Cubic critical

Percentage Accurate: 51.8% → 86.1%

Time: 13.7s

Alternatives: 12

Speedup: 11.6×

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: 51.8% 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: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{+153}:\\ \;\;\;\;-0.5 \cdot \frac{-c}{b} - 0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-94}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.9e+153)
   (- (* -0.5 (/ (- c) b)) (* 0.6666666666666666 (/ b a)))
   (if (<= b 1.45e-94)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.9e+153) {
		tmp = (-0.5 * (-c / b)) - (0.6666666666666666 * (b / a));
	} else if (b <= 1.45e-94) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.9d+153)) then
        tmp = ((-0.5d0) * (-c / b)) - (0.6666666666666666d0 * (b / a))
    else if (b <= 1.45d-94) then
        tmp = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.9e+153) {
		tmp = (-0.5 * (-c / b)) - (0.6666666666666666 * (b / a));
	} else if (b <= 1.45e-94) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.9e+153:
		tmp = (-0.5 * (-c / b)) - (0.6666666666666666 * (b / a))
	elif b <= 1.45e-94:
		tmp = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.9e+153)
		tmp = Float64(Float64(-0.5 * Float64(Float64(-c) / b)) - Float64(0.6666666666666666 * Float64(b / a)));
	elseif (b <= 1.45e-94)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.9e+153)
		tmp = (-0.5 * (-c / b)) - (0.6666666666666666 * (b / a));
	elseif (b <= 1.45e-94)
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.9e+153], N[(N[(-0.5 * N[((-c) / b), $MachinePrecision]), $MachinePrecision] - N[(0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.45e-94], 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[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.89999999999999983e153

   1. Initial program 42.8%

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

      1. sqr-neg 42.8%

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

      2. sqr-neg 42.8%

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

      3. associate-*l* 42.8%

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

   3. Simplified 42.8%

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

   5. Taylor expanded in b around -inf 99.3%

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

   6. Taylor expanded in c around 0 99.6%

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

   if -1.89999999999999983e153 < b < 1.44999999999999998e-94

   1. Initial program 84.2%

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

   if 1.44999999999999998e-94 < b

   1. Initial program 19.2%

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

      1. sqr-neg 19.2%

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

      2. sqr-neg 19.2%

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

      3. associate-*l* 19.2%

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

   3. Simplified 19.2%

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

   5. Taylor expanded in b around inf 86.6%

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

      1. *-commutative 86.6%

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

   7. Simplified 86.6%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{+153}:\\ \;\;\;\;-0.5 \cdot \frac{-c}{b} - 0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-94}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{+153}:\\ \;\;\;\;-0.5 \cdot \frac{-c}{b} - 0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-98}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.4e+153)
   (- (* -0.5 (/ (- c) b)) (* 0.6666666666666666 (/ b a)))
   (if (<= b 1.12e-98)
     (/ (- (sqrt (- (* b b) (* 3.0 (* c a)))) b) (* a 3.0))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.4e+153) {
		tmp = (-0.5 * (-c / b)) - (0.6666666666666666 * (b / a));
	} else if (b <= 1.12e-98) {
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.4d+153)) then
        tmp = ((-0.5d0) * (-c / b)) - (0.6666666666666666d0 * (b / a))
    else if (b <= 1.12d-98) then
        tmp = (sqrt(((b * b) - (3.0d0 * (c * a)))) - b) / (a * 3.0d0)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.4e+153) {
		tmp = (-0.5 * (-c / b)) - (0.6666666666666666 * (b / a));
	} else if (b <= 1.12e-98) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.4e+153:
		tmp = (-0.5 * (-c / b)) - (0.6666666666666666 * (b / a))
	elif b <= 1.12e-98:
		tmp = (math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.4e+153)
		tmp = Float64(Float64(-0.5 * Float64(Float64(-c) / b)) - Float64(0.6666666666666666 * Float64(b / a)));
	elseif (b <= 1.12e-98)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(c * a)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.4e+153)
		tmp = (-0.5 * (-c / b)) - (0.6666666666666666 * (b / a));
	elseif (b <= 1.12e-98)
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.4e+153], N[(N[(-0.5 * N[((-c) / b), $MachinePrecision]), $MachinePrecision] - N[(0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.12e-98], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.39999999999999992e153

