Cubic critical

Percentage Accurate: 52.1% → 86.0%

Time: 13.5s

Alternatives: 11

Speedup: 11.6×

• Could not converge on a ground truth

  1. reg0 = (bf "-2.433668981810911697289152734319378471642e249")
  2. reg1 = (bf "4.379319667604497961945953425416589565839e302")
  3. reg2 = (bf #e2.361657223582877279636594806671203979133e78)

Specification

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: 52.1% 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: 86.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{+94}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.6e+94)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 3e-65)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.6e+94) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 3e-65) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= (-3.6d+94)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 3d-65) then
        tmp = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.6e+94) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 3e-65) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.6e+94:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 3e-65:
		tmp = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.6e+94)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 3e-65)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.6e+94)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 3e-65)
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.6e+94], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3e-65], 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], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.59999999999999992e94

   1. Initial program 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in b around -inf 93.0%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 93.0%

         \[\leadsto \color{blue}{-b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)} \]

      2. *-commutative 93.0%

         \[\leadsto -\color{blue}{\left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot b} \]

      3. distribute-rgt-neg-in 93.0%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot \left(-b\right)} \]

      4. fma-define 93.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, 0.6666666666666666 \cdot \frac{1}{a}\right)} \cdot \left(-b\right) \]

      5. associate-*r/ 93.0%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \color{blue}{\frac{0.6666666666666666 \cdot 1}{a}}\right) \cdot \left(-b\right) \]

      6. metadata-eval 93.0%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \frac{\color{blue}{0.6666666666666666}}{a}\right) \cdot \left(-b\right) \]

   7. Simplified 93.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)} \]

   8. Taylor expanded in c around 0 93.4%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -3.59999999999999992e94 < b < 2.99999999999999998e-65

   1. Initial program 83.4%

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

   if 2.99999999999999998e-65 < b

   1. Initial program 11.4%

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

   3. Simplified 11.4%

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

   5. Taylor expanded in b around inf 85.0%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{+94}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{+94}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-64}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.5e+94)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 1.1e-64)
     (/ (- (sqrt (- (* b b) (* a (* c 3.0)))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.5e+94) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 1.1e-64) {
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= (-7.5d+94)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 1.1d-64) then
        tmp = (sqrt(((b * b) - (a * (c * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.5e+94) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 1.1e-64) {
		tmp = (Math.sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.5e+94:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 1.1e-64:
		tmp = (math.sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.5e+94)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 1.1e-64)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(a * Float64(c * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.5e+94)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 1.1e-64)
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.5e+94], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.1e-64], 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[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.49999999999999978e94

   1. Initial program 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in b around -inf 93.0%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 93.0%

         \[\leadsto \color{blue}{-b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)} \]

      2. *-commutative 93.0%

         \[\leadsto -\color{blue}{\left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot b} \]

      3. distribute-rgt-neg-in 93.0%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot \left(-b\right)} \]

      4. fma-define 93.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, 0.6666666666666666 \cdot \frac{1}{a}\right)} \cdot \left(-b\right) \]

      5. associate-*r/ 93.0%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \color{blue}{\frac{0.6666666666666666 \cdot 1}{a}}\right) \cdot \left(-b\right) \]

      6. metadata-eval 93.0%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \frac{\color{blue}{0.6666666666666666}}{a}\right) \cdot \left(-b\right) \]

   7. Simplified 93.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)} \]

   8. Taylor expanded in c around 0 93.4%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -7.49999999999999978e94 < b < 1.1e-64

   1. Initial program 83.4%

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

      1. sqr-neg 83.4%

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

      2. sqr-neg 83.4%

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

      3. associate-*l* 83.3%

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

   3. Simplified 83.3%

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

      1. add-cube-cbrt 83.0%

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

      2. pow3 83.0%

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

   6. Applied egg-rr 83.0%

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

   7. Taylor expanded in a around 0 83.3%

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

      1. *-commutative 83.3%

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

      2. associate-*r* 83.4%

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

      3. *-commutative 83.4%

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

   9. Simplified 83.4%

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

   if 1.1e-64 < b

   1. Initial program 11.4%

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

   3. Simplified 11.4%

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

   5. Taylor expanded in b around inf 85.0%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{+94}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-64}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{+94}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 10^{-65}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.25e+94)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 1e-65)
     (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.25e+94) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 1e-65) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= (-2.25d+94)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 1d-65) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.25e+94) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 1e-65) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.25e+94:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 1e-65:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.25e+94)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 1e-65)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.25e+94)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 1e-65)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.25e+94], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e-65], 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 / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.24999999999999986e94

