quadp (p42, positive)

Percentage Accurate: 52.4% → 88.0%

Time: 14.2s

Alternatives: 14

Speedup: 12.9×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * 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) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * 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) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 88.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(4 \cdot c\right)\\ \mathbf{if}\;b \leq -6 \cdot 10^{+138}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{+33}:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - {b}^{2}\right) - t\_0}{b + \sqrt{{b}^{2} - t\_0}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* 4.0 c))))
   (if (<= b -6e+138)
     (/ b (- a))
     (if (<= b 4.6e-112)
       (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
       (if (<= b 2.05e+33)
         (/
          (/
           (- (- (pow b 2.0) (pow b 2.0)) t_0)
           (+ b (sqrt (- (pow b 2.0) t_0))))
          (* a 2.0))
         (/ c (- b)))))))

C

double code(double a, double b, double c) {
	double t_0 = a * (4.0 * c);
	double tmp;
	if (b <= -6e+138) {
		tmp = b / -a;
	} else if (b <= 4.6e-112) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else if (b <= 2.05e+33) {
		tmp = (((pow(b, 2.0) - pow(b, 2.0)) - t_0) / (b + sqrt((pow(b, 2.0) - t_0)))) / (a * 2.0);
	} else {
		tmp = 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) :: t_0
    real(8) :: tmp
    t_0 = a * (4.0d0 * c)
    if (b <= (-6d+138)) then
        tmp = b / -a
    else if (b <= 4.6d-112) then
        tmp = (sqrt(((b * b) - (4.0d0 * (a * c)))) - b) / (a * 2.0d0)
    else if (b <= 2.05d+33) then
        tmp = ((((b ** 2.0d0) - (b ** 2.0d0)) - t_0) / (b + sqrt(((b ** 2.0d0) - t_0)))) / (a * 2.0d0)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = a * (4.0 * c);
	double tmp;
	if (b <= -6e+138) {
		tmp = b / -a;
	} else if (b <= 4.6e-112) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else if (b <= 2.05e+33) {
		tmp = (((Math.pow(b, 2.0) - Math.pow(b, 2.0)) - t_0) / (b + Math.sqrt((Math.pow(b, 2.0) - t_0)))) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = a * (4.0 * c)
	tmp = 0
	if b <= -6e+138:
		tmp = b / -a
	elif b <= 4.6e-112:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	elif b <= 2.05e+33:
		tmp = (((math.pow(b, 2.0) - math.pow(b, 2.0)) - t_0) / (b + math.sqrt((math.pow(b, 2.0) - t_0)))) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(4.0 * c))
	tmp = 0.0
	if (b <= -6e+138)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 4.6e-112)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	elseif (b <= 2.05e+33)
		tmp = Float64(Float64(Float64(Float64((b ^ 2.0) - (b ^ 2.0)) - t_0) / Float64(b + sqrt(Float64((b ^ 2.0) - t_0)))) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = a * (4.0 * c);
	tmp = 0.0;
	if (b <= -6e+138)
		tmp = b / -a;
	elseif (b <= 4.6e-112)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	elseif (b <= 2.05e+33)
		tmp = ((((b ^ 2.0) - (b ^ 2.0)) - t_0) / (b + sqrt(((b ^ 2.0) - t_0)))) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(4.0 * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -6e+138], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 4.6e-112], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.05e+33], N[(N[(N[(N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(b + N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -6.0000000000000002e138

   1. Initial program 46.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 46.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 46.6%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 94.4%

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

      1. associate-*r/ 94.4%

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

      2. mul-1-neg 94.4%

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

   7. Simplified 94.4%

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

   if -6.0000000000000002e138 < b < 4.59999999999999981e-112

   1. Initial program 85.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   if 4.59999999999999981e-112 < b < 2.04999999999999997e33

   1. Initial program 49.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 49.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 49.9%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 49.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}\right) \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

      2. pow3 49.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

      3. *-commutative 49.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\sqrt[3]{\color{blue}{\left(a \cdot c\right) \cdot 4}}\right)}^{3}}}{a \cdot 2} \]

      4. associate-*l* 49.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\sqrt[3]{\color{blue}{a \cdot \left(c \cdot 4\right)}}\right)}^{3}}}{a \cdot 2} \]

   6. Applied egg-rr 49.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\sqrt[3]{a \cdot \left(c \cdot 4\right)}\right)}^{3}}}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. flip-+ 49.3%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{a \cdot \left(c \cdot 4\right)}\right)}^{3}} \cdot \sqrt{b \cdot b - {\left(\sqrt[3]{a \cdot \left(c \cdot 4\right)}\right)}^{3}}}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{a \cdot \left(c \cdot 4\right)}\right)}^{3}}}}}{a \cdot 2} \]

      2. pow2 49.3%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - {\left(\sqrt[3]{a \cdot \left(c \cdot 4\right)}\right)}^{3}} \cdot \sqrt{b \cdot b - {\left(\sqrt[3]{a \cdot \left(c \cdot 4\right)}\right)}^{3}}}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{a \cdot \left(c \cdot 4\right)}\right)}^{3}}}}{a \cdot 2} \]

      3. pow2 49.3%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \sqrt{\color{blue}{{b}^{2}} - {\left(\sqrt[3]{a \cdot \left(c \cdot 4\right)}\right)}^{3}} \cdot \sqrt{b \cdot b - {\left(\sqrt[3]{a \cdot \left(c \cdot 4\right)}\right)}^{3}}}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{a \cdot \left(c \cdot 4\right)}\right)}^{3}}}}{a \cdot 2} \]

      4. pow2 49.3%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \sqrt{{b}^{2} - {\left(\sqrt[3]{a \cdot \left(c \cdot 4\right)}\right)}^{3}} \cdot \sqrt{\color{blue}{{b}^{2}} - {\left(\sqrt[3]{a \cdot \left(c \cdot 4\right)}\right)}^{3}}}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{a \cdot \left(c \cdot 4\right)}\right)}^{3}}}}{a \cdot 2} \]

      5. add-sqr-sqrt 49.4%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left({b}^{2} - {\left(\sqrt[3]{a \cdot \left(c \cdot 4\right)}\right)}^{3}\right)}}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{a \cdot \left(c \cdot 4\right)}\right)}^{3}}}}{a \cdot 2} \]

      6. unpow3 49.5%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\left(\sqrt[3]{a \cdot \left(c \cdot 4\right)} \cdot \sqrt[3]{a \cdot \left(c \cdot 4\right)}\right) \cdot \sqrt[3]{a \cdot \left(c \cdot 4\right)}}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{a \cdot \left(c \cdot 4\right)}\right)}^{3}}}}{a \cdot 2} \]

      7. add-cube-cbrt 49.6%

         \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{a \cdot \left(c \cdot 4\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{a \cdot \left(c \cdot 4\right)}\right)}^{3}}}}{a \cdot 2} \]

   8. Applied egg-rr 49.9%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - a \cdot \left(c \cdot 4\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}}}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. associate--r- 84.6%

         \[\leadsto \frac{\frac{\color{blue}{\left({\left(-b\right)}^{2} - {b}^{2}\right) + a \cdot \left(c \cdot 4\right)}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}}}{a \cdot 2} \]

      2. unpow2 84.6%

         \[\leadsto \frac{\frac{\left(\color{blue}{\left(-b\right) \cdot \left(-b\right)} - {b}^{2}\right) + a \cdot \left(c \cdot 4\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}}}{a \cdot 2} \]

      3. sqr-neg 84.6%

         \[\leadsto \frac{\frac{\left(\color{blue}{b \cdot b} - {b}^{2}\right) + a \cdot \left(c \cdot 4\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}}}{a \cdot 2} \]

      4. unpow2 84.6%

         \[\leadsto \frac{\frac{\left(\color{blue}{{b}^{2}} - {b}^{2}\right) + a \cdot \left(c \cdot 4\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}}}{a \cdot 2} \]

   10. Simplified 84.6%

       \[\leadsto \frac{\color{blue}{\frac{\left({b}^{2} - {b}^{2}\right) + a \cdot \left(c \cdot 4\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}}}}{a \cdot 2} \]

   if 2.04999999999999997e33 < b

   1. Initial program 8.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 8.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 8.2%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 88.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 88.7%

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

      2. neg-mul-1 88.7%

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

   7. Simplified 88.7%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+138}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{+33}:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - {b}^{2}\right) - a \cdot \left(4 \cdot c\right)}{b + \sqrt{{b}^{2} - a \cdot \left(4 \cdot c\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+140}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+140)
   (/ b (- a))
   (if (<= b 8.5e-15)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+140) {
		tmp = b / -a;
	} else if (b <= 8.5e-15) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = 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 <= (-1d+140)) then
        tmp = b / -a
    else if (b <= 8.5d-15) then
        tmp = (sqrt(((b * b) - (4.0d0 * (a * c)))) - b) / (a * 2.0d0)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+140) {
		tmp = b / -a;
	} else if (b <= 8.5e-15) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e+140:
		tmp = b / -a
	elif b <= 8.5e-15:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e+140)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 8.5e-15)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e+140)
		tmp = b / -a;
	elseif (b <= 8.5e-15)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+140], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 8.5e-15], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.00000000000000006e140