   1. Initial program 42.8%

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

      1. sqr-neg 42.8%

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

      2. sqr-neg 42.8%

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

      3. associate-*l* 42.8%

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

   3. Simplified 42.8%

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

   5. Taylor expanded in b around -inf 99.3%

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

   6. Taylor expanded in c around 0 99.6%

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

   if -2.39999999999999992e153 < b < 1.12e-98

   1. Initial program 84.2%

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

      1. sqr-neg 84.2%

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

      2. sqr-neg 84.2%

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

      3. associate-*l* 84.1%

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

   3. Simplified 84.1%

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

   if 1.12e-98 < b

   1. Initial program 19.2%

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

      1. sqr-neg 19.2%

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

      2. sqr-neg 19.2%

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

      3. associate-*l* 19.2%

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

   3. Simplified 19.2%

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

   5. Taylor expanded in b around inf 86.6%

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

      1. *-commutative 86.6%

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

   7. Simplified 86.6%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{+153}:\\ \;\;\;\;-0.5 \cdot \frac{-c}{b} - 0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-98}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{-63}:\\ \;\;\;\;\frac{b}{a \cdot -3} - \frac{\frac{b}{3}}{a}\\ \mathbf{elif}\;b \leq 8.4 \cdot 10^{-97}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.1e-63)
   (- (/ b (* a -3.0)) (/ (/ b 3.0) a))
   (if (<= b 8.4e-97)
     (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.1e-63) {
		tmp = (b / (a * -3.0)) - ((b / 3.0) / a);
	} else if (b <= 8.4e-97) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.1d-63)) then
        tmp = (b / (a * (-3.0d0))) - ((b / 3.0d0) / a)
    else if (b <= 8.4d-97) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.1e-63) {
		tmp = (b / (a * -3.0)) - ((b / 3.0) / a);
	} else if (b <= 8.4e-97) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.1e-63:
		tmp = (b / (a * -3.0)) - ((b / 3.0) / a)
	elif b <= 8.4e-97:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (a * 3.0)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.1e-63)
		tmp = Float64(Float64(b / Float64(a * -3.0)) - Float64(Float64(b / 3.0) / a));
	elseif (b <= 8.4e-97)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.1e-63)
		tmp = (b / (a * -3.0)) - ((b / 3.0) / a);
	elseif (b <= 8.4e-97)
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.1e-63], N[(N[(b / N[(a * -3.0), $MachinePrecision]), $MachinePrecision] - N[(N[(b / 3.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.4e-97], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.1e-63

   1. Initial program 72.1%

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

      1. sqr-neg 72.1%

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

      2. sqr-neg 72.1%

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

      3. associate-*l* 72.1%

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

   3. Simplified 72.1%

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

   6. Applied egg-rr 60.2%

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

   7. Taylor expanded in b around -inf 85.5%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b}{a}} - \frac{\frac{b}{3}}{a} \]
   8. Step-by-step derivation

      1. metadata-eval 85.5%

         \[\leadsto \color{blue}{\frac{-1}{3}} \cdot \frac{b}{a} - \frac{\frac{b}{3}}{a} \]

      2. times-frac 85.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot b}{3 \cdot a}} - \frac{\frac{b}{3}}{a} \]

      3. neg-mul-1 85.6%

         \[\leadsto \frac{\color{blue}{-b}}{3 \cdot a} - \frac{\frac{b}{3}}{a} \]

      4. *-commutative 85.6%

         \[\leadsto \frac{-b}{\color{blue}{a \cdot 3}} - \frac{\frac{b}{3}}{a} \]

      5. distribute-neg-frac 85.6%

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

      6. distribute-neg-frac2 85.6%

         \[\leadsto \color{blue}{\frac{b}{-a \cdot 3}} - \frac{\frac{b}{3}}{a} \]

      7. distribute-rgt-neg-in 85.6%

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

      8. metadata-eval 85.6%

         \[\leadsto \frac{b}{a \cdot \color{blue}{-3}} - \frac{\frac{b}{3}}{a} \]

   9. Simplified 85.6%

      \[\leadsto \color{blue}{\frac{b}{a \cdot -3}} - \frac{\frac{b}{3}}{a} \]

   if -2.1e-63 < b < 8.4000000000000005e-97

   1. Initial program 76.1%

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

      1. sqr-neg 76.1%

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

      2. sqr-neg 76.1%

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

      3. associate-*l* 76.0%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in b around 0 70.8%