   1. Initial program 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in b around -inf 93.0%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 93.0%

         \[\leadsto \color{blue}{-b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)} \]

      2. *-commutative 93.0%

         \[\leadsto -\color{blue}{\left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot b} \]

      3. distribute-rgt-neg-in 93.0%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot \left(-b\right)} \]

      4. fma-define 93.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, 0.6666666666666666 \cdot \frac{1}{a}\right)} \cdot \left(-b\right) \]

      5. associate-*r/ 93.0%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \color{blue}{\frac{0.6666666666666666 \cdot 1}{a}}\right) \cdot \left(-b\right) \]

      6. metadata-eval 93.0%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \frac{\color{blue}{0.6666666666666666}}{a}\right) \cdot \left(-b\right) \]

   7. Simplified 93.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)} \]

   8. Taylor expanded in c around 0 93.4%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -2.24999999999999986e94 < b < 9.99999999999999923e-66

   1. Initial program 83.4%

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

      1. sqr-neg 83.4%

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

      2. sqr-neg 83.4%

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

      3. associate-*l* 83.3%

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

   3. Simplified 83.3%

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

   if 9.99999999999999923e-66 < b

   1. Initial program 11.4%

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

   3. Simplified 11.4%

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

   5. Taylor expanded in b around inf 85.0%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{+94}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 10^{-65}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-54}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9e-54)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 2.1e-66)
     (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9e-54) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 2.1e-66) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= (-9d-54)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 2.1d-66) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -9e-54) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 2.1e-66) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9e-54:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 2.1e-66:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9e-54)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 2.1e-66)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -9e-54)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 2.1e-66)
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9e-54], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.1e-66], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.9999999999999997e-54

   1. Initial program 66.8%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in b around -inf 86.2%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 86.2%

         \[\leadsto \color{blue}{-b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)} \]

      2. *-commutative 86.2%

         \[\leadsto -\color{blue}{\left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot b} \]

      3. distribute-rgt-neg-in 86.2%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot \left(-b\right)} \]

      4. fma-define 86.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, 0.6666666666666666 \cdot \frac{1}{a}\right)} \cdot \left(-b\right) \]

      5. associate-*r/ 86.3%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \color{blue}{\frac{0.6666666666666666 \cdot 1}{a}}\right) \cdot \left(-b\right) \]

      6. metadata-eval 86.3%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \frac{\color{blue}{0.6666666666666666}}{a}\right) \cdot \left(-b\right) \]

   7. Simplified 86.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)} \]

   8. Taylor expanded in c around 0 86.5%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -8.9999999999999997e-54 < b < 2.1e-66

   1. Initial program 79.2%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in b around 0 77.8%

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

      1. associate-*r* 78.0%

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

      2. *-commutative 78.0%

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

   7. Simplified 78.0%

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

   if 2.1e-66 < b

   1. Initial program 11.4%

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

   3. Simplified 11.4%

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

   5. Taylor expanded in b around inf 85.0%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-54}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{-53}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-70}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.7e-53)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 2.9e-70)
     (* 0.3333333333333333 (/ (- (sqrt (* a (* c -3.0))) b) a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.7e-53) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 2.9e-70) {
		tmp = 0.3333333333333333 * ((sqrt((a * (c * -3.0))) - b) / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= (-1.7d-53)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 2.9d-70) then
        tmp = 0.3333333333333333d0 * ((sqrt((a * (c * (-3.0d0)))) - b) / a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.7e-53) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 2.9e-70) {
		tmp = 0.3333333333333333 * ((Math.sqrt((a * (c * -3.0))) - b) / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.7e-53:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 2.9e-70:
		tmp = 0.3333333333333333 * ((math.sqrt((a * (c * -3.0))) - b) / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.7e-53)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 2.9e-70)
		tmp = Float64(0.3333333333333333 * Float64(Float64(sqrt(Float64(a * Float64(c * -3.0))) - b) / a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.7e-53)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 2.9e-70)
		tmp = 0.3333333333333333 * ((sqrt((a * (c * -3.0))) - b) / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.7e-53], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.9e-70], N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.7e-53

