 = (-0.75 * t_0) + (t_0 * 0.375);
	t_2 = (a * c) * 3.0;
	tmp = 0.0;
	if (b <= 0.009)
		tmp = (((-b ^ 2.0) + (t_2 - (b ^ 2.0))) / (-b - sqrt(((b ^ 2.0) - t_2)))) / (a * 3.0);
	else
		tmp = ((-3.0 * (((-0.75 * (a * (c * t_1))) + ((-0.2222222222222222 * (((5.0625 * ((a ^ 4.0) * (c ^ 4.0))) + ((-1.125 * ((a ^ 2.0) * (c ^ 2.0))) ^ 2.0)) / (a * (c ^ 2.0)))) + (0.5625 * ((c ^ 2.0) * (a ^ 3.0))))) / (b ^ 5.0))) + ((-3.0 * (t_1 / (b ^ 3.0))) + ((-2.0 * (b / c)) + (1.5 * (a / b))))) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-0.75 * t$95$0), $MachinePrecision] + N[(t$95$0 * 0.375), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * c), $MachinePrecision] * 3.0), $MachinePrecision]}, If[LessEqual[b, 0.009], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$2 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(-3.0 * N[(N[(N[(-0.75 * N[(a * N[(c * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.2222222222222222 * N[(N[(N[(5.0625 * N[(N[Power[a, 4.0], $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(-1.125 * N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5625 * N[(N[Power[c, 2.0], $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-3.0 * N[(t$95$1 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 2: 92.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot c\right) \cdot 3\\ \mathbf{if}\;b \leq 0.0096:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(t_0 - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - t_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* a c) 3.0)))
   (if (<= b 0.0096)
     (/
      (/
       (+ (pow (- b) 2.0) (- t_0 (pow b 2.0)))
       (- (- b) (sqrt (- (pow b 2.0) t_0))))
      (* a 3.0))
     (+
      (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
      (+
       (* -0.5 (/ c b))
       (+
        (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))
        (*
         -0.16666666666666666
         (* (/ (pow (* a c) 4.0) a) (/ 6.328125 (pow b 7.0))))))))))

C

double code(double a, double b, double c) {
	double t_0 = (a * c) * 3.0;
	double tmp;
	if (b <= 0.0096) {
		tmp = ((pow(-b, 2.0) + (t_0 - pow(b, 2.0))) / (-b - sqrt((pow(b, 2.0) - t_0)))) / (a * 3.0);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (-0.16666666666666666 * ((pow((a * c), 4.0) / a) * (6.328125 / pow(b, 7.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a * c) * 3.0d0
    if (b <= 0.0096d0) then
        tmp = (((-b ** 2.0d0) + (t_0 - (b ** 2.0d0))) / (-b - sqrt(((b ** 2.0d0) - t_0)))) / (a * 3.0d0)
    else
        tmp = ((-0.5625d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + (((-0.5d0) * (c / b)) + (((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0))) + ((-0.16666666666666666d0) * ((((a * c) ** 4.0d0) / a) * (6.328125d0 / (b ** 7.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (a * c) * 3.0;
	double tmp;
	if (b <= 0.0096) {
		tmp = ((Math.pow(-b, 2.0) + (t_0 - Math.pow(b, 2.0))) / (-b - Math.sqrt((Math.pow(b, 2.0) - t_0)))) / (a * 3.0);
	} else {
		tmp = (-0.5625 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))) + (-0.16666666666666666 * ((Math.pow((a * c), 4.0) / a) * (6.328125 / Math.pow(b, 7.0))))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (a * c) * 3.0
	tmp = 0
	if b <= 0.0096:
		tmp = ((math.pow(-b, 2.0) + (t_0 - math.pow(b, 2.0))) / (-b - math.sqrt((math.pow(b, 2.0) - t_0)))) / (a * 3.0)
	else:
		tmp = (-0.5625 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))) + (-0.16666666666666666 * ((math.pow((a * c), 4.0) / a) * (6.328125 / math.pow(b, 7.0))))))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(a * c) * 3.0)
	tmp = 0.0
	if (b <= 0.0096)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) + Float64(t_0 - (b ^ 2.0))) / Float64(Float64(-b) - sqrt(Float64((b ^ 2.0) - t_0)))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-0.16666666666666666 * Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(6.328125 / (b ^ 7.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (a * c) * 3.0;
	tmp = 0.0;
	if (b <= 0.0096)
		tmp = (((-b ^ 2.0) + (t_0 - (b ^ 2.0))) / (-b - sqrt(((b ^ 2.0) - t_0)))) / (a * 3.0);
	else
		tmp = (-0.5625 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0))) + (-0.16666666666666666 * ((((a * c) ^ 4.0) / a) * (6.328125 / (b ^ 7.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] * 3.0), $MachinePrecision]}, If[LessEqual[b, 0.0096], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $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], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(6.328125 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 3: 90.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot {a}^{2}\\ t_1 := \mathsf{fma}\left(-3, a \cdot c, {b}^{2}\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -5:\\ \;\;\;\;\frac{\frac{{b}^{2} - t_1}{\left(-b\right) - \sqrt{t_1}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;{\left(-3 \cdot \frac{-0.75 \cdot t_0 + t_0 \cdot 0.375}{{b}^{3}} + \left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (pow a 2.0))) (t_1 (fma -3.0 (* a c) (pow b 2.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -5.0)
     (/ (/ (- (pow b 2.0) t_1) (- (- b) (sqrt t_1))) (* a 3.0))
     (pow
      (+
       (* -3.0 (/ (+ (* -0.75 t_0) (* t_0 0.375)) (pow b 3.0)))
       (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))))
      -1.0))))