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

      1. *-commutative 70.8%

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

      2. *-commutative 70.8%

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

      3. associate-*r* 71.0%

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

   7. Simplified 71.0%

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

   if 8.4000000000000005e-97 < b

   1. Initial program 19.2%

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

      1. sqr-neg 19.2%

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

      2. sqr-neg 19.2%

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

      3. associate-*l* 19.2%

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

   3. Simplified 19.2%

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

   5. Taylor expanded in b around inf 86.6%

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

      1. *-commutative 86.6%

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

   7. Simplified 86.6%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{-63}:\\ \;\;\;\;\frac{b}{a \cdot -3} - \frac{\frac{b}{3}}{a}\\ \mathbf{elif}\;b \leq 8.4 \cdot 10^{-97}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-65}:\\ \;\;\;\;\frac{b}{a \cdot -3} - \frac{\frac{b}{3}}{a}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.9e-65)
   (- (/ b (* a -3.0)) (/ (/ b 3.0) a))
   (if (<= b 4.5e-99)
     (/ (- (sqrt (* a (* c -3.0))) b) (* a 3.0))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.9e-65) {
		tmp = (b / (a * -3.0)) - ((b / 3.0) / a);
	} else if (b <= 4.5e-99) {
		tmp = (sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.9d-65)) then
        tmp = (b / (a * (-3.0d0))) - ((b / 3.0d0) / a)
    else if (b <= 4.5d-99) then
        tmp = (sqrt((a * (c * (-3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.9e-65) {
		tmp = (b / (a * -3.0)) - ((b / 3.0) / a);
	} else if (b <= 4.5e-99) {
		tmp = (Math.sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.9e-65:
		tmp = (b / (a * -3.0)) - ((b / 3.0) / a)
	elif b <= 4.5e-99:
		tmp = (math.sqrt((a * (c * -3.0))) - b) / (a * 3.0)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.9e-65)
		tmp = Float64(Float64(b / Float64(a * -3.0)) - Float64(Float64(b / 3.0) / a));
	elseif (b <= 4.5e-99)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.9e-65)
		tmp = (b / (a * -3.0)) - ((b / 3.0) / a);
	elseif (b <= 4.5e-99)
		tmp = (sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.9e-65], N[(N[(b / N[(a * -3.0), $MachinePrecision]), $MachinePrecision] - N[(N[(b / 3.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.5e-99], N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.8999999999999998e-65

   1. Initial program 72.1%

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

      1. sqr-neg 72.1%

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

      2. sqr-neg 72.1%

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

      3. associate-*l* 72.1%

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

   3. Simplified 72.1%

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

   6. Applied egg-rr 60.2%

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

   7. Taylor expanded in b around -inf 85.5%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b}{a}} - \frac{\frac{b}{3}}{a} \]
   8. Step-by-step derivation

      1. metadata-eval 85.5%

         \[\leadsto \color{blue}{\frac{-1}{3}} \cdot \frac{b}{a} - \frac{\frac{b}{3}}{a} \]

      2. times-frac 85.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot b}{3 \cdot a}} - \frac{\frac{b}{3}}{a} \]

      3. neg-mul-1 85.6%

         \[\leadsto \frac{\color{blue}{-b}}{3 \cdot a} - \frac{\frac{b}{3}}{a} \]

      4. *-commutative 85.6%

         \[\leadsto \frac{-b}{\color{blue}{a \cdot 3}} - \frac{\frac{b}{3}}{a} \]

      5. distribute-neg-frac 85.6%

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

      6. distribute-neg-frac2 85.6%

         \[\leadsto \color{blue}{\frac{b}{-a \cdot 3}} - \frac{\frac{b}{3}}{a} \]

      7. distribute-rgt-neg-in 85.6%

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

      8. metadata-eval 85.6%

         \[\leadsto \frac{b}{a \cdot \color{blue}{-3}} - \frac{\frac{b}{3}}{a} \]

   9. Simplified 85.6%

      \[\leadsto \color{blue}{\frac{b}{a \cdot -3}} - \frac{\frac{b}{3}}{a} \]

   if -2.8999999999999998e-65 < b < 4.5000000000000003e-99

   1. Initial program 76.1%

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

      1. sqr-neg 76.1%

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

      2. sqr-neg 76.1%

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

      3. associate-*l* 76.0%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in b around 0 70.8%

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

      1. *-commutative 70.8%

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

      2. associate-*r* 70.9%

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

   7. Simplified 70.9%

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

   if 4.5000000000000003e-99 < b

   1. Initial program 19.2%

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

      1. sqr-neg 19.2%

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

      2. sqr-neg 19.2%

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

      3. associate-*l* 19.2%

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

   3. Simplified 19.2%

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

   5. Taylor expanded in b around inf 86.6%

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

      1. *-commutative 86.6%

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

   7. Simplified 86.6%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-65}:\\ \;\;\;\;\frac{b}{a \cdot -3} - \frac{\frac{b}{3}}{a}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-69}:\\ \;\;\;\;\frac{b}{a \cdot -3} - \frac{\frac{b}{3}}{a}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-98}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot a\right) \cdot -3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.2e-69)
   (- (/ b (* a -3.0)) (/ (/ b 3.0) a))
   (if (<= b 5.8e-98)
     (/ (- (sqrt (* (* c a) -3.0)) b) (* a 3.0))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.2e-69) {
		tmp = (b / (a * -3.0)) - ((b / 3.0) / a);
	} else if (b <= 5.8e-98) {
		tmp = (sqrt(((c * a) * -3.0)) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4.2d-69)) then
        tmp = (b / (a * (-3.0d0))) - ((b / 3.0d0) / a)
    else if (b <= 5.8d-98) then
        tmp = (sqrt(((c * a) * (-3.0d0))) - b) / (a * 3.0d0)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.2e-69) {
		tmp = (b / (a * -3.0)) - ((b / 3.0) / a);
	} else if (b <= 5.8e-98) {
		tmp = (Math.sqrt(((c * a) * -3.0)) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.2e-69:
		tmp = (b / (a * -3.0)) - ((b / 3.0) / a)
	elif b <= 5.8e-98:
		tmp = (math.sqrt(((c * a) * -3.0)) - b) / (a * 3.0)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.2e-69)
		tmp = Float64(Float64(b / Float64(a * -3.0)) - Float64(Float64(b / 3.0) / a));
	elseif (b <= 5.8e-98)
		tmp = Float64(Float64(sqrt(Float64(Float64(c * a) * -3.0)) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.2e-69)
		tmp = (b / (a * -3.0)) - ((b / 3.0) / a);
	elseif (b <= 5.8e-98)
		tmp = (sqrt(((c * a) * -3.0)) - b) / (a * 3.0);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.2e-69], N[(N[(b / N[(a * -3.0), $MachinePrecision]), $MachinePrecision] - N[(N[(b / 3.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.8e-98], N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.1999999999999999e-69