   1. Initial program 46.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 46.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 46.6%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 94.4%

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

      1. associate-*r/ 94.4%

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

      2. mul-1-neg 94.4%

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

   7. Simplified 94.4%

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

   if -1.00000000000000006e140 < b < 8.50000000000000007e-15

   1. Initial program 80.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   if 8.50000000000000007e-15 < b

   1. Initial program 10.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 10.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 10.3%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 87.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 87.7%

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

      2. neg-mul-1 87.7%

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

   7. Simplified 87.7%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+140}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-97}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \mathbf{elif}\;b \leq -1.05 \cdot 10^{-241}:\\ \;\;\;\;\sqrt{\left(-4 \cdot \frac{c}{a}\right) \cdot 0.25}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-163}:\\ \;\;\;\;-0.5 \cdot \frac{1}{\sqrt{-0.25 \cdot \frac{a}{c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.2e-97)
   (* b (+ (/ c (pow b 2.0)) (/ -1.0 a)))
   (if (<= b -1.05e-241)
     (sqrt (* (* -4.0 (/ c a)) 0.25))
     (if (<= b 9e-163)
       (* -0.5 (/ 1.0 (sqrt (* -0.25 (/ a c)))))
       (/ 1.0 (- (/ a b) (/ b c)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.2e-97) {
		tmp = b * ((c / pow(b, 2.0)) + (-1.0 / a));
	} else if (b <= -1.05e-241) {
		tmp = sqrt(((-4.0 * (c / a)) * 0.25));
	} else if (b <= 9e-163) {
		tmp = -0.5 * (1.0 / sqrt((-0.25 * (a / c))));
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	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 <= (-8.2d-97)) then
        tmp = b * ((c / (b ** 2.0d0)) + ((-1.0d0) / a))
    else if (b <= (-1.05d-241)) then
        tmp = sqrt((((-4.0d0) * (c / a)) * 0.25d0))
    else if (b <= 9d-163) then
        tmp = (-0.5d0) * (1.0d0 / sqrt(((-0.25d0) * (a / c))))
    else
        tmp = 1.0d0 / ((a / b) - (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.2e-97) {
		tmp = b * ((c / Math.pow(b, 2.0)) + (-1.0 / a));
	} else if (b <= -1.05e-241) {
		tmp = Math.sqrt(((-4.0 * (c / a)) * 0.25));
	} else if (b <= 9e-163) {
		tmp = -0.5 * (1.0 / Math.sqrt((-0.25 * (a / c))));
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.2e-97:
		tmp = b * ((c / math.pow(b, 2.0)) + (-1.0 / a))
	elif b <= -1.05e-241:
		tmp = math.sqrt(((-4.0 * (c / a)) * 0.25))
	elif b <= 9e-163:
		tmp = -0.5 * (1.0 / math.sqrt((-0.25 * (a / c))))
	else:
		tmp = 1.0 / ((a / b) - (b / c))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.2e-97)
		tmp = Float64(b * Float64(Float64(c / (b ^ 2.0)) + Float64(-1.0 / a)));
	elseif (b <= -1.05e-241)
		tmp = sqrt(Float64(Float64(-4.0 * Float64(c / a)) * 0.25));
	elseif (b <= 9e-163)
		tmp = Float64(-0.5 * Float64(1.0 / sqrt(Float64(-0.25 * Float64(a / c)))));
	else
		tmp = Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.2e-97)
		tmp = b * ((c / (b ^ 2.0)) + (-1.0 / a));
	elseif (b <= -1.05e-241)
		tmp = sqrt(((-4.0 * (c / a)) * 0.25));
	elseif (b <= 9e-163)
		tmp = -0.5 * (1.0 / sqrt((-0.25 * (a / c))));
	else
		tmp = 1.0 / ((a / b) - (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.2e-97], N[(b * N[(N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -1.05e-241], N[Sqrt[N[(N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]], $MachinePrecision], If[LessEqual[b, 9e-163], N[(-0.5 * N[(1.0 / N[Sqrt[N[(-0.25 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -8.19999999999999986e-97

   1. Initial program 70.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 70.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 70.0%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 82.9%

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

      1. mul-1-neg 82.9%

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

      2. distribute-rgt-neg-in 82.9%

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

      3. +-commutative 82.9%

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

      4. mul-1-neg 82.9%

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

      5. unsub-neg 82.9%

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

   7. Simplified 82.9%

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

   if -8.19999999999999986e-97 < b < -1.05e-241

   1. Initial program 88.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 88.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 88.9%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 88.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}\right) \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

      2. pow3 88.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

      3. *-commutative 88.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\sqrt[3]{\color{blue}{\left(a \cdot c\right) \cdot 4}}\right)}^{3}}}{a \cdot 2} \]

      4. associate-*l* 88.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\sqrt[3]{\color{blue}{a \cdot \left(c \cdot 4\right)}}\right)}^{3}}}{a \cdot 2} \]

   6. Applied egg-rr 88.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\sqrt[3]{a \cdot \left(c \cdot 4\right)}\right)}^{3}}}}{a \cdot 2} \]

   7. Taylor expanded in a around -inf 0.0%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto -0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right) \]

      3. rem-square-sqrt 57.3%

         \[\leadsto -0.5 \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right) \]

      4. rem-cube-cbrt 57.7%

         \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\frac{c \cdot \color{blue}{-4}}{a}}\right) \]

      5. associate-/l* 57.7%

         \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\color{blue}{c \cdot \frac{-4}{a}}}\right) \]

   9. Simplified 57.7%

      \[\leadsto \color{blue}{-0.5 \cdot \left(-1 \cdot \sqrt{c \cdot \frac{-4}{a}}\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ 57.7%

          \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\color{blue}{\frac{c \cdot -4}{a}}}\right) \]

   11. Applied egg-rr 57.7%

       \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\color{blue}{\frac{c \cdot -4}{a}}}\right) \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 57.5%

          \[\leadsto \color{blue}{\sqrt{-0.5 \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)} \cdot \sqrt{-0.5 \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)}} \]

       2. sqrt-unprod 57.7%

          \[\leadsto \color{blue}{\sqrt{\left(-0.5 \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)\right) \cdot \left(-0.5 \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)\right)}} \]

       3. *-commutative 57.7%

          \[\leadsto \sqrt{\color{blue}{\left(\left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right) \cdot -0.5\right)} \cdot \left(-0.5 \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)\right)} \]

       4. *-commutative 57.7%

          \[\leadsto \sqrt{\left(\left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right) \cdot -0.5\right) \cdot \color{blue}{\left(\left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right) \cdot -0.5\right)}} \]

       5. swap-sqr 57.7%

          \[\leadsto \sqrt{\color{blue}{\left(\left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right) \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)\right) \cdot \left(-0.5 \cdot -0.5\right)}} \]

       6. mul-1-neg 57.7%

          \[\leadsto \sqrt{\left(\color{blue}{\left(-\sqrt{\frac{c \cdot -4}{a}}\right)} \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)\right) \cdot \left(-0.5 \cdot -0.5\right)} \]

       7. mul-1-neg 57.7%

          \[\leadsto \sqrt{\left(\left(-\sqrt{\frac{c \cdot -4}{a}}\right) \cdot \color{blue}{\left(-\sqrt{\frac{c \cdot -4}{a}}\right)}\right) \cdot \left(-0.5 \cdot -0.5\right)} \]

       8. sqr-neg 57.7%

          \[\leadsto \sqrt{\color{blue}{\left(\sqrt{\frac{c \cdot -4}{a}} \cdot \sqrt{\frac{c \cdot -4}{a}}\right)} \cdot \left(-0.5 \cdot -0.5\right)} \]

       9. add-sqr-sqrt 57.7%

          \[\leadsto \sqrt{\color{blue}{\frac{c \cdot -4}{a}} \cdot \left(-0.5 \cdot -0.5\right)} \]

       10. *-commutative 57.7%

           \[\leadsto \sqrt{\frac{\color{blue}{-4 \cdot c}}{a} \cdot \left(-0.5 \cdot -0.5\right)} \]

       11. associate-/l* 57.7%

           \[\leadsto \sqrt{\color{blue}{\left(-4 \cdot \frac{c}{a}\right)} \cdot \left(-0.5 \cdot -0.5\right)} \]

       12. metadata-eval 57.7%

           \[\leadsto \sqrt{\left(-4 \cdot \frac{c}{a}\right) \cdot \color{blue}{0.25}} \]

   13. Applied egg-rr 57.7%

       \[\leadsto \color{blue}{\sqrt{\left(-4 \cdot \frac{c}{a}\right) \cdot 0.25}} \]

   if -1.05e-241 < b < 8.9999999999999995e-163

   1. Initial program 82.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 82.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 82.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 82.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}\right) \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