   1. Initial program 66.8%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in b around -inf 86.2%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 86.2%

         \[\leadsto \color{blue}{-b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)} \]

      2. *-commutative 86.2%

         \[\leadsto -\color{blue}{\left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot b} \]

      3. distribute-rgt-neg-in 86.2%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot \left(-b\right)} \]

      4. fma-define 86.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, 0.6666666666666666 \cdot \frac{1}{a}\right)} \cdot \left(-b\right) \]

      5. associate-*r/ 86.3%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \color{blue}{\frac{0.6666666666666666 \cdot 1}{a}}\right) \cdot \left(-b\right) \]

      6. metadata-eval 86.3%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \frac{\color{blue}{0.6666666666666666}}{a}\right) \cdot \left(-b\right) \]

   7. Simplified 86.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)} \]

   8. Taylor expanded in c around 0 86.5%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -1.7e-53 < b < 2.89999999999999971e-70

   1. Initial program 79.2%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in b around 0 77.8%

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

      1. associate-*r* 78.0%

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

      2. *-commutative 78.0%

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

   7. Simplified 78.0%

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

      1. *-un-lft-identity 78.0%

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

      2. times-frac 77.9%

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

      3. metadata-eval 77.9%

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

      4. associate-*l* 77.9%

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

   9. Applied egg-rr 77.9%

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

   if 2.89999999999999971e-70 < b

   1. Initial program 11.4%

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

   3. Simplified 11.4%

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

   5. Taylor expanded in b around inf 85.0%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{-53}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-70}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-67}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-203}:\\ \;\;\;\;\sqrt{c \cdot \frac{-3}{a}} \cdot \left(--0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.6e-67)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 5e-203)
     (* (sqrt (* c (/ -3.0 a))) (- -0.3333333333333333))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.6e-67) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 5e-203) {
		tmp = sqrt((c * (-3.0 / a))) * -(-0.3333333333333333);
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= (-3.6d-67)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 5d-203) then
        tmp = sqrt((c * ((-3.0d0) / a))) * -(-0.3333333333333333d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.6e-67) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 5e-203) {
		tmp = Math.sqrt((c * (-3.0 / a))) * -(-0.3333333333333333);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.6e-67:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 5e-203:
		tmp = math.sqrt((c * (-3.0 / a))) * -(-0.3333333333333333)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.6e-67)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 5e-203)
		tmp = Float64(sqrt(Float64(c * Float64(-3.0 / a))) * Float64(-(-0.3333333333333333)));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.6e-67)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 5e-203)
		tmp = sqrt((c * (-3.0 / a))) * -(-0.3333333333333333);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.6e-67], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e-203], N[(N[Sqrt[N[(c * N[(-3.0 / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (--0.3333333333333333)), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.59999999999999999e-67

   1. Initial program 66.8%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in b around -inf 86.2%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 86.2%

         \[\leadsto \color{blue}{-b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)} \]

      2. *-commutative 86.2%

         \[\leadsto -\color{blue}{\left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot b} \]

      3. distribute-rgt-neg-in 86.2%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot \left(-b\right)} \]

      4. fma-define 86.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, 0.6666666666666666 \cdot \frac{1}{a}\right)} \cdot \left(-b\right) \]

      5. associate-*r/ 86.3%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \color{blue}{\frac{0.6666666666666666 \cdot 1}{a}}\right) \cdot \left(-b\right) \]

      6. metadata-eval 86.3%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \frac{\color{blue}{0.6666666666666666}}{a}\right) \cdot \left(-b\right) \]

   7. Simplified 86.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)} \]

   8. Taylor expanded in c around 0 86.5%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -3.59999999999999999e-67 < b < 5.0000000000000002e-203