   1. Initial program 72.1%

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

      1. sqr-neg 72.1%

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

      2. sqr-neg 72.1%

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

      3. associate-*l* 72.1%

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

   3. Simplified 72.1%

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

   6. Applied egg-rr 60.2%

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

   7. Taylor expanded in b around -inf 85.5%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b}{a}} - \frac{\frac{b}{3}}{a} \]
   8. Step-by-step derivation

      1. metadata-eval 85.5%

         \[\leadsto \color{blue}{\frac{-1}{3}} \cdot \frac{b}{a} - \frac{\frac{b}{3}}{a} \]

      2. times-frac 85.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot b}{3 \cdot a}} - \frac{\frac{b}{3}}{a} \]

      3. neg-mul-1 85.6%

         \[\leadsto \frac{\color{blue}{-b}}{3 \cdot a} - \frac{\frac{b}{3}}{a} \]

      4. *-commutative 85.6%

         \[\leadsto \frac{-b}{\color{blue}{a \cdot 3}} - \frac{\frac{b}{3}}{a} \]

      5. distribute-neg-frac 85.6%

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

      6. distribute-neg-frac2 85.6%

         \[\leadsto \color{blue}{\frac{b}{-a \cdot 3}} - \frac{\frac{b}{3}}{a} \]

      7. distribute-rgt-neg-in 85.6%

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

      8. metadata-eval 85.6%

         \[\leadsto \frac{b}{a \cdot \color{blue}{-3}} - \frac{\frac{b}{3}}{a} \]

   9. Simplified 85.6%

      \[\leadsto \color{blue}{\frac{b}{a \cdot -3}} - \frac{\frac{b}{3}}{a} \]

   if -4.1999999999999999e-69 < b < 5.8e-98

   1. Initial program 76.1%

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

      1. sqr-neg 76.1%

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

      2. sqr-neg 76.1%

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

      3. associate-*l* 76.0%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in b around 0 70.8%

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

   if 5.8e-98 < b

   1. Initial program 19.2%

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

      1. sqr-neg 19.2%

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

      2. sqr-neg 19.2%

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

      3. associate-*l* 19.2%

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

   3. Simplified 19.2%

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

   5. Taylor expanded in b around inf 86.6%

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

      1. *-commutative 86.6%

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

   7. Simplified 86.6%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-69}:\\ \;\;\;\;\frac{b}{a \cdot -3} - \frac{\frac{b}{3}}{a}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-98}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot a\right) \cdot -3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{-c}{b} - 0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310)
   (- (* -0.5 (/ (- c) b)) (* 0.6666666666666666 (/ b a)))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = (-0.5 * (-c / b)) - (0.6666666666666666 * (b / a));
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = ((-0.5d0) * (-c / b)) - (0.6666666666666666d0 * (b / a))
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = (-0.5 * (-c / b)) - (0.6666666666666666 * (b / a));
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = (-0.5 * (-c / b)) - (0.6666666666666666 * (b / a))
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(Float64(-0.5 * Float64(Float64(-c) / b)) - Float64(0.6666666666666666 * Float64(b / a)));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-310)
		tmp = (-0.5 * (-c / b)) - (0.6666666666666666 * (b / a));
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(N[(-0.5 * N[((-c) / b), $MachinePrecision]), $MachinePrecision] - N[(0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 74.7%

