quadp (p42, positive)

Percentage Accurate: 51.7% → 84.1%

Time: 13.1s

Alternatives: 7

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: 51.7% 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: 84.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{-b}\\ t_1 := \frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{if}\;b \leq -1.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-127}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 10^{-58}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 2.25 \cdot 10^{+207}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot 2, \left(c \cdot \frac{-4}{{b}^{3}}\right) \cdot -0.125, \frac{1}{b}\right), \frac{b}{-c}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ c (- b)))
        (t_1 (/ (- (sqrt (- (* b b) (* 4.0 (* c a)))) b) (* a 2.0))))
   (if (<= b -1.5e+87)
     (- (/ c b) (/ b a))
     (if (<= b 4e-127)
       t_1
       (if (<= b 1e-58)
         t_0
         (if (<= b 3.5e-15)
           t_1
           (if (<= b 2.25e+207)
             (/
              1.0
              (fma
               a
               (fma (* a 2.0) (* (* c (/ -4.0 (pow b 3.0))) -0.125) (/ 1.0 b))
               (/ b (- c))))
             t_0)))))))

C

double code(double a, double b, double c) {
	double t_0 = c / -b;
	double t_1 = (sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	double tmp;
	if (b <= -1.5e+87) {
		tmp = (c / b) - (b / a);
	} else if (b <= 4e-127) {
		tmp = t_1;
	} else if (b <= 1e-58) {
		tmp = t_0;
	} else if (b <= 3.5e-15) {
		tmp = t_1;
	} else if (b <= 2.25e+207) {
		tmp = 1.0 / fma(a, fma((a * 2.0), ((c * (-4.0 / pow(b, 3.0))) * -0.125), (1.0 / b)), (b / -c));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(c / Float64(-b))
	t_1 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (b <= -1.5e+87)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 4e-127)
		tmp = t_1;
	elseif (b <= 1e-58)
		tmp = t_0;
	elseif (b <= 3.5e-15)
		tmp = t_1;
	elseif (b <= 2.25e+207)
		tmp = Float64(1.0 / fma(a, fma(Float64(a * 2.0), Float64(Float64(c * Float64(-4.0 / (b ^ 3.0))) * -0.125), Float64(1.0 / b)), Float64(b / Float64(-c))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c / (-b)), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.5e+87], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4e-127], t$95$1, If[LessEqual[b, 1e-58], t$95$0, If[LessEqual[b, 3.5e-15], t$95$1, If[LessEqual[b, 2.25e+207], N[(1.0 / N[(a * N[(N[(a * 2.0), $MachinePrecision] * N[(N[(c * N[(-4.0 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision] + N[(b / (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.4999999999999999e87

   1. Initial program 49.0%

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

      1. *-commutative 49.0%

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

   3. Simplified 49.0%

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

   5. Taylor expanded in a around 0 1.8%

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

      1. add-cube-cbrt 1.8%

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

      2. *-un-lft-identity 1.8%

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

      3. times-frac 1.8%

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

   7. Applied egg-rr 90.7%

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

      1. /-rgt-identity 90.7%

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

      2. associate-*r/ 90.7%

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

      3. unpow2 90.7%

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

      4. rem-3cbrt-lft 92.0%

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

   9. Simplified 92.0%

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

   10. Taylor expanded in a around inf 92.0%

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

       1. +-commutative 92.0%

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

       2. mul-1-neg 92.0%

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

       3. unsub-neg 92.0%

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

   12. Simplified 92.0%

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

   if -1.4999999999999999e87 < b < 4.0000000000000001e-127 or 1e-58 < b < 3.5000000000000001e-15

   1. Initial program 83.8%

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

   if 4.0000000000000001e-127 < b < 1e-58 or 2.25000000000000002e207 < b

   1. Initial program 15.3%

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

      1. *-commutative 15.3%

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

   3. Simplified 15.3%

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

   5. Taylor expanded in b around inf 85.1%

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

      1. associate-*r/ 85.1%

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

      2. neg-mul-1 85.1%

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

   7. Simplified 85.1%

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

   if 3.5000000000000001e-15 < b < 2.25000000000000002e207

   1. Initial program 19.6%

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

      1. *-commutative 19.6%

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

   3. Simplified 19.6%

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

   6. Applied egg-rr 24.8%

      \[\leadsto \color{blue}{\left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right) \cdot \frac{1}{a \cdot -2}} \]
   7. Step-by-step derivation

      1. un-div-inv 24.8%

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

      2. clear-num 24.8%

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

   8. Applied egg-rr 24.8%

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

   9. Taylor expanded in a around 0 0.0%

      \[\leadsto \frac{1}{\color{blue}{4 \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}} + a \cdot \left(2 \cdot \left(a \cdot \left(-0.25 \cdot \frac{c \cdot {\left(\sqrt{-4}\right)}^{2}}{{b}^{3}} + 0.125 \cdot \frac{c \cdot {\left(\sqrt{-4}\right)}^{2}}{{b}^{3}}\right)\right) + \frac{1}{b}\right)}} \]
   10. Step-by-step derivation