C

double code(double a, double b, double c) {
	double t_0 = c * pow(a, 2.0);
	double t_1 = fma(-3.0, (a * c), pow(b, 2.0));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -5.0) {
		tmp = ((pow(b, 2.0) - t_1) / (-b - sqrt(t_1))) / (a * 3.0);
	} else {
		tmp = pow(((-3.0 * (((-0.75 * t_0) + (t_0 * 0.375)) / pow(b, 3.0))) + ((-2.0 * (b / c)) + (1.5 * (a / b)))), -1.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(c * (a ^ 2.0))
	t_1 = fma(-3.0, Float64(a * c), (b ^ 2.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -5.0)
		tmp = Float64(Float64(Float64((b ^ 2.0) - t_1) / Float64(Float64(-b) - sqrt(t_1))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-3.0 * Float64(Float64(Float64(-0.75 * t_0) + Float64(t_0 * 0.375)) / (b ^ 3.0))) + Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b)))) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-3.0 * N[(a * c), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, 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], -5.0], N[(N[(N[(N[Power[b, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(-3.0 * N[(N[(N[(-0.75 * t$95$0), $MachinePrecision] + N[(t$95$0 * 0.375), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 4: 90.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot {a}^{2}\\ t_1 := \left(a \cdot c\right) \cdot 3\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -5:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(t_1 - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - t_1}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;{\left(-3 \cdot \frac{-0.75 \cdot t_0 + t_0 \cdot 0.375}{{b}^{3}} + \left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (pow a 2.0))) (t_1 (* (* a c) 3.0)))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -5.0)
     (/
      (/
       (+ (pow (- b) 2.0) (- t_1 (pow b 2.0)))
       (- (- b) (sqrt (- (pow b 2.0) t_1))))
      (* a 3.0))
     (pow
      (+
       (* -3.0 (/ (+ (* -0.75 t_0) (* t_0 0.375)) (pow b 3.0)))
       (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))))
      -1.0))))