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

      1. sqr-neg 74.7%

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

      2. sqr-neg 74.7%

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

      3. associate-*l* 74.7%

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

   3. Simplified 74.7%

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

   5. Taylor expanded in b around -inf 63.6%

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

   6. Taylor expanded in c around 0 63.8%

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

   if -1.999999999999994e-310 < b

   1. Initial program 28.9%

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

      1. sqr-neg 28.9%

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

      2. sqr-neg 28.9%

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

      3. associate-*l* 28.8%

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

   3. Simplified 28.8%

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

   5. Taylor expanded in b around inf 71.9%

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

      1. *-commutative 71.9%

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

   7. Simplified 71.9%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{-c}{b} - 0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.6% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-303}:\\ \;\;\;\;\frac{b \cdot 2}{-3} \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3e-303) (* (/ (* b 2.0) -3.0) (/ 1.0 a)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e-303) {
		tmp = ((b * 2.0) / -3.0) * (1.0 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 3d-303) then
        tmp = ((b * 2.0d0) / (-3.0d0)) * (1.0d0 / a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e-303) {
		tmp = ((b * 2.0) / -3.0) * (1.0 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3e-303:
		tmp = ((b * 2.0) / -3.0) * (1.0 / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3e-303)
		tmp = Float64(Float64(Float64(b * 2.0) / -3.0) * Float64(1.0 / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3e-303)
		tmp = ((b * 2.0) / -3.0) * (1.0 / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3e-303], N[(N[(N[(b * 2.0), $MachinePrecision] / -3.0), $MachinePrecision] * N[(1.0 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.00000000000000028e-303

   1. Initial program 74.9%

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

      1. sqr-neg 74.9%

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

      2. sqr-neg 74.9%

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

      3. associate-*l* 74.9%

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

   3. Simplified 74.9%

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

   6. Applied egg-rr 66.7%

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

   7. Taylor expanded in b around -inf 63.1%

      \[\leadsto \frac{\color{blue}{2 \cdot b}}{-3} \cdot \frac{1}{a} \]
   8. Step-by-step derivation

      1. *-commutative 63.1%

         \[\leadsto \frac{\color{blue}{b \cdot 2}}{-3} \cdot \frac{1}{a} \]

   9. Simplified 63.1%

      \[\leadsto \frac{\color{blue}{b \cdot 2}}{-3} \cdot \frac{1}{a} \]

   if 3.00000000000000028e-303 < b

   1. Initial program 28.3%

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

      1. sqr-neg 28.3%

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

      2. sqr-neg 28.3%

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

      3. associate-*l* 28.2%

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

   3. Simplified 28.2%

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

   5. Taylor expanded in b around inf 72.5%

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

      1. *-commutative 72.5%

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

   7. Simplified 72.5%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-303}:\\ \;\;\;\;\frac{b \cdot 2}{-3} \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-303}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.9e-303) (/ (* b -2.0) (* a 3.0)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.9e-303) {
		tmp = (b * -2.0) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.9d-303) then
        tmp = (b * (-2.0d0)) / (a * 3.0d0)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.9e-303) {
		tmp = (b * -2.0) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.9e-303:
		tmp = (b * -2.0) / (a * 3.0)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.9e-303)
		tmp = Float64(Float64(b * -2.0) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.9e-303)
		tmp = (b * -2.0) / (a * 3.0);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.9e-303], N[(N[(b * -2.0), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.90000000000000014e-303

   1. Initial program 74.9%

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

      1. sqr-neg 74.9%

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

      2. sqr-neg 74.9%

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

      3. associate-*l* 74.9%

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

   3. Simplified 74.9%

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

   5. Taylor expanded in b around -inf 63.1%

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. *-commutative 63.1%

         \[\leadsto \frac{\color{blue}{b \cdot -2}}{3 \cdot a} \]

   7. Simplified 63.1%

      \[\leadsto \frac{\color{blue}{b \cdot -2}}{3 \cdot a} \]

   if 2.90000000000000014e-303 < b

   1. Initial program 28.3%

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

      1. sqr-neg 28.3%

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

      2. sqr-neg 28.3%

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

      3. associate-*l* 28.2%

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

   3. Simplified 28.2%

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

   5. Taylor expanded in b around inf 72.5%

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

      1. *-commutative 72.5%

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

   7. Simplified 72.5%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-303}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.6% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-303}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.9e-303) (* b (/ -0.6666666666666666 a)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.9e-303) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.9d-303) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.9e-303) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.9e-303:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.9e-303)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.9e-303)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.9e-303], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.90000000000000014e-303

   1. Initial program 74.9%

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

      1. sqr-neg 74.9%

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

      2. sqr-neg 74.9%

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

      3. associate-*l* 74.9%

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

   3. Simplified 74.9%

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

   6. Applied egg-rr 66.7%

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

   7. Taylor expanded in b around -inf 62.9%

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

      1. *-commutative 62.9%

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

      2. associate-*l/ 63.0%

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

      3. associate-/l* 63.0%

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

   9. Simplified 63.0%

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

   if 2.90000000000000014e-303 < b

   1. Initial program 28.3%

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

      1. sqr-neg 28.3%

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

      2. sqr-neg 28.3%

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

      3. associate-*l* 28.2%

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

   3. Simplified 28.2%

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

   5. Taylor expanded in b around inf 72.5%

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

      1. *-commutative 72.5%

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

   7. Simplified 72.5%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-303}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.5% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-303}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.9e-303) (* b (/ -0.6666666666666666 a)) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.9e-303) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.9d-303) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = c * ((-0.5d0) / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.9e-303) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.9e-303:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.9e-303)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(c * Float64(-0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.9e-303)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.9e-303], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.90000000000000014e-303