      2. pow3 82.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

      3. *-commutative 82.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\sqrt[3]{\color{blue}{\left(a \cdot c\right) \cdot 4}}\right)}^{3}}}{a \cdot 2} \]

      4. associate-*l* 82.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\sqrt[3]{\color{blue}{a \cdot \left(c \cdot 4\right)}}\right)}^{3}}}{a \cdot 2} \]

   6. Applied egg-rr 82.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\sqrt[3]{a \cdot \left(c \cdot 4\right)}\right)}^{3}}}}{a \cdot 2} \]

   7. Taylor expanded in a around -inf 0.0%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto -0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right) \]

      3. rem-square-sqrt 28.0%

         \[\leadsto -0.5 \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right) \]

      4. rem-cube-cbrt 28.1%

         \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\frac{c \cdot \color{blue}{-4}}{a}}\right) \]

      5. associate-/l* 28.0%

         \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\color{blue}{c \cdot \frac{-4}{a}}}\right) \]

   9. Simplified 28.0%

      \[\leadsto \color{blue}{-0.5 \cdot \left(-1 \cdot \sqrt{c \cdot \frac{-4}{a}}\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ 28.1%

          \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\color{blue}{\frac{c \cdot -4}{a}}}\right) \]

   11. Applied egg-rr 28.1%

       \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\color{blue}{\frac{c \cdot -4}{a}}}\right) \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 0.8%

          \[\leadsto -0.5 \cdot \color{blue}{\left(\sqrt{-1 \cdot \sqrt{\frac{c \cdot -4}{a}}} \cdot \sqrt{-1 \cdot \sqrt{\frac{c \cdot -4}{a}}}\right)} \]

       2. sqrt-unprod 42.9%

          \[\leadsto -0.5 \cdot \color{blue}{\sqrt{\left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right) \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)}} \]

       3. mul-1-neg 42.9%

          \[\leadsto -0.5 \cdot \sqrt{\color{blue}{\left(-\sqrt{\frac{c \cdot -4}{a}}\right)} \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)} \]

       4. mul-1-neg 42.9%

          \[\leadsto -0.5 \cdot \sqrt{\left(-\sqrt{\frac{c \cdot -4}{a}}\right) \cdot \color{blue}{\left(-\sqrt{\frac{c \cdot -4}{a}}\right)}} \]

       5. sqr-neg 42.9%

          \[\leadsto -0.5 \cdot \sqrt{\color{blue}{\sqrt{\frac{c \cdot -4}{a}} \cdot \sqrt{\frac{c \cdot -4}{a}}}} \]

       6. add-sqr-sqrt 42.9%

          \[\leadsto -0.5 \cdot \sqrt{\color{blue}{\frac{c \cdot -4}{a}}} \]

       7. clear-num 42.8%

          \[\leadsto -0.5 \cdot \sqrt{\color{blue}{\frac{1}{\frac{a}{c \cdot -4}}}} \]

       8. sqrt-div 45.0%

          \[\leadsto -0.5 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{a}{c \cdot -4}}}} \]

       9. metadata-eval 45.0%

          \[\leadsto -0.5 \cdot \frac{\color{blue}{1}}{\sqrt{\frac{a}{c \cdot -4}}} \]

       10. *-un-lft-identity 45.0%

           \[\leadsto -0.5 \cdot \frac{1}{\sqrt{\frac{\color{blue}{1 \cdot a}}{c \cdot -4}}} \]

       11. *-commutative 45.0%

           \[\leadsto -0.5 \cdot \frac{1}{\sqrt{\frac{1 \cdot a}{\color{blue}{-4 \cdot c}}}} \]

       12. times-frac 45.0%

           \[\leadsto -0.5 \cdot \frac{1}{\sqrt{\color{blue}{\frac{1}{-4} \cdot \frac{a}{c}}}} \]

       13. metadata-eval 45.0%

           \[\leadsto -0.5 \cdot \frac{1}{\sqrt{\color{blue}{-0.25} \cdot \frac{a}{c}}} \]

   13. Applied egg-rr 45.0%

       \[\leadsto -0.5 \cdot \color{blue}{\frac{1}{\sqrt{-0.25 \cdot \frac{a}{c}}}} \]

   if 8.9999999999999995e-163 < b

   1. Initial program 21.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 21.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 21.6%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-exp-log 14.3%

         \[\leadsto \color{blue}{e^{\log \left(\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\right)}} \]

      2. frac-2neg 14.3%

         \[\leadsto e^{\log \color{blue}{\left(\frac{-\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{-a \cdot 2}\right)}} \]

   6. Applied egg-rr 14.3%

      \[\leadsto \color{blue}{e^{\log \left(\frac{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}}{a \cdot -2}\right)}} \]
   7. Step-by-step derivation

      1. rem-exp-log 21.6%

         \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}}{a \cdot -2}} \]

      2. clear-num 21.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -2}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}}}} \]

   8. Applied egg-rr 21.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -2}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}}}} \]

   9. Taylor expanded in a around 0 76.3%

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

       1. +-commutative 76.3%

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

       2. mul-1-neg 76.3%

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

       3. unsub-neg 76.3%

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

   11. Simplified 76.3%

       \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} - \frac{b}{c}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-97}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \mathbf{elif}\;b \leq -1.05 \cdot 10^{-241}:\\ \;\;\;\;\sqrt{\left(-4 \cdot \frac{c}{a}\right) \cdot 0.25}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-163}:\\ \;\;\;\;-0.5 \cdot \frac{1}{\sqrt{-0.25 \cdot \frac{a}{c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.4 \cdot 10^{-97}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq -2.4 \cdot 10^{-242}:\\ \;\;\;\;\sqrt{\left(-4 \cdot \frac{c}{a}\right) \cdot 0.25}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-162}:\\ \;\;\;\;-0.5 \cdot \frac{1}{\sqrt{-0.25 \cdot \frac{a}{c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.4e-97)
   (/ b (- a))
   (if (<= b -2.4e-242)
     (sqrt (* (* -4.0 (/ c a)) 0.25))
     (if (<= b 1.05e-162)
       (* -0.5 (/ 1.0 (sqrt (* -0.25 (/ a c)))))
       (/ 1.0 (- (/ a b) (/ b c)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.4e-97) {
		tmp = b / -a;
	} else if (b <= -2.4e-242) {
		tmp = sqrt(((-4.0 * (c / a)) * 0.25));
	} else if (b <= 1.05e-162) {
		tmp = -0.5 * (1.0 / sqrt((-0.25 * (a / c))));
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	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 <= (-8.4d-97)) then
        tmp = b / -a
    else if (b <= (-2.4d-242)) then
        tmp = sqrt((((-4.0d0) * (c / a)) * 0.25d0))
    else if (b <= 1.05d-162) then
        tmp = (-0.5d0) * (1.0d0 / sqrt(((-0.25d0) * (a / c))))
    else
        tmp = 1.0d0 / ((a / b) - (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.4e-97) {
		tmp = b / -a;
	} else if (b <= -2.4e-242) {
		tmp = Math.sqrt(((-4.0 * (c / a)) * 0.25));
	} else if (b <= 1.05e-162) {
		tmp = -0.5 * (1.0 / Math.sqrt((-0.25 * (a / c))));
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.4e-97:
		tmp = b / -a
	elif b <= -2.4e-242:
		tmp = math.sqrt(((-4.0 * (c / a)) * 0.25))
	elif b <= 1.05e-162:
		tmp = -0.5 * (1.0 / math.sqrt((-0.25 * (a / c))))
	else:
		tmp = 1.0 / ((a / b) - (b / c))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.4e-97)
		tmp = Float64(b / Float64(-a));
	elseif (b <= -2.4e-242)
		tmp = sqrt(Float64(Float64(-4.0 * Float64(c / a)) * 0.25));
	elseif (b <= 1.05e-162)
		tmp = Float64(-0.5 * Float64(1.0 / sqrt(Float64(-0.25 * Float64(a / c)))));
	else
		tmp = Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.4e-97)
		tmp = b / -a;
	elseif (b <= -2.4e-242)
		tmp = sqrt(((-4.0 * (c / a)) * 0.25));
	elseif (b <= 1.05e-162)
		tmp = -0.5 * (1.0 / sqrt((-0.25 * (a / c))));
	else
		tmp = 1.0 / ((a / b) - (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.4e-97], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, -2.4e-242], N[Sqrt[N[(N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]], $MachinePrecision], If[LessEqual[b, 1.05e-162], N[(-0.5 * N[(1.0 / N[Sqrt[N[(-0.25 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -8.4000000000000005e-97

   1. Initial program 70.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 70.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 70.0%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 82.7%

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

      1. associate-*r/ 82.7%

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

      2. mul-1-neg 82.7%

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

   7. Simplified 82.7%

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

   if -8.4000000000000005e-97 < b < -2.4000000000000001e-242

   1. Initial program 88.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 88.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 88.9%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 88.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}\right) \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

      2. pow3 88.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

      3. *-commutative 88.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\sqrt[3]{\color{blue}{\left(a \cdot c\right) \cdot 4}}\right)}^{3}}}{a \cdot 2} \]