   1. Initial program 81.6%

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

      1. add-cube-cbrt 80.9%

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

      2. pow3 80.9%

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

   4. Applied egg-rr 80.9%

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

   5. Taylor expanded in a around -inf 0.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left(\sqrt{\frac{c \cdot {\left(\sqrt[3]{-3}\right)}^{3}}{a}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-3}\right)}^{3}}{a}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto -0.3333333333333333 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-3}\right)}^{3}}{a}}\right) \]

      3. rem-square-sqrt 36.6%

         \[\leadsto -0.3333333333333333 \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-3}\right)}^{3}}{a}}\right) \]

      4. rem-cube-cbrt 36.9%

         \[\leadsto -0.3333333333333333 \cdot \left(-1 \cdot \sqrt{\frac{c \cdot \color{blue}{-3}}{a}}\right) \]

      5. associate-/l* 36.9%

         \[\leadsto -0.3333333333333333 \cdot \left(-1 \cdot \sqrt{\color{blue}{c \cdot \frac{-3}{a}}}\right) \]

   7. Simplified 36.9%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left(-1 \cdot \sqrt{c \cdot \frac{-3}{a}}\right)} \]

   if 5.0000000000000002e-203 < b

   1. Initial program 18.4%

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

   3. Simplified 18.4%

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

   5. Taylor expanded in b around inf 78.1%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-67}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-203}:\\ \;\;\;\;\sqrt{c \cdot \frac{-3}{a}} \cdot \left(--0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.1% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= (-5d-310)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 71.6%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in b around -inf 62.9%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 62.9%

         \[\leadsto \color{blue}{-b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)} \]

      2. *-commutative 62.9%

         \[\leadsto -\color{blue}{\left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot b} \]

      3. distribute-rgt-neg-in 62.9%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right) \cdot \left(-b\right)} \]

      4. fma-define 62.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, 0.6666666666666666 \cdot \frac{1}{a}\right)} \cdot \left(-b\right) \]

      5. associate-*r/ 63.0%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \color{blue}{\frac{0.6666666666666666 \cdot 1}{a}}\right) \cdot \left(-b\right) \]

      6. metadata-eval 63.0%

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \frac{\color{blue}{0.6666666666666666}}{a}\right) \cdot \left(-b\right) \]

   7. Simplified 63.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{{b}^{2}}, \frac{0.6666666666666666}{a}\right) \cdot \left(-b\right)} \]

   8. Taylor expanded in c around 0 64.9%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -4.999999999999985e-310 < b

   1. Initial program 24.0%

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

   3. Simplified 24.0%

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

   5. Taylor expanded in b around inf 70.8%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-300}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.15e-300) (/ (* b -0.6666666666666666) a) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.15e-300) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= 1.15d-300) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.15e-300) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.15e-300:
		tmp = (b * -0.6666666666666666) / a
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.15e-300)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.15e-300)
		tmp = (b * -0.6666666666666666) / a;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.15e-300], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.15e-300

   1. Initial program 71.8%

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

      1. sqr-neg 71.8%

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

      2. sqr-neg 71.8%

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

      3. associate-*l* 71.8%

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

   3. Simplified 71.8%

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

      1. add-cube-cbrt 71.6%

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

      2. pow3 71.6%

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

   6. Applied egg-rr 71.6%

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

   7. Taylor expanded in b around -inf 64.0%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   8. Step-by-step derivation

      1. associate-*r/ 64.1%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

      2. *-commutative 64.1%

         \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

   9. Simplified 64.1%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]

   if 1.15e-300 < b

   1. Initial program 23.4%

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

   3. Simplified 23.4%

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

   5. Taylor expanded in b around inf 71.3%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{-300}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-301}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.8e-301) (* b (/ -0.6666666666666666 a)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.8e-301) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= 2.8d-301) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.8e-301) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.8e-301:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.8e-301)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.8e-301)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.8e-301], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.8000000000000001e-301

   1. Initial program 71.8%

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

   3. Simplified 71.8%

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

   5. Taylor expanded in b around -inf 64.0%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   6. Step-by-step derivation

      1. *-commutative 64.0%

         \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

   7. Simplified 64.0%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

   8. Taylor expanded in b around 0 64.0%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   9. Step-by-step derivation

      1. *-commutative 64.0%

         \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

      2. associate-*l/ 64.1%

         \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]