   1. Initial program 74.9%

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

      1. sqr-neg 74.9%

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

      2. sqr-neg 74.9%

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

      3. associate-*l* 74.9%

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

   3. Simplified 74.9%

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

   6. Applied egg-rr 66.7%

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

   7. Taylor expanded in b around -inf 62.9%

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

      1. *-commutative 62.9%

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

      2. associate-*l/ 63.0%

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

      3. associate-/l* 63.0%

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

   9. Simplified 63.0%

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

   if 2.90000000000000014e-303 < b

   1. Initial program 28.3%

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

      1. sqr-neg 28.3%

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

      2. sqr-neg 28.3%

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

      3. associate-*l* 28.2%

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

   3. Simplified 28.2%

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

   6. Applied egg-rr 34.4%

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

      1. un-div-inv 34.4%

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

      2. div-inv 34.4%

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

      3. metadata-eval 34.4%

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

   8. Applied egg-rr 34.4%

      \[\leadsto \color{blue}{\frac{\left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)\right) \cdot -0.3333333333333333}{a}} \]
   9. Step-by-step derivation

      1. associate-/l* 34.4%

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

   10. Applied egg-rr 34.4%

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

   11. Taylor expanded in b around inf 0.0%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}} \]
   12. Step-by-step derivation

       1. *-commutative 0.0%

          \[\leadsto \color{blue}{\frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b} \cdot 0.16666666666666666} \]

       2. associate-/l* 0.0%

          \[\leadsto \color{blue}{\left(c \cdot \frac{{\left(\sqrt{-3}\right)}^{2}}{b}\right)} \cdot 0.16666666666666666 \]

       3. associate-*r* 0.0%

          \[\leadsto \color{blue}{c \cdot \left(\frac{{\left(\sqrt{-3}\right)}^{2}}{b} \cdot 0.16666666666666666\right)} \]

       4. *-commutative 0.0%

          \[\leadsto c \cdot \color{blue}{\left(0.16666666666666666 \cdot \frac{{\left(\sqrt{-3}\right)}^{2}}{b}\right)} \]

       5. associate-*r/ 0.0%

          \[\leadsto c \cdot \color{blue}{\frac{0.16666666666666666 \cdot {\left(\sqrt{-3}\right)}^{2}}{b}} \]

       6. unpow2 0.0%

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

       7. rem-square-sqrt 72.3%

          \[\leadsto c \cdot \frac{0.16666666666666666 \cdot \color{blue}{-3}}{b} \]

       8. metadata-eval 72.3%

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

   13. Simplified 72.3%

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

Alternative 11: 43.7% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.5 \cdot 10^{+14}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.5e+14) (* b (/ -0.6666666666666666 a)) (* (/ c b) 0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.5e+14) {
		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 <= 4.5d+14) 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 <= 4.5e+14) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (c / b) * 0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.5e+14:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = (c / b) * 0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.5e+14)
		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 <= 4.5e+14)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = (c / b) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.5e+14], 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 < 4.5e14

   1. Initial program 69.1%

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

      1. sqr-neg 69.1%

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

      2. sqr-neg 69.1%

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

      3. associate-*l* 69.0%

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

   3. Simplified 69.0%

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

   6. Applied egg-rr 62.9%

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

   7. Taylor expanded in b around -inf 47.6%

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

      1. *-commutative 47.6%

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

      2. associate-*l/ 47.6%

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

      3. associate-/l* 47.6%

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

   9. Simplified 47.6%

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

   if 4.5e14 < b

   1. Initial program 13.6%

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

      1. sqr-neg 13.6%

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

      2. sqr-neg 13.6%

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

      3. associate-*l* 13.6%

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

   3. Simplified 13.6%

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

   5. Taylor expanded in b around inf 76.5%

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. *-commutative 76.5%

         \[\leadsto \frac{\color{blue}{\frac{a \cdot c}{b} \cdot -1.5}}{3 \cdot a} \]

      2. associate-/l* 78.0%

         \[\leadsto \frac{\color{blue}{\left(a \cdot \frac{c}{b}\right)} \cdot -1.5}{3 \cdot a} \]

      3. associate-*r* 78.0%

         \[\leadsto \frac{\color{blue}{a \cdot \left(\frac{c}{b} \cdot -1.5\right)}}{3 \cdot a} \]

      4. *-commutative 78.0%

         \[\leadsto \frac{a \cdot \color{blue}{\left(-1.5 \cdot \frac{c}{b}\right)}}{3 \cdot a} \]

      5. associate-*r/ 77.9%

         \[\leadsto \frac{a \cdot \color{blue}{\frac{-1.5 \cdot c}{b}}}{3 \cdot a} \]