      4. associate-*l* 88.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\sqrt[3]{\color{blue}{a \cdot \left(c \cdot 4\right)}}\right)}^{3}}}{a \cdot 2} \]

   6. Applied egg-rr 88.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\sqrt[3]{a \cdot \left(c \cdot 4\right)}\right)}^{3}}}}{a \cdot 2} \]

   7. Taylor expanded in a around -inf 0.0%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto -0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right) \]

      3. rem-square-sqrt 57.3%

         \[\leadsto -0.5 \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right) \]

      4. rem-cube-cbrt 57.7%

         \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\frac{c \cdot \color{blue}{-4}}{a}}\right) \]

      5. associate-/l* 57.7%

         \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\color{blue}{c \cdot \frac{-4}{a}}}\right) \]

   9. Simplified 57.7%

      \[\leadsto \color{blue}{-0.5 \cdot \left(-1 \cdot \sqrt{c \cdot \frac{-4}{a}}\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ 57.7%

          \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\color{blue}{\frac{c \cdot -4}{a}}}\right) \]

   11. Applied egg-rr 57.7%

       \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\color{blue}{\frac{c \cdot -4}{a}}}\right) \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 57.5%

          \[\leadsto \color{blue}{\sqrt{-0.5 \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)} \cdot \sqrt{-0.5 \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)}} \]

       2. sqrt-unprod 57.7%

          \[\leadsto \color{blue}{\sqrt{\left(-0.5 \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)\right) \cdot \left(-0.5 \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)\right)}} \]

       3. *-commutative 57.7%

          \[\leadsto \sqrt{\color{blue}{\left(\left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right) \cdot -0.5\right)} \cdot \left(-0.5 \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)\right)} \]

       4. *-commutative 57.7%

          \[\leadsto \sqrt{\left(\left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right) \cdot -0.5\right) \cdot \color{blue}{\left(\left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right) \cdot -0.5\right)}} \]

       5. swap-sqr 57.7%

          \[\leadsto \sqrt{\color{blue}{\left(\left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right) \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)\right) \cdot \left(-0.5 \cdot -0.5\right)}} \]

       6. mul-1-neg 57.7%

          \[\leadsto \sqrt{\left(\color{blue}{\left(-\sqrt{\frac{c \cdot -4}{a}}\right)} \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)\right) \cdot \left(-0.5 \cdot -0.5\right)} \]

       7. mul-1-neg 57.7%

          \[\leadsto \sqrt{\left(\left(-\sqrt{\frac{c \cdot -4}{a}}\right) \cdot \color{blue}{\left(-\sqrt{\frac{c \cdot -4}{a}}\right)}\right) \cdot \left(-0.5 \cdot -0.5\right)} \]

       8. sqr-neg 57.7%

          \[\leadsto \sqrt{\color{blue}{\left(\sqrt{\frac{c \cdot -4}{a}} \cdot \sqrt{\frac{c \cdot -4}{a}}\right)} \cdot \left(-0.5 \cdot -0.5\right)} \]

       9. add-sqr-sqrt 57.7%

          \[\leadsto \sqrt{\color{blue}{\frac{c \cdot -4}{a}} \cdot \left(-0.5 \cdot -0.5\right)} \]

       10. *-commutative 57.7%

           \[\leadsto \sqrt{\frac{\color{blue}{-4 \cdot c}}{a} \cdot \left(-0.5 \cdot -0.5\right)} \]

       11. associate-/l* 57.7%

           \[\leadsto \sqrt{\color{blue}{\left(-4 \cdot \frac{c}{a}\right)} \cdot \left(-0.5 \cdot -0.5\right)} \]

       12. metadata-eval 57.7%

           \[\leadsto \sqrt{\left(-4 \cdot \frac{c}{a}\right) \cdot \color{blue}{0.25}} \]

   13. Applied egg-rr 57.7%

       \[\leadsto \color{blue}{\sqrt{\left(-4 \cdot \frac{c}{a}\right) \cdot 0.25}} \]

   if -2.4000000000000001e-242 < b < 1.05e-162

   1. Initial program 82.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 82.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 82.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 82.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}\right) \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

      2. pow3 82.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

      3. *-commutative 82.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\sqrt[3]{\color{blue}{\left(a \cdot c\right) \cdot 4}}\right)}^{3}}}{a \cdot 2} \]

      4. associate-*l* 82.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\sqrt[3]{\color{blue}{a \cdot \left(c \cdot 4\right)}}\right)}^{3}}}{a \cdot 2} \]

   6. Applied egg-rr 82.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\sqrt[3]{a \cdot \left(c \cdot 4\right)}\right)}^{3}}}}{a \cdot 2} \]

   7. Taylor expanded in a around -inf 0.0%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto -0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right) \]

      3. rem-square-sqrt 28.0%

         \[\leadsto -0.5 \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right) \]

      4. rem-cube-cbrt 28.1%

         \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\frac{c \cdot \color{blue}{-4}}{a}}\right) \]

      5. associate-/l* 28.0%

         \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\color{blue}{c \cdot \frac{-4}{a}}}\right) \]

   9. Simplified 28.0%

      \[\leadsto \color{blue}{-0.5 \cdot \left(-1 \cdot \sqrt{c \cdot \frac{-4}{a}}\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ 28.1%

          \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\color{blue}{\frac{c \cdot -4}{a}}}\right) \]

   11. Applied egg-rr 28.1%

       \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\color{blue}{\frac{c \cdot -4}{a}}}\right) \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 0.8%

          \[\leadsto -0.5 \cdot \color{blue}{\left(\sqrt{-1 \cdot \sqrt{\frac{c \cdot -4}{a}}} \cdot \sqrt{-1 \cdot \sqrt{\frac{c \cdot -4}{a}}}\right)} \]

       2. sqrt-unprod 42.9%

          \[\leadsto -0.5 \cdot \color{blue}{\sqrt{\left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right) \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)}} \]

       3. mul-1-neg 42.9%

          \[\leadsto -0.5 \cdot \sqrt{\color{blue}{\left(-\sqrt{\frac{c \cdot -4}{a}}\right)} \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)} \]

       4. mul-1-neg 42.9%

          \[\leadsto -0.5 \cdot \sqrt{\left(-\sqrt{\frac{c \cdot -4}{a}}\right) \cdot \color{blue}{\left(-\sqrt{\frac{c \cdot -4}{a}}\right)}} \]

       5. sqr-neg 42.9%

          \[\leadsto -0.5 \cdot \sqrt{\color{blue}{\sqrt{\frac{c \cdot -4}{a}} \cdot \sqrt{\frac{c \cdot -4}{a}}}} \]

       6. add-sqr-sqrt 42.9%

          \[\leadsto -0.5 \cdot \sqrt{\color{blue}{\frac{c \cdot -4}{a}}} \]

       7. clear-num 42.8%

          \[\leadsto -0.5 \cdot \sqrt{\color{blue}{\frac{1}{\frac{a}{c \cdot -4}}}} \]

       8. sqrt-div 45.0%

          \[\leadsto -0.5 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{a}{c \cdot -4}}}} \]

       9. metadata-eval 45.0%

          \[\leadsto -0.5 \cdot \frac{\color{blue}{1}}{\sqrt{\frac{a}{c \cdot -4}}} \]

       10. *-un-lft-identity 45.0%

           \[\leadsto -0.5 \cdot \frac{1}{\sqrt{\frac{\color{blue}{1 \cdot a}}{c \cdot -4}}} \]

       11. *-commutative 45.0%

           \[\leadsto -0.5 \cdot \frac{1}{\sqrt{\frac{1 \cdot a}{\color{blue}{-4 \cdot c}}}} \]

       12. times-frac 45.0%

           \[\leadsto -0.5 \cdot \frac{1}{\sqrt{\color{blue}{\frac{1}{-4} \cdot \frac{a}{c}}}} \]

       13. metadata-eval 45.0%

           \[\leadsto -0.5 \cdot \frac{1}{\sqrt{\color{blue}{-0.25} \cdot \frac{a}{c}}} \]

   13. Applied egg-rr 45.0%

       \[\leadsto -0.5 \cdot \color{blue}{\frac{1}{\sqrt{-0.25 \cdot \frac{a}{c}}}} \]

   if 1.05e-162 < b

   1. Initial program 21.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 21.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 21.6%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-exp-log 14.3%

         \[\leadsto \color{blue}{e^{\log \left(\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\right)}} \]

      2. frac-2neg 14.3%

         \[\leadsto e^{\log \color{blue}{\left(\frac{-\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{-a \cdot 2}\right)}} \]

   6. Applied egg-rr 14.3%

      \[\leadsto \color{blue}{e^{\log \left(\frac{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}}{a \cdot -2}\right)}} \]
   7. Step-by-step derivation

      1. rem-exp-log 21.6%

         \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}}{a \cdot -2}} \]

      2. clear-num 21.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -2}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}}}} \]

   8. Applied egg-rr 21.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -2}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}}}} \]