      3. associate-/l* 64.0%

         \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]

   10. Simplified 64.0%

       \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]

   if 2.8000000000000001e-301 < b

   1. Initial program 23.4%

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

   3. Simplified 23.4%

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

   5. Taylor expanded in b around inf 71.3%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-301}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.6% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.1%

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

3. Simplified 49.1%

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

5. Taylor expanded in b around inf 34.6%

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

6. Final simplification 34.6%

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

Alternative 11: 11.0% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 49.1%

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

   1. sqr-neg 49.1%

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

   2. sqr-neg 49.1%

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

   3. associate-*l* 49.1%

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

3. Simplified 49.1%

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

   1. add-cube-cbrt 48.9%

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

   2. pow3 48.9%

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

6. Applied egg-rr 48.9%

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

   1. div-inv 48.9%

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

   2. neg-mul-1 48.9%

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

   3. fma-define 48.9%

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

   4. cancel-sign-sub-inv 48.9%

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

   5. fma-define 48.9%

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

   6. metadata-eval 48.9%

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

   7. unpow3 48.9%

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

   8. add-cube-cbrt 49.0%

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

8. Applied egg-rr 49.0%

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

   1. associate-/r* 49.0%

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

   2. metadata-eval 49.0%

      \[\leadsto \mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}\right) \cdot \frac{\color{blue}{0.3333333333333333}}{a} \]

   3. metadata-eval 49.0%

      \[\leadsto \mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}\right) \cdot \frac{\color{blue}{0.3333333333333333 \cdot 1}}{a} \]

   4. associate-*r/ 48.9%

      \[\leadsto \mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{a}\right)} \]

   5. *-commutative 48.9%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{a}\right) \cdot \mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}\right)} \]

   6. associate-*r/ 49.0%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot 1}{a}} \cdot \mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}\right) \]

   7. metadata-eval 49.0%

      \[\leadsto \frac{\color{blue}{0.3333333333333333}}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}\right) \]

   8. fma-undefine 49.0%

      \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{\color{blue}{b \cdot b + -3 \cdot \left(a \cdot c\right)}}\right) \]

   9. unpow2 49.0%

      \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{2}} + -3 \cdot \left(a \cdot c\right)}\right) \]

   10. metadata-eval 49.0%

       \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} + \color{blue}{\left(-3\right)} \cdot \left(a \cdot c\right)}\right) \]

   11. cancel-sign-sub-inv 49.0%

       \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}\right) \]

   12. associate-*r* 49.1%

       \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{\left(3 \cdot a\right) \cdot c}}\right) \]

   13. *-commutative 49.1%

       \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{c \cdot \left(3 \cdot a\right)}}\right) \]

   14. unpow2 49.1%

       \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{\color{blue}{b \cdot b} - c \cdot \left(3 \cdot a\right)}\right) \]

   15. fma-neg 49.1%

       \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -c \cdot \left(3 \cdot a\right)\right)}}\right) \]

   16. *-commutative 49.1%

       \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(3 \cdot a\right) \cdot c}\right)}\right) \]

   17. associate-*r* 49.0%

       \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{3 \cdot \left(a \cdot c\right)}\right)}\right) \]

   18. distribute-lft-neg-in 49.0%

       \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right) \cdot \left(a \cdot c\right)}\right)}\right) \]

   19. metadata-eval 49.0%

       \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3} \cdot \left(a \cdot c\right)\right)}\right) \]

   20. associate-*r* 49.1%

       \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3 \cdot a\right) \cdot c}\right)}\right) \]

   21. *-commutative 49.1%

       \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot -3\right)} \cdot c\right)}\right) \]

10. Simplified 49.1%

    \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)}\right)} \]

11. Taylor expanded in a around 0 12.5%

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

    1. associate-*r/ 12.5%

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

    2. distribute-rgt1-in 12.5%

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

    3. metadata-eval 12.5%

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

    4. mul0-lft 12.5%

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

    5. metadata-eval 12.5%

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

    6. div0 12.5%

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

13. Simplified 12.5%

    \[\leadsto \color{blue}{0} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))