   7. Simplified 77.9%

      \[\leadsto \frac{\color{blue}{a \cdot \frac{-1.5 \cdot c}{b}}}{3 \cdot a} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 54.1%

         \[\leadsto \frac{a \cdot \color{blue}{\left(\sqrt{\frac{-1.5 \cdot c}{b}} \cdot \sqrt{\frac{-1.5 \cdot c}{b}}\right)}}{3 \cdot a} \]

      2. sqrt-unprod 47.9%

         \[\leadsto \frac{a \cdot \color{blue}{\sqrt{\frac{-1.5 \cdot c}{b} \cdot \frac{-1.5 \cdot c}{b}}}}{3 \cdot a} \]

      3. pow2 47.9%

         \[\leadsto \frac{a \cdot \sqrt{\color{blue}{{\left(\frac{-1.5 \cdot c}{b}\right)}^{2}}}}{3 \cdot a} \]

      4. associate-/l* 47.9%

         \[\leadsto \frac{a \cdot \sqrt{{\color{blue}{\left(-1.5 \cdot \frac{c}{b}\right)}}^{2}}}{3 \cdot a} \]

   9. Applied egg-rr 47.9%

      \[\leadsto \frac{a \cdot \color{blue}{\sqrt{{\left(-1.5 \cdot \frac{c}{b}\right)}^{2}}}}{3 \cdot a} \]
   10. Step-by-step derivation

       1. unpow2 47.9%

          \[\leadsto \frac{a \cdot \sqrt{\color{blue}{\left(-1.5 \cdot \frac{c}{b}\right) \cdot \left(-1.5 \cdot \frac{c}{b}\right)}}}{3 \cdot a} \]

       2. rem-sqrt-square 57.3%

          \[\leadsto \frac{a \cdot \color{blue}{\left|-1.5 \cdot \frac{c}{b}\right|}}{3 \cdot a} \]

       3. *-commutative 57.3%

          \[\leadsto \frac{a \cdot \left|\color{blue}{\frac{c}{b} \cdot -1.5}\right|}{3 \cdot a} \]

   11. Simplified 57.3%

       \[\leadsto \frac{a \cdot \color{blue}{\left|\frac{c}{b} \cdot -1.5\right|}}{3 \cdot a} \]

   12. Taylor expanded in a around 0 69.7%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \left|-1.5 \cdot \frac{c}{b}\right|} \]
   13. Step-by-step derivation

       1. pow1 69.7%

          \[\leadsto \color{blue}{{\left(0.3333333333333333 \cdot \left|-1.5 \cdot \frac{c}{b}\right|\right)}^{1}} \]

       2. fabs-mul 69.7%

          \[\leadsto {\left(0.3333333333333333 \cdot \color{blue}{\left(\left|-1.5\right| \cdot \left|\frac{c}{b}\right|\right)}\right)}^{1} \]

       3. add-sqr-sqrt 24.4%

          \[\leadsto {\left(0.3333333333333333 \cdot \left(\left|-1.5\right| \cdot \left|\color{blue}{\sqrt{\frac{c}{b}} \cdot \sqrt{\frac{c}{b}}}\right|\right)\right)}^{1} \]

       4. fabs-sqr 24.4%

          \[\leadsto {\left(0.3333333333333333 \cdot \left(\left|-1.5\right| \cdot \color{blue}{\left(\sqrt{\frac{c}{b}} \cdot \sqrt{\frac{c}{b}}\right)}\right)\right)}^{1} \]

       5. add-sqr-sqrt 27.6%

          \[\leadsto {\left(0.3333333333333333 \cdot \left(\left|-1.5\right| \cdot \color{blue}{\frac{c}{b}}\right)\right)}^{1} \]

       6. associate-*r* 27.6%

          \[\leadsto {\color{blue}{\left(\left(0.3333333333333333 \cdot \left|-1.5\right|\right) \cdot \frac{c}{b}\right)}}^{1} \]

       7. metadata-eval 27.6%

          \[\leadsto {\left(\left(0.3333333333333333 \cdot \color{blue}{1.5}\right) \cdot \frac{c}{b}\right)}^{1} \]

       8. metadata-eval 27.6%

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

   14. Applied egg-rr 27.6%

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

       1. unpow1 27.6%

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

   16. Simplified 27.6%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.5 \cdot 10^{+14}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 11.4% 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 52.9%

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

   1. sqr-neg 52.9%

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

   2. sqr-neg 52.9%

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

   3. associate-*l* 52.8%

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

3. Simplified 52.8%

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

5. Taylor expanded in b around inf 27.5%

   \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
6. Step-by-step derivation

   1. *-commutative 27.5%

      \[\leadsto \frac{\color{blue}{\frac{a \cdot c}{b} \cdot -1.5}}{3 \cdot a} \]

   2. associate-/l* 29.0%

      \[\leadsto \frac{\color{blue}{\left(a \cdot \frac{c}{b}\right)} \cdot -1.5}{3 \cdot a} \]

   3. associate-*r* 29.0%

      \[\leadsto \frac{\color{blue}{a \cdot \left(\frac{c}{b} \cdot -1.5\right)}}{3 \cdot a} \]