   9. Taylor expanded in a around 0 76.3%

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

       1. +-commutative 76.3%

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

       2. mul-1-neg 76.3%

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

       3. unsub-neg 76.3%

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

   11. Simplified 76.3%

       \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} - \frac{b}{c}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.4 \cdot 10^{-97}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq -2.4 \cdot 10^{-242}:\\ \;\;\;\;\sqrt{\left(-4 \cdot \frac{c}{a}\right) \cdot 0.25}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-162}:\\ \;\;\;\;-0.5 \cdot \frac{1}{\sqrt{-0.25 \cdot \frac{a}{c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -4 \cdot \frac{c}{a}\\ \mathbf{if}\;b \leq -7.5 \cdot 10^{-97}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq -4.1 \cdot 10^{-241}:\\ \;\;\;\;\sqrt{t\_0 \cdot 0.25}\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-163}:\\ \;\;\;\;-0.5 \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* -4.0 (/ c a))))
   (if (<= b -7.5e-97)
     (/ b (- a))
     (if (<= b -4.1e-241)
       (sqrt (* t_0 0.25))
       (if (<= b 8.2e-163) (* -0.5 (sqrt t_0)) (/ 1.0 (- (/ a b) (/ b c))))))))

C

double code(double a, double b, double c) {
	double t_0 = -4.0 * (c / a);
	double tmp;
	if (b <= -7.5e-97) {
		tmp = b / -a;
	} else if (b <= -4.1e-241) {
		tmp = sqrt((t_0 * 0.25));
	} else if (b <= 8.2e-163) {
		tmp = -0.5 * sqrt(t_0);
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-4.0d0) * (c / a)
    if (b <= (-7.5d-97)) then
        tmp = b / -a
    else if (b <= (-4.1d-241)) then
        tmp = sqrt((t_0 * 0.25d0))
    else if (b <= 8.2d-163) then
        tmp = (-0.5d0) * sqrt(t_0)
    else
        tmp = 1.0d0 / ((a / b) - (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -4.0 * (c / a);
	double tmp;
	if (b <= -7.5e-97) {
		tmp = b / -a;
	} else if (b <= -4.1e-241) {
		tmp = Math.sqrt((t_0 * 0.25));
	} else if (b <= 8.2e-163) {
		tmp = -0.5 * Math.sqrt(t_0);
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = -4.0 * (c / a)
	tmp = 0
	if b <= -7.5e-97:
		tmp = b / -a
	elif b <= -4.1e-241:
		tmp = math.sqrt((t_0 * 0.25))
	elif b <= 8.2e-163:
		tmp = -0.5 * math.sqrt(t_0)
	else:
		tmp = 1.0 / ((a / b) - (b / c))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(-4.0 * Float64(c / a))
	tmp = 0.0
	if (b <= -7.5e-97)
		tmp = Float64(b / Float64(-a));
	elseif (b <= -4.1e-241)
		tmp = sqrt(Float64(t_0 * 0.25));
	elseif (b <= 8.2e-163)
		tmp = Float64(-0.5 * sqrt(t_0));
	else
		tmp = Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = -4.0 * (c / a);
	tmp = 0.0;
	if (b <= -7.5e-97)
		tmp = b / -a;
	elseif (b <= -4.1e-241)
		tmp = sqrt((t_0 * 0.25));
	elseif (b <= 8.2e-163)
		tmp = -0.5 * sqrt(t_0);
	else
		tmp = 1.0 / ((a / b) - (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -7.5e-97], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, -4.1e-241], N[Sqrt[N[(t$95$0 * 0.25), $MachinePrecision]], $MachinePrecision], If[LessEqual[b, 8.2e-163], N[(-0.5 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -7.5e-97

   1. Initial program 70.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 70.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 70.0%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 82.7%

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

      1. associate-*r/ 82.7%

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

      2. mul-1-neg 82.7%

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

   7. Simplified 82.7%

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

   if -7.5e-97 < b < -4.0999999999999999e-241

   1. Initial program 88.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 88.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 88.9%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 88.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}\right) \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

      2. pow3 88.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

      3. *-commutative 88.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\sqrt[3]{\color{blue}{\left(a \cdot c\right) \cdot 4}}\right)}^{3}}}{a \cdot 2} \]

      4. associate-*l* 88.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\sqrt[3]{\color{blue}{a \cdot \left(c \cdot 4\right)}}\right)}^{3}}}{a \cdot 2} \]

   6. Applied egg-rr 88.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\sqrt[3]{a \cdot \left(c \cdot 4\right)}\right)}^{3}}}}{a \cdot 2} \]

   7. Taylor expanded in a around -inf 0.0%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto -0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right) \]

      3. rem-square-sqrt 57.3%

         \[\leadsto -0.5 \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right) \]

      4. rem-cube-cbrt 57.7%

         \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\frac{c \cdot \color{blue}{-4}}{a}}\right) \]

      5. associate-/l* 57.7%

         \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\color{blue}{c \cdot \frac{-4}{a}}}\right) \]

   9. Simplified 57.7%

      \[\leadsto \color{blue}{-0.5 \cdot \left(-1 \cdot \sqrt{c \cdot \frac{-4}{a}}\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ 57.7%

          \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\color{blue}{\frac{c \cdot -4}{a}}}\right) \]

   11. Applied egg-rr 57.7%

       \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\color{blue}{\frac{c \cdot -4}{a}}}\right) \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 57.5%

          \[\leadsto \color{blue}{\sqrt{-0.5 \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)} \cdot \sqrt{-0.5 \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)}} \]

       2. sqrt-unprod 57.7%

          \[\leadsto \color{blue}{\sqrt{\left(-0.5 \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)\right) \cdot \left(-0.5 \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)\right)}} \]

       3. *-commutative 57.7%

          \[\leadsto \sqrt{\color{blue}{\left(\left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right) \cdot -0.5\right)} \cdot \left(-0.5 \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)\right)} \]

       4. *-commutative 57.7%

          \[\leadsto \sqrt{\left(\left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right) \cdot -0.5\right) \cdot \color{blue}{\left(\left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right) \cdot -0.5\right)}} \]

       5. swap-sqr 57.7%

          \[\leadsto \sqrt{\color{blue}{\left(\left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right) \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)\right) \cdot \left(-0.5 \cdot -0.5\right)}} \]

       6. mul-1-neg 57.7%

          \[\leadsto \sqrt{\left(\color{blue}{\left(-\sqrt{\frac{c \cdot -4}{a}}\right)} \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)\right) \cdot \left(-0.5 \cdot -0.5\right)} \]

       7. mul-1-neg 57.7%

          \[\leadsto \sqrt{\left(\left(-\sqrt{\frac{c \cdot -4}{a}}\right) \cdot \color{blue}{\left(-\sqrt{\frac{c \cdot -4}{a}}\right)}\right) \cdot \left(-0.5 \cdot -0.5\right)} \]

       8. sqr-neg 57.7%

          \[\leadsto \sqrt{\color{blue}{\left(\sqrt{\frac{c \cdot -4}{a}} \cdot \sqrt{\frac{c \cdot -4}{a}}\right)} \cdot \left(-0.5 \cdot -0.5\right)} \]

       9. add-sqr-sqrt 57.7%

          \[\leadsto \sqrt{\color{blue}{\frac{c \cdot -4}{a}} \cdot \left(-0.5 \cdot -0.5\right)} \]

       10. *-commutative 57.7%

           \[\leadsto \sqrt{\frac{\color{blue}{-4 \cdot c}}{a} \cdot \left(-0.5 \cdot -0.5\right)} \]

       11. associate-/l* 57.7%

           \[\leadsto \sqrt{\color{blue}{\left(-4 \cdot \frac{c}{a}\right)} \cdot \left(-0.5 \cdot -0.5\right)} \]

       12. metadata-eval 57.7%

           \[\leadsto \sqrt{\left(-4 \cdot \frac{c}{a}\right) \cdot \color{blue}{0.25}} \]

   13. Applied egg-rr 57.7%

       \[\leadsto \color{blue}{\sqrt{\left(-4 \cdot \frac{c}{a}\right) \cdot 0.25}} \]

   if -4.0999999999999999e-241 < b < 8.19999999999999965e-163

   1. Initial program 82.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 82.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 82.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 82.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}\right) \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

      2. pow3 82.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

      3. *-commutative 82.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\sqrt[3]{\color{blue}{\left(a \cdot c\right) \cdot 4}}\right)}^{3}}}{a \cdot 2} \]

      4. associate-*l* 82.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\sqrt[3]{\color{blue}{a \cdot \left(c \cdot 4\right)}}\right)}^{3}}}{a \cdot 2} \]

   6. Applied egg-rr 82.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\sqrt[3]{a \cdot \left(c \cdot 4\right)}\right)}^{3}}}}{a \cdot 2} \]

   7. Taylor expanded in a around -inf 0.0%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto -0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right) \]

      3. rem-square-sqrt 28.0%

         \[\leadsto -0.5 \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right) \]