C

double code(double a, double b, double c) {
	double t_0 = c * pow(a, 2.0);
	double t_1 = (a * c) * 3.0;
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -5.0) {
		tmp = ((pow(-b, 2.0) + (t_1 - pow(b, 2.0))) / (-b - sqrt((pow(b, 2.0) - t_1)))) / (a * 3.0);
	} else {
		tmp = pow(((-3.0 * (((-0.75 * t_0) + (t_0 * 0.375)) / pow(b, 3.0))) + ((-2.0 * (b / c)) + (1.5 * (a / b)))), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c * (a ** 2.0d0)
    t_1 = (a * c) * 3.0d0
    if (((sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)) <= (-5.0d0)) then
        tmp = (((-b ** 2.0d0) + (t_1 - (b ** 2.0d0))) / (-b - sqrt(((b ** 2.0d0) - t_1)))) / (a * 3.0d0)
    else
        tmp = (((-3.0d0) * ((((-0.75d0) * t_0) + (t_0 * 0.375d0)) / (b ** 3.0d0))) + (((-2.0d0) * (b / c)) + (1.5d0 * (a / b)))) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * Math.pow(a, 2.0);
	double t_1 = (a * c) * 3.0;
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -5.0) {
		tmp = ((Math.pow(-b, 2.0) + (t_1 - Math.pow(b, 2.0))) / (-b - Math.sqrt((Math.pow(b, 2.0) - t_1)))) / (a * 3.0);
	} else {
		tmp = Math.pow(((-3.0 * (((-0.75 * t_0) + (t_0 * 0.375)) / Math.pow(b, 3.0))) + ((-2.0 * (b / c)) + (1.5 * (a / b)))), -1.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c * math.pow(a, 2.0)
	t_1 = (a * c) * 3.0
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -5.0:
		tmp = ((math.pow(-b, 2.0) + (t_1 - math.pow(b, 2.0))) / (-b - math.sqrt((math.pow(b, 2.0) - t_1)))) / (a * 3.0)
	else:
		tmp = math.pow(((-3.0 * (((-0.75 * t_0) + (t_0 * 0.375)) / math.pow(b, 3.0))) + ((-2.0 * (b / c)) + (1.5 * (a / b)))), -1.0)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c * (a ^ 2.0))
	t_1 = Float64(Float64(a * c) * 3.0)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -5.0)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) + Float64(t_1 - (b ^ 2.0))) / Float64(Float64(-b) - sqrt(Float64((b ^ 2.0) - t_1)))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-3.0 * Float64(Float64(Float64(-0.75 * t_0) + Float64(t_0 * 0.375)) / (b ^ 3.0))) + Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b)))) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = c * (a ^ 2.0);
	t_1 = (a * c) * 3.0;
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -5.0)
		tmp = (((-b ^ 2.0) + (t_1 - (b ^ 2.0))) / (-b - sqrt(((b ^ 2.0) - t_1)))) / (a * 3.0);
	else
		tmp = ((-3.0 * (((-0.75 * t_0) + (t_0 * 0.375)) / (b ^ 3.0))) + ((-2.0 * (b / c)) + (1.5 * (a / b)))) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * c), $MachinePrecision] * 3.0), $MachinePrecision]}, 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], -5.0], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$1 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(-3.0 * N[(N[(N[(-0.75 * t$95$0), $MachinePrecision] + N[(t$95$0 * 0.375), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 5: 86.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot c\right) \cdot 3\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.005:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} + \left(t_0 - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - t_0}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;{\left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* a c) 3.0)))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.005)
     (/
      (/
       (+ (pow (- b) 2.0) (- t_0 (pow b 2.0)))
       (- (- b) (sqrt (- (pow b 2.0) t_0))))
      (* a 3.0))
     (pow (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))) -1.0))))