   4. *-commutative 29.0%

      \[\leadsto \frac{a \cdot \color{blue}{\left(-1.5 \cdot \frac{c}{b}\right)}}{3 \cdot a} \]

   5. associate-*r/ 29.0%

      \[\leadsto \frac{a \cdot \color{blue}{\frac{-1.5 \cdot c}{b}}}{3 \cdot a} \]

7. Simplified 29.0%

   \[\leadsto \frac{\color{blue}{a \cdot \frac{-1.5 \cdot c}{b}}}{3 \cdot a} \]
8. Step-by-step derivation

   1. add-sqr-sqrt 19.1%

      \[\leadsto \frac{a \cdot \color{blue}{\left(\sqrt{\frac{-1.5 \cdot c}{b}} \cdot \sqrt{\frac{-1.5 \cdot c}{b}}\right)}}{3 \cdot a} \]

   2. sqrt-unprod 18.0%

      \[\leadsto \frac{a \cdot \color{blue}{\sqrt{\frac{-1.5 \cdot c}{b} \cdot \frac{-1.5 \cdot c}{b}}}}{3 \cdot a} \]

   3. pow2 18.0%

      \[\leadsto \frac{a \cdot \sqrt{\color{blue}{{\left(\frac{-1.5 \cdot c}{b}\right)}^{2}}}}{3 \cdot a} \]

   4. associate-/l* 18.0%

      \[\leadsto \frac{a \cdot \sqrt{{\color{blue}{\left(-1.5 \cdot \frac{c}{b}\right)}}^{2}}}{3 \cdot a} \]

9. Applied egg-rr 18.0%

   \[\leadsto \frac{a \cdot \color{blue}{\sqrt{{\left(-1.5 \cdot \frac{c}{b}\right)}^{2}}}}{3 \cdot a} \]
10. Step-by-step derivation

    1. unpow2 18.0%

       \[\leadsto \frac{a \cdot \sqrt{\color{blue}{\left(-1.5 \cdot \frac{c}{b}\right) \cdot \left(-1.5 \cdot \frac{c}{b}\right)}}}{3 \cdot a} \]

    2. rem-sqrt-square 21.2%

       \[\leadsto \frac{a \cdot \color{blue}{\left|-1.5 \cdot \frac{c}{b}\right|}}{3 \cdot a} \]

    3. *-commutative 21.2%

       \[\leadsto \frac{a \cdot \left|\color{blue}{\frac{c}{b} \cdot -1.5}\right|}{3 \cdot a} \]

11. Simplified 21.2%

    \[\leadsto \frac{a \cdot \color{blue}{\left|\frac{c}{b} \cdot -1.5\right|}}{3 \cdot a} \]

12. Taylor expanded in a around 0 25.5%

    \[\leadsto \color{blue}{0.3333333333333333 \cdot \left|-1.5 \cdot \frac{c}{b}\right|} \]
13. Step-by-step derivation

    1. pow1 25.5%

       \[\leadsto \color{blue}{{\left(0.3333333333333333 \cdot \left|-1.5 \cdot \frac{c}{b}\right|\right)}^{1}} \]

    2. fabs-mul 25.5%

       \[\leadsto {\left(0.3333333333333333 \cdot \color{blue}{\left(\left|-1.5\right| \cdot \left|\frac{c}{b}\right|\right)}\right)}^{1} \]

    3. add-sqr-sqrt 8.4%

       \[\leadsto {\left(0.3333333333333333 \cdot \left(\left|-1.5\right| \cdot \left|\color{blue}{\sqrt{\frac{c}{b}} \cdot \sqrt{\frac{c}{b}}}\right|\right)\right)}^{1} \]

    4. fabs-sqr 8.4%

       \[\leadsto {\left(0.3333333333333333 \cdot \left(\left|-1.5\right| \cdot \color{blue}{\left(\sqrt{\frac{c}{b}} \cdot \sqrt{\frac{c}{b}}\right)}\right)\right)}^{1} \]

    5. add-sqr-sqrt 10.4%

       \[\leadsto {\left(0.3333333333333333 \cdot \left(\left|-1.5\right| \cdot \color{blue}{\frac{c}{b}}\right)\right)}^{1} \]

    6. associate-*r* 10.4%

       \[\leadsto {\color{blue}{\left(\left(0.3333333333333333 \cdot \left|-1.5\right|\right) \cdot \frac{c}{b}\right)}}^{1} \]

    7. metadata-eval 10.4%

       \[\leadsto {\left(\left(0.3333333333333333 \cdot \color{blue}{1.5}\right) \cdot \frac{c}{b}\right)}^{1} \]

    8. metadata-eval 10.4%

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

14. Applied egg-rr 10.4%

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

    1. unpow1 10.4%

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

16. Simplified 10.4%

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

17. Final simplification 10.4%

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

Reproduce

herbie shell --seed 2024108 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))