      4. rem-cube-cbrt 28.1%

         \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\frac{c \cdot \color{blue}{-4}}{a}}\right) \]

      5. associate-/l* 28.0%

         \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\color{blue}{c \cdot \frac{-4}{a}}}\right) \]

   9. Simplified 28.0%

      \[\leadsto \color{blue}{-0.5 \cdot \left(-1 \cdot \sqrt{c \cdot \frac{-4}{a}}\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ 28.1%

          \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\color{blue}{\frac{c \cdot -4}{a}}}\right) \]

   11. Applied egg-rr 28.1%

       \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\color{blue}{\frac{c \cdot -4}{a}}}\right) \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 0.8%

          \[\leadsto -0.5 \cdot \color{blue}{\left(\sqrt{-1 \cdot \sqrt{\frac{c \cdot -4}{a}}} \cdot \sqrt{-1 \cdot \sqrt{\frac{c \cdot -4}{a}}}\right)} \]

       2. sqrt-unprod 42.9%

          \[\leadsto -0.5 \cdot \color{blue}{\sqrt{\left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right) \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)}} \]

       3. mul-1-neg 42.9%

          \[\leadsto -0.5 \cdot \sqrt{\color{blue}{\left(-\sqrt{\frac{c \cdot -4}{a}}\right)} \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)} \]

       4. mul-1-neg 42.9%

          \[\leadsto -0.5 \cdot \sqrt{\left(-\sqrt{\frac{c \cdot -4}{a}}\right) \cdot \color{blue}{\left(-\sqrt{\frac{c \cdot -4}{a}}\right)}} \]

       5. sqr-neg 42.9%

          \[\leadsto -0.5 \cdot \sqrt{\color{blue}{\sqrt{\frac{c \cdot -4}{a}} \cdot \sqrt{\frac{c \cdot -4}{a}}}} \]

       6. add-sqr-sqrt 42.9%

          \[\leadsto -0.5 \cdot \sqrt{\color{blue}{\frac{c \cdot -4}{a}}} \]

       7. *-commutative 42.9%

          \[\leadsto -0.5 \cdot \sqrt{\frac{\color{blue}{-4 \cdot c}}{a}} \]

       8. associate-/l* 42.9%

          \[\leadsto -0.5 \cdot \sqrt{\color{blue}{-4 \cdot \frac{c}{a}}} \]

   13. Applied egg-rr 42.9%

       \[\leadsto -0.5 \cdot \color{blue}{\sqrt{-4 \cdot \frac{c}{a}}} \]

   if 8.19999999999999965e-163 < b

   1. Initial program 21.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 21.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 21.6%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-exp-log 14.3%

         \[\leadsto \color{blue}{e^{\log \left(\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\right)}} \]

      2. frac-2neg 14.3%

         \[\leadsto e^{\log \color{blue}{\left(\frac{-\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{-a \cdot 2}\right)}} \]

   6. Applied egg-rr 14.3%

      \[\leadsto \color{blue}{e^{\log \left(\frac{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}}{a \cdot -2}\right)}} \]
   7. Step-by-step derivation

      1. rem-exp-log 21.6%

         \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}}{a \cdot -2}} \]

      2. clear-num 21.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -2}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}}}} \]

   8. Applied egg-rr 21.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -2}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}}}} \]

   9. Taylor expanded in a around 0 76.3%

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

       1. +-commutative 76.3%

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

       2. mul-1-neg 76.3%

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

       3. unsub-neg 76.3%

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

   11. Simplified 76.3%

       \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} - \frac{b}{c}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-97}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq -4.1 \cdot 10^{-241}:\\ \;\;\;\;\sqrt{\left(-4 \cdot \frac{c}{a}\right) \cdot 0.25}\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-163}:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{-92}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.4e-92)
   (* b (+ (/ c (pow b 2.0)) (/ -1.0 a)))
   (if (<= b 6e-9) (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0)) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.4e-92) {
		tmp = b * ((c / pow(b, 2.0)) + (-1.0 / a));
	} else if (b <= 6e-9) {
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = 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 <= (-5.4d-92)) then
        tmp = b * ((c / (b ** 2.0d0)) + ((-1.0d0) / a))
    else if (b <= 6d-9) then
        tmp = (sqrt((c * (a * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.4e-92) {
		tmp = b * ((c / Math.pow(b, 2.0)) + (-1.0 / a));
	} else if (b <= 6e-9) {
		tmp = (Math.sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.4e-92:
		tmp = b * ((c / math.pow(b, 2.0)) + (-1.0 / a))
	elif b <= 6e-9:
		tmp = (math.sqrt((c * (a * -4.0))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.4e-92)
		tmp = Float64(b * Float64(Float64(c / (b ^ 2.0)) + Float64(-1.0 / a)));
	elseif (b <= 6e-9)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.4e-92)
		tmp = b * ((c / (b ^ 2.0)) + (-1.0 / a));
	elseif (b <= 6e-9)
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.4e-92], N[(b * N[(N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6e-9], N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.3999999999999999e-92

   1. Initial program 69.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 69.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 69.7%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 83.7%

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

      1. mul-1-neg 83.7%

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

      2. distribute-rgt-neg-in 83.7%

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

      3. +-commutative 83.7%

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

      4. mul-1-neg 83.7%

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

      5. unsub-neg 83.7%

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

   7. Simplified 83.7%

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

   if -5.3999999999999999e-92 < b < 5.99999999999999996e-9

   1. Initial program 74.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 74.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 74.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 69.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. associate-*r* 69.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(-4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

      2. *-commutative 69.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{c \cdot \left(-4 \cdot a\right)}}}{a \cdot 2} \]

      3. *-commutative 69.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{c \cdot \color{blue}{\left(a \cdot -4\right)}}}{a \cdot 2} \]

   7. Simplified 69.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}}{a \cdot 2} \]

   if 5.99999999999999996e-9 < b

   1. Initial program 10.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 10.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 10.3%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 87.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 87.7%

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

      2. neg-mul-1 87.7%

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

   7. Simplified 87.7%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{-92}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-96}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{\left(-4 \cdot \frac{c}{a}\right) \cdot 0.25}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.15e-96)
   (/ b (- a))
   (if (<= b 6.2e-160)
     (sqrt (* (* -4.0 (/ c a)) 0.25))
     (/ 1.0 (- (/ a b) (/ b c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.15e-96) {
		tmp = b / -a;
	} else if (b <= 6.2e-160) {
		tmp = sqrt(((-4.0 * (c / a)) * 0.25));
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	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-96)) then
        tmp = b / -a
    else if (b <= 6.2d-160) then
        tmp = sqrt((((-4.0d0) * (c / a)) * 0.25d0))
    else
        tmp = 1.0d0 / ((a / b) - (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.15e-96) {
		tmp = b / -a;
	} else if (b <= 6.2e-160) {
		tmp = Math.sqrt(((-4.0 * (c / a)) * 0.25));
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.15e-96:
		tmp = b / -a
	elif b <= 6.2e-160:
		tmp = math.sqrt(((-4.0 * (c / a)) * 0.25))
	else:
		tmp = 1.0 / ((a / b) - (b / c))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.15e-96)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 6.2e-160)
		tmp = sqrt(Float64(Float64(-4.0 * Float64(c / a)) * 0.25));
	else
		tmp = Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.15e-96)
		tmp = b / -a;
	elseif (b <= 6.2e-160)
		tmp = sqrt(((-4.0 * (c / a)) * 0.25));
	else
		tmp = 1.0 / ((a / b) - (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.15e-96], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 6.2e-160], N[Sqrt[N[(N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.15e-96

   1. Initial program 70.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 70.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 70.0%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 82.7%

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

      1. associate-*r/ 82.7%

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

      2. mul-1-neg 82.7%

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

   7. Simplified 82.7%

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

   if -1.15e-96 < b < 6.2000000000000001e-160

   1. Initial program 85.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 85.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 85.1%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 84.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}\right) \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

      2. pow3 84.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

      3. *-commutative 84.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\sqrt[3]{\color{blue}{\left(a \cdot c\right) \cdot 4}}\right)}^{3}}}{a \cdot 2} \]

      4. associate-*l* 84.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\sqrt[3]{\color{blue}{a \cdot \left(c \cdot 4\right)}}\right)}^{3}}}{a \cdot 2} \]

   6. Applied egg-rr 84.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\sqrt[3]{a \cdot \left(c \cdot 4\right)}\right)}^{3}}}}{a \cdot 2} \]

   7. Taylor expanded in a around -inf 0.0%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto -0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right) \]

      3. rem-square-sqrt 38.9%

         \[\leadsto -0.5 \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right) \]

      4. rem-cube-cbrt 39.2%

         \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\frac{c \cdot \color{blue}{-4}}{a}}\right) \]

      5. associate-/l* 39.1%

         \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\color{blue}{c \cdot \frac{-4}{a}}}\right) \]

   9. Simplified 39.1%

      \[\leadsto \color{blue}{-0.5 \cdot \left(-1 \cdot \sqrt{c \cdot \frac{-4}{a}}\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ 39.2%