C

double code(double a, double b, double c) {
	double t_0 = (a * c) * 3.0;
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.005) {
		tmp = ((pow(-b, 2.0) + (t_0 - pow(b, 2.0))) / (-b - sqrt((pow(b, 2.0) - t_0)))) / (a * 3.0);
	} else {
		tmp = pow(((-2.0 * (b / c)) + (1.5 * (a / b))), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a * c) * 3.0d0
    if (((sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)) <= (-0.005d0)) then
        tmp = (((-b ** 2.0d0) + (t_0 - (b ** 2.0d0))) / (-b - sqrt(((b ** 2.0d0) - t_0)))) / (a * 3.0d0)
    else
        tmp = (((-2.0d0) * (b / c)) + (1.5d0 * (a / b))) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (a * c) * 3.0;
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.005) {
		tmp = ((Math.pow(-b, 2.0) + (t_0 - Math.pow(b, 2.0))) / (-b - Math.sqrt((Math.pow(b, 2.0) - t_0)))) / (a * 3.0);
	} else {
		tmp = Math.pow(((-2.0 * (b / c)) + (1.5 * (a / b))), -1.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (a * c) * 3.0
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.005:
		tmp = ((math.pow(-b, 2.0) + (t_0 - math.pow(b, 2.0))) / (-b - math.sqrt((math.pow(b, 2.0) - t_0)))) / (a * 3.0)
	else:
		tmp = math.pow(((-2.0 * (b / c)) + (1.5 * (a / b))), -1.0)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(a * c) * 3.0)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.005)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) + Float64(t_0 - (b ^ 2.0))) / Float64(Float64(-b) - sqrt(Float64((b ^ 2.0) - t_0)))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (a * c) * 3.0;
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.005)
		tmp = (((-b ^ 2.0) + (t_0 - (b ^ 2.0))) / (-b - sqrt(((b ^ 2.0) - t_0)))) / (a * 3.0);
	else
		tmp = ((-2.0 * (b / c)) + (1.5 * (a / b))) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] * 3.0), $MachinePrecision]}, 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], -0.005], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $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], N[Power[N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 6: 85.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.3:\\ \;\;\;\;\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot 3}\right) \cdot \frac{1}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;{\left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.3)
   (* (fma -1.0 b (sqrt (- (pow b 2.0) (* (* a c) 3.0)))) (/ 1.0 (* a 3.0)))
   (pow (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))) -1.0)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.3) {
		tmp = fma(-1.0, b, sqrt((pow(b, 2.0) - ((a * c) * 3.0)))) * (1.0 / (a * 3.0));
	} else {
		tmp = pow(((-2.0 * (b / c)) + (1.5 * (a / b))), -1.0);
	}
	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)) <= -0.3)
		tmp = Float64(fma(-1.0, b, sqrt(Float64((b ^ 2.0) - Float64(Float64(a * c) * 3.0)))) * Float64(1.0 / Float64(a * 3.0)));
	else
		tmp = Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))) ^ -1.0;
	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], -0.3], N[(N[(-1.0 * b + N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] - N[(N[(a * c), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 7: 85.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.3:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{{b}^{2} - \left(a \cdot c\right) \cdot 3} - b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.3)
   (* (/ 0.3333333333333333 a) (- (sqrt (- (pow b 2.0) (* (* a c) 3.0))) b))
   (pow (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))) -1.0)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.3) {
		tmp = (0.3333333333333333 / a) * (sqrt((pow(b, 2.0) - ((a * c) * 3.0))) - b);
	} else {
		tmp = pow(((-2.0 * (b / c)) + (1.5 * (a / b))), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (((sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)) <= (-0.3d0)) then
        tmp = (0.3333333333333333d0 / a) * (sqrt(((b ** 2.0d0) - ((a * c) * 3.0d0))) - b)
    else
        tmp = (((-2.0d0) * (b / c)) + (1.5d0 * (a / b))) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.3) {
		tmp = (0.3333333333333333 / a) * (Math.sqrt((Math.pow(b, 2.0) - ((a * c) * 3.0))) - b);
	} else {
		tmp = Math.pow(((-2.0 * (b / c)) + (1.5 * (a / b))), -1.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.3:
		tmp = (0.3333333333333333 / a) * (math.sqrt((math.pow(b, 2.0) - ((a * c) * 3.0))) - b)
	else:
		tmp = math.pow(((-2.0 * (b / c)) + (1.5 * (a / b))), -1.0)
	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)) <= -0.3)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(Float64((b ^ 2.0) - Float64(Float64(a * c) * 3.0))) - b));
	else
		tmp = Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.3)
		tmp = (0.3333333333333333 / a) * (sqrt(((b ^ 2.0) - ((a * c) * 3.0))) - b);
	else
		tmp = ((-2.0 * (b / c)) + (1.5 * (a / b))) ^ -1.0;
	end
	tmp_2 = 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], -0.3], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] - N[(N[(a * c), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 8: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t_0 \leq -0.3:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;{\left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -0.3) t_0 (pow (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))) -1.0))))