          \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\color{blue}{\frac{c \cdot -4}{a}}}\right) \]

   11. Applied egg-rr 39.2%

       \[\leadsto -0.5 \cdot \left(-1 \cdot \sqrt{\color{blue}{\frac{c \cdot -4}{a}}}\right) \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 38.9%

          \[\leadsto \color{blue}{\sqrt{-0.5 \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)} \cdot \sqrt{-0.5 \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)}} \]

       2. sqrt-unprod 39.2%

          \[\leadsto \color{blue}{\sqrt{\left(-0.5 \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)\right) \cdot \left(-0.5 \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)\right)}} \]

       3. *-commutative 39.2%

          \[\leadsto \sqrt{\color{blue}{\left(\left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right) \cdot -0.5\right)} \cdot \left(-0.5 \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)\right)} \]

       4. *-commutative 39.2%

          \[\leadsto \sqrt{\left(\left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right) \cdot -0.5\right) \cdot \color{blue}{\left(\left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right) \cdot -0.5\right)}} \]

       5. swap-sqr 39.2%

          \[\leadsto \sqrt{\color{blue}{\left(\left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right) \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)\right) \cdot \left(-0.5 \cdot -0.5\right)}} \]

       6. mul-1-neg 39.2%

          \[\leadsto \sqrt{\left(\color{blue}{\left(-\sqrt{\frac{c \cdot -4}{a}}\right)} \cdot \left(-1 \cdot \sqrt{\frac{c \cdot -4}{a}}\right)\right) \cdot \left(-0.5 \cdot -0.5\right)} \]

       7. mul-1-neg 39.2%

          \[\leadsto \sqrt{\left(\left(-\sqrt{\frac{c \cdot -4}{a}}\right) \cdot \color{blue}{\left(-\sqrt{\frac{c \cdot -4}{a}}\right)}\right) \cdot \left(-0.5 \cdot -0.5\right)} \]

       8. sqr-neg 39.2%

          \[\leadsto \sqrt{\color{blue}{\left(\sqrt{\frac{c \cdot -4}{a}} \cdot \sqrt{\frac{c \cdot -4}{a}}\right)} \cdot \left(-0.5 \cdot -0.5\right)} \]

       9. add-sqr-sqrt 39.2%

          \[\leadsto \sqrt{\color{blue}{\frac{c \cdot -4}{a}} \cdot \left(-0.5 \cdot -0.5\right)} \]

       10. *-commutative 39.2%

           \[\leadsto \sqrt{\frac{\color{blue}{-4 \cdot c}}{a} \cdot \left(-0.5 \cdot -0.5\right)} \]

       11. associate-/l* 39.2%

           \[\leadsto \sqrt{\color{blue}{\left(-4 \cdot \frac{c}{a}\right)} \cdot \left(-0.5 \cdot -0.5\right)} \]

       12. metadata-eval 39.2%

           \[\leadsto \sqrt{\left(-4 \cdot \frac{c}{a}\right) \cdot \color{blue}{0.25}} \]

   13. Applied egg-rr 39.2%

       \[\leadsto \color{blue}{\sqrt{\left(-4 \cdot \frac{c}{a}\right) \cdot 0.25}} \]

   if 6.2000000000000001e-160 < b

   1. Initial program 20.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 20.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 20.9%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-exp-log 13.6%

         \[\leadsto \color{blue}{e^{\log \left(\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\right)}} \]

      2. frac-2neg 13.6%

         \[\leadsto e^{\log \color{blue}{\left(\frac{-\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{-a \cdot 2}\right)}} \]

   6. Applied egg-rr 13.6%

      \[\leadsto \color{blue}{e^{\log \left(\frac{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}}{a \cdot -2}\right)}} \]
   7. Step-by-step derivation

      1. rem-exp-log 20.9%

         \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}}{a \cdot -2}} \]

      2. clear-num 20.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -2}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}}}} \]

   8. Applied egg-rr 20.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -2}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}}}} \]

   9. Taylor expanded in a around 0 76.9%

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

       1. +-commutative 76.9%

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

       2. mul-1-neg 76.9%

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

       3. unsub-neg 76.9%

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

   11. Simplified 76.9%

       \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} - \frac{b}{c}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-96}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{\left(-4 \cdot \frac{c}{a}\right) \cdot 0.25}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.3% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (/ b (- a)) (/ 1.0 (- (/ a b) (/ b c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = b / -a;
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	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 <= (-1d-310)) then
        tmp = b / -a
    else
        tmp = 1.0d0 / ((a / b) - (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = b / -a;
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = b / -a
	else:
		tmp = 1.0 / ((a / b) - (b / c))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-310)
		tmp = b / -a;
	else
		tmp = 1.0 / ((a / b) - (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(b / (-a)), $MachinePrecision], N[(1.0 / N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 74.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 74.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 74.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 65.8%

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

      1. associate-*r/ 65.8%

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

      2. mul-1-neg 65.8%

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

   7. Simplified 65.8%

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

   if -9.999999999999969e-311 < b

   1. Initial program 30.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 30.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 30.9%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-exp-log 18.2%

         \[\leadsto \color{blue}{e^{\log \left(\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\right)}} \]

      2. frac-2neg 18.2%

         \[\leadsto e^{\log \color{blue}{\left(\frac{-\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{-a \cdot 2}\right)}} \]

   6. Applied egg-rr 18.2%

      \[\leadsto \color{blue}{e^{\log \left(\frac{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}}{a \cdot -2}\right)}} \]
   7. Step-by-step derivation

      1. rem-exp-log 30.9%

         \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}}{a \cdot -2}} \]

      2. clear-num 30.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -2}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}}}} \]

   8. Applied egg-rr 30.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -2}{b - \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}}}} \]

   9. Taylor expanded in a around 0 65.4%

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

       1. +-commutative 65.4%

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

       2. mul-1-neg 65.4%

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

       3. unsub-neg 65.4%

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

   11. Simplified 65.4%

       \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} - \frac{b}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.5% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{-272}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.7e-272) (/ b (- a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.7e-272) {
		tmp = b / -a;
	} else {
		tmp = 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 <= 2.7d-272) then
        tmp = b / -a
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.7e-272) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.7e-272:
		tmp = b / -a
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.7e-272)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.7e-272)
		tmp = b / -a;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.7e-272], N[(b / (-a)), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.69999999999999993e-272

   1. Initial program 75.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 75.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 75.7%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 63.4%

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

      1. associate-*r/ 63.4%

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

      2. mul-1-neg 63.4%

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

   7. Simplified 63.4%

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

   if 2.69999999999999993e-272 < b

   1. Initial program 28.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 28.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 28.3%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 67.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 67.6%

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

      2. neg-mul-1 67.6%

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

   7. Simplified 67.6%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{-272}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.1% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 3.8e-7) (/ b (- a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.8e-7) {
		tmp = b / -a;
	} else {
		tmp = 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 <= 3.8d-7) then
        tmp = b / -a
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.8e-7) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.8e-7:
		tmp = b / -a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.8e-7)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3.8e-7)
		tmp = b / -a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.8e-7], N[(b / (-a)), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.80000000000000015e-7

   1. Initial program 72.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 72.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 72.1%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 47.8%

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

      1. associate-*r/ 47.8%

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

      2. mul-1-neg 47.8%

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

   7. Simplified 47.8%

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

   if 3.80000000000000015e-7 < b

   1. Initial program 10.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 10.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 10.3%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 87.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 87.7%

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

      2. neg-mul-1 87.7%

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

   7. Simplified 87.7%

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

      1. add-sqr-sqrt 41.8%

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

      2. sqrt-unprod 41.2%

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

      3. sqr-neg 41.2%

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

      4. sqrt-unprod 14.9%

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

      5. add-sqr-sqrt 23.4%

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

      6. *-un-lft-identity 23.4%

         \[\leadsto \color{blue}{1 \cdot \frac{c}{b}} \]

   9. Applied egg-rr 23.4%

      \[\leadsto \color{blue}{1 \cdot \frac{c}{b}} \]
   10. Step-by-step derivation

       1. *-lft-identity 23.4%

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

   11. Simplified 23.4%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 12.4% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{+70}:\\ \;\;\;\;b \cdot \left(-c\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 1.05e+70) (* b (- c)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.05e+70) {
		tmp = b * -c;
	} else {
		tmp = 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 <= 1.05d+70) then
        tmp = b * -c
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.05e+70) {
		tmp = b * -c;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.05e+70:
		tmp = b * -c
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.05e+70)
		tmp = Float64(b * Float64(-c));
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.05e+70)
		tmp = b * -c;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.05e+70], N[(b * (-c)), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.05000000000000004e70

   1. Initial program 66.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 66.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 66.0%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 17.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 17.3%

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

      2. neg-mul-1 17.3%

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

   7. Simplified 17.3%

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

      1. frac-2neg 17.3%

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

      2. div-inv 17.2%

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

      3. remove-double-neg 17.2%

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

      4. add-exp-log 1.3%

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

      5. metadata-eval 1.3%

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

      6. frac-2neg 1.3%

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

      7. add-sqr-sqrt 0.4%

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

      8. sqrt-unprod 4.5%

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

      9. sqr-neg 4.5%

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

      10. sqrt-unprod 4.1%

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

      11. add-sqr-sqrt 4.7%

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

      12. rec-exp 4.7%

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

      13. add-exp-log 7.0%

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

      14. frac-2neg 7.0%

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

      15. metadata-eval 7.0%

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

      16. remove-double-div 7.0%

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

   9. Applied egg-rr 7.0%

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

   if 1.05000000000000004e70 < b

   1. Initial program 7.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 7.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 7.4%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 90.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. associate-*r/ 90.9%

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

      2. neg-mul-1 90.9%

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

   7. Simplified 90.9%

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

      1. add-sqr-sqrt 43.4%

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

      2. sqrt-unprod 48.9%

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

      3. sqr-neg 48.9%

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

      4. sqrt-unprod 19.4%

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

      5. add-sqr-sqrt 30.3%

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

      6. *-un-lft-identity 30.3%

         \[\leadsto \color{blue}{1 \cdot \frac{c}{b}} \]

   9. Applied egg-rr 30.3%

      \[\leadsto \color{blue}{1 \cdot \frac{c}{b}} \]
   10. Step-by-step derivation

       1. *-lft-identity 30.3%

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

   11. Simplified 30.3%

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

4. Final simplification 12.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{+70}:\\ \;\;\;\;b \cdot \left(-c\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 11.1% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 51.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

3. Simplified 51.8%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in b around inf 35.1%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
6. Step-by-step derivation

   1. associate-*r/ 35.1%

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

   2. neg-mul-1 35.1%

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

7. Simplified 35.1%

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

   1. add-sqr-sqrt 17.0%

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

   2. sqrt-unprod 17.9%

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

   3. sqr-neg 17.9%

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

   4. sqrt-unprod 5.7%

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

   5. add-sqr-sqrt 9.8%

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

   6. *-un-lft-identity 9.8%

      \[\leadsto \color{blue}{1 \cdot \frac{c}{b}} \]

9. Applied egg-rr 9.8%

   \[\leadsto \color{blue}{1 \cdot \frac{c}{b}} \]
10. Step-by-step derivation

    1. *-lft-identity 9.8%

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

11. Simplified 9.8%

    \[\leadsto \color{blue}{\frac{c}{b}} \]
12. Add Preprocessing

Alternative 13: 2.5% accurate, 38.7× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return b / 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 / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 51.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

3. Simplified 51.8%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in b around -inf 32.9%

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

   1. associate-*r/ 32.9%

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

   2. mul-1-neg 32.9%

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

7. Simplified 32.9%

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

   1. add-sqr-sqrt 31.2%

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

   2. sqrt-unprod 25.0%

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

   3. sqr-neg 25.0%

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

   4. sqrt-prod 1.9%

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

   5. add-sqr-sqrt 2.7%

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

   6. add-cbrt-cube 2.3%

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

   7. pow1/3 11.2%

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

   8. pow3 11.2%

      \[\leadsto \frac{{\color{blue}{\left({b}^{3}\right)}}^{0.3333333333333333}}{a} \]

9. Applied egg-rr 11.2%

   \[\leadsto \frac{\color{blue}{{\left({b}^{3}\right)}^{0.3333333333333333}}}{a} \]

10. Taylor expanded in b around 0 2.7%

    \[\leadsto \color{blue}{\frac{b}{a}} \]
11. Add Preprocessing

Alternative 14: 2.3% accurate, 38.7× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return b * 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 * a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 51.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{a \cdot 2}} \]

3. Simplified 51.8%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in b around -inf 32.9%

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

   1. associate-*r/ 32.9%

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

   2. mul-1-neg 32.9%

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

7. Simplified 32.9%

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

   1. add-sqr-sqrt 31.2%

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

   2. sqrt-unprod 25.0%

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

   3. sqr-neg 25.0%

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

   4. sqrt-prod 1.9%

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

   5. add-sqr-sqrt 2.7%

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

   6. add-cbrt-cube 2.3%

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

   7. pow1/3 11.2%

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

   8. pow3 11.2%

      \[\leadsto \frac{{\color{blue}{\left({b}^{3}\right)}}^{0.3333333333333333}}{a} \]

9. Applied egg-rr 11.2%

   \[\leadsto \frac{\color{blue}{{\left({b}^{3}\right)}^{0.3333333333333333}}}{a} \]
10. Step-by-step derivation

    1. expm1-log1p-u 5.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left({b}^{3}\right)}^{0.3333333333333333}}{a}\right)\right)} \]

    2. *-un-lft-identity 5.6%

       \[\leadsto \color{blue}{1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left({b}^{3}\right)}^{0.3333333333333333}}{a}\right)\right)} \]

    3. unpow1/3 2.0%

       \[\leadsto 1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{\sqrt[3]{{b}^{3}}}}{a}\right)\right) \]

    4. rem-cbrt-cube 2.2%

       \[\leadsto 1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{b}}{a}\right)\right) \]

    5. div-inv 2.2%

       \[\leadsto 1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{b \cdot \frac{1}{a}}\right)\right) \]

    6. expm1-log1p-u 2.7%

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

    7. rem-exp-log 1.4%

       \[\leadsto 1 \cdot \left(b \cdot \color{blue}{e^{\log \left(\frac{1}{a}\right)}}\right) \]

    8. neg-log 1.4%

       \[\leadsto 1 \cdot \left(b \cdot e^{\color{blue}{-\log a}}\right) \]

    9. add-sqr-sqrt 0.5%

       \[\leadsto 1 \cdot \left(b \cdot e^{\color{blue}{\sqrt{-\log a} \cdot \sqrt{-\log a}}}\right) \]

    10. sqrt-unprod 1.2%

        \[\leadsto 1 \cdot \left(b \cdot e^{\color{blue}{\sqrt{\left(-\log a\right) \cdot \left(-\log a\right)}}}\right) \]

    11. sqr-neg 1.2%

        \[\leadsto 1 \cdot \left(b \cdot e^{\sqrt{\color{blue}{\log a \cdot \log a}}}\right) \]

    12. sqrt-unprod 0.7%

        \[\leadsto 1 \cdot \left(b \cdot e^{\color{blue}{\sqrt{\log a} \cdot \sqrt{\log a}}}\right) \]

    13. add-sqr-sqrt 1.3%

        \[\leadsto 1 \cdot \left(b \cdot e^{\color{blue}{\log a}}\right) \]

    14. add-exp-log 2.4%

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

11. Applied egg-rr 2.4%

    \[\leadsto \color{blue}{1 \cdot \left(b \cdot a\right)} \]
12. Step-by-step derivation

    1. *-lft-identity 2.4%

       \[\leadsto \color{blue}{b \cdot a} \]

13. Simplified 2.4%

    \[\leadsto \color{blue}{b \cdot a} \]
14. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t\_0 - t\_1} \cdot \sqrt{t\_0 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t\_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{t\_2 - \frac{b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{\frac{b}{2} + t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fabs (/ b 2.0)))
        (t_1 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_2
         (if (== (copysign a c) a)
           (* (sqrt (- t_0 t_1)) (sqrt (+ t_0 t_1)))
           (hypot (/ b 2.0) t_1))))
   (if (< b 0.0) (/ (- t_2 (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) t_2)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs((b / 2.0));
	double t_1 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	} else {
		tmp = hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs((b / 2.0));
	double t_1 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((t_0 - t_1)) * Math.sqrt((t_0 + t_1));
	} else {
		tmp = Math.hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.fabs((b / 2.0))
	t_1 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((t_0 - t_1)) * math.sqrt((t_0 + t_1))
	else:
		tmp = math.hypot((b / 2.0), t_1)
	t_2 = tmp
	tmp_1 = 0
	if b < 0.0:
		tmp_1 = (t_2 - (b / 2.0)) / a
	else:
		tmp_1 = -c / ((b / 2.0) + t_2)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = abs(Float64(b / 2.0))
	t_1 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(t_0 - t_1)) * sqrt(Float64(t_0 + t_1)));
	else
		tmp = hypot(Float64(b / 2.0), t_1);
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b < 0.0)
		tmp_1 = Float64(Float64(t_2 - Float64(b / 2.0)) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(Float64(b / 2.0) + t_2));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = abs((b / 2.0));
	t_1 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	else
		tmp = hypot((b / 2.0), t_1);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b < 0.0)
		tmp_2 = (t_2 - (b / 2.0)) / a;
	else
		tmp_2 = -c / ((b / 2.0) + t_2);
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Abs[N[(b / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(b / 2.0), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]]}, If[Less[b, 0.0], N[(N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024133 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2)) x)) (sqrt (+ (fabs (/ b 2)) x))) (hypot (/ b 2) x))))) (if (< b 0) (/ (- sqtD (/ b 2)) a) (/ (- c) (+ (/ b 2) sqtD)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))