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 <= -0.3) {
		tmp = t_0;
	} else {
		tmp = pow(((-2.0 * (b / c)) + (1.5 * (a / b))), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    if (t_0 <= (-0.3d0)) then
        tmp = t_0
    else
        tmp = (((-2.0d0) * (b / c)) + (1.5d0 * (a / b))) ** (-1.0d0)
    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 <= -0.3) {
		tmp = t_0;
	} else {
		tmp = Math.pow(((-2.0 * (b / c)) + (1.5 * (a / b))), -1.0);
	}
	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 <= -0.3:
		tmp = t_0
	else:
		tmp = math.pow(((-2.0 * (b / c)) + (1.5 * (a / b))), -1.0)
	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 <= -0.3)
		tmp = t_0;
	else
		tmp = Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))) ^ -1.0;
	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 <= -0.3)
		tmp = t_0;
	else
		tmp = ((-2.0 * (b / c)) + (1.5 * (a / b))) ^ -1.0;
	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, -0.3], t$95$0, N[Power[N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 9: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.3:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;{\left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.3)
   (/ (- (sqrt (- (* b b) (* (* a c) 3.0))) b) (* a 3.0))
   (pow (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))) -1.0)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.3) {
		tmp = (sqrt(((b * b) - ((a * c) * 3.0))) - b) / (a * 3.0);
	} else {
		tmp = pow(((-2.0 * (b / c)) + (1.5 * (a / b))), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (((sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)) <= (-0.3d0)) then
        tmp = (sqrt(((b * b) - ((a * c) * 3.0d0))) - b) / (a * 3.0d0)
    else
        tmp = (((-2.0d0) * (b / c)) + (1.5d0 * (a / b))) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.3) {
		tmp = (Math.sqrt(((b * b) - ((a * c) * 3.0))) - b) / (a * 3.0);
	} else {
		tmp = Math.pow(((-2.0 * (b / c)) + (1.5 * (a / b))), -1.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.3:
		tmp = (math.sqrt(((b * b) - ((a * c) * 3.0))) - b) / (a * 3.0)
	else:
		tmp = math.pow(((-2.0 * (b / c)) + (1.5 * (a / b))), -1.0)
	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)) <= -0.3)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * c) * 3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.3)
		tmp = (sqrt(((b * b) - ((a * c) * 3.0))) - b) / (a * 3.0);
	else
		tmp = ((-2.0 * (b / c)) + (1.5 * (a / b))) ^ -1.0;
	end
	tmp_2 = 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], -0.3], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 10: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 13:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;{\left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 13.0)
   (/ (- (sqrt (- (* b b) (* a (* c 3.0)))) b) (* a 3.0))
   (pow (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))) -1.0)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 13.0) {
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = pow(((-2.0 * (b / c)) + (1.5 * (a / b))), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 13.0d0) then
        tmp = (sqrt(((b * b) - (a * (c * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (((-2.0d0) * (b / c)) + (1.5d0 * (a / b))) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 13.0) {
		tmp = (Math.sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = Math.pow(((-2.0 * (b / c)) + (1.5 * (a / b))), -1.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 13.0:
		tmp = (math.sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = math.pow(((-2.0 * (b / c)) + (1.5 * (a / b))), -1.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 13.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(a * Float64(c * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 13.0)
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	else
		tmp = ((-2.0 * (b / c)) + (1.5 * (a / b))) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 13.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 11: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}\right)}^{-1} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (pow (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))) -1.0))

C

double code(double a, double b, double c) {
	return pow(((-2.0 * (b / c)) + (1.5 * (a / b))), -1.0);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((-2.0d0) * (b / c)) + (1.5d0 * (a / b))) ** (-1.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return Math.pow(((-2.0 * (b / c)) + (1.5 * (a / b))), -1.0);
}

Python

def code(a, b, c):
	return math.pow(((-2.0 * (b / c)) + (1.5 * (a / b))), -1.0)

Julia

function code(a, b, c)
	return Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))) ^ -1.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((-2.0 * (b / c)) + (1.5 * (a / b))) ^ -1.0;
end

Wolfram

code[a_, b_, c_] := N[Power[N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 12: 11.7% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ \frac{b}{a} \cdot -0.3333333333333333 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* (/ b a) -0.3333333333333333))

C

double code(double a, double b, double c) {
	return (b / a) * -0.3333333333333333;
}

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 / a) * (-0.3333333333333333d0)
end function

Java

public static double code(double a, double b, double c) {
	return (b / a) * -0.3333333333333333;
}

Python

def code(a, b, c):
	return (b / a) * -0.3333333333333333

Julia

function code(a, b, c)
	return Float64(Float64(b / a) * -0.3333333333333333)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (b / a) * -0.3333333333333333;
end

Wolfram

code[a_, b_, c_] := N[(N[(b / a), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 13: 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

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 14: 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

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Reproduce

herbie shell --seed 2023340 
(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)))