       1. +-commutative 0.0%

          \[\leadsto \frac{1}{\color{blue}{a \cdot \left(2 \cdot \left(a \cdot \left(-0.25 \cdot \frac{c \cdot {\left(\sqrt{-4}\right)}^{2}}{{b}^{3}} + 0.125 \cdot \frac{c \cdot {\left(\sqrt{-4}\right)}^{2}}{{b}^{3}}\right)\right) + \frac{1}{b}\right) + 4 \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}}} \]

       2. fma-define 0.0%

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, 2 \cdot \left(a \cdot \left(-0.25 \cdot \frac{c \cdot {\left(\sqrt{-4}\right)}^{2}}{{b}^{3}} + 0.125 \cdot \frac{c \cdot {\left(\sqrt{-4}\right)}^{2}}{{b}^{3}}\right)\right) + \frac{1}{b}, 4 \cdot \frac{b}{c \cdot {\left(\sqrt{-4}\right)}^{2}}\right)}} \]

   11. Simplified 85.2%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(2 \cdot a, \left(c \cdot \frac{-4}{{b}^{3}}\right) \cdot -0.125, \frac{1}{b}\right), \frac{-b}{c}\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 10^{-58}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 2.25 \cdot 10^{+207}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot 2, \left(c \cdot \frac{-4}{{b}^{3}}\right) \cdot -0.125, \frac{1}{b}\right), \frac{b}{-c}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.5e+87)
   (- (/ c b) (/ b a))
   (if (<= b 4e-127)
     (/ (- (sqrt (- (* b b) (* 4.0 (* c a)))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e+87) {
		tmp = (c / b) - (b / a);
	} else if (b <= 4e-127) {
		tmp = (sqrt(((b * b) - (4.0 * (c * a)))) - 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 <= (-1.5d+87)) then
        tmp = (c / b) - (b / a)
    else if (b <= 4d-127) then
        tmp = (sqrt(((b * b) - (4.0d0 * (c * a)))) - 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 <= -1.5e+87) {
		tmp = (c / b) - (b / a);
	} else if (b <= 4e-127) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.5e+87:
		tmp = (c / b) - (b / a)
	elif b <= 4e-127:
		tmp = (math.sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.5e+87)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 4e-127)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a)))) - 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 <= -1.5e+87)
		tmp = (c / b) - (b / a);
	elseif (b <= 4e-127)
		tmp = (sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.5e+87], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4e-127], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $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 < -1.4999999999999999e87

   1. Initial program 49.0%

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

      1. *-commutative 49.0%

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

   3. Simplified 49.0%

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

   5. Taylor expanded in a around 0 1.8%

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

      1. add-cube-cbrt 1.8%

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

      2. *-un-lft-identity 1.8%

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

      3. times-frac 1.8%

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

   7. Applied egg-rr 90.7%

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

      1. /-rgt-identity 90.7%

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

      2. associate-*r/ 90.7%

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

      3. unpow2 90.7%

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

      4. rem-3cbrt-lft 92.0%

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

   9. Simplified 92.0%

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

   10. Taylor expanded in a around inf 92.0%

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

       1. +-commutative 92.0%

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

       2. mul-1-neg 92.0%

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

       3. unsub-neg 92.0%

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

   12. Simplified 92.0%

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

   if -1.4999999999999999e87 < b < 4.0000000000000001e-127

   1. Initial program 84.5%

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

   if 4.0000000000000001e-127 < b

   1. Initial program 22.4%

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

      1. *-commutative 22.4%

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

   3. Simplified 22.4%

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

   5. Taylor expanded in b around inf 80.0%

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

      1. associate-*r/ 80.0%

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

      2. neg-mul-1 80.0%

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

   7. Simplified 80.0%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.3e-5)
   (- (/ c b) (/ b a))
   (if (<= b 4e-127)
     (* (/ -0.5 a) (- b (sqrt (* a (* c -4.0)))))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.3e-5) {
		tmp = (c / b) - (b / a);
	} else if (b <= 4e-127) {
		tmp = (-0.5 / a) * (b - sqrt((a * (c * -4.0))));
	} 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 <= (-3.3d-5)) then
        tmp = (c / b) - (b / a)
    else if (b <= 4d-127) then
        tmp = ((-0.5d0) / a) * (b - sqrt((a * (c * (-4.0d0)))))
    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 <= -3.3e-5) {
		tmp = (c / b) - (b / a);
	} else if (b <= 4e-127) {
		tmp = (-0.5 / a) * (b - Math.sqrt((a * (c * -4.0))));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.3e-5:
		tmp = (c / b) - (b / a)
	elif b <= 4e-127:
		tmp = (-0.5 / a) * (b - math.sqrt((a * (c * -4.0))))
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.3e-5)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 4e-127)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(Float64(a * Float64(c * -4.0)))));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.3e-5)
		tmp = (c / b) - (b / a);
	elseif (b <= 4e-127)
		tmp = (-0.5 / a) * (b - sqrt((a * (c * -4.0))));
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.3e-5], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4e-127], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.3000000000000003e-5

   1. Initial program 57.5%

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

      1. *-commutative 57.5%

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

   3. Simplified 57.5%

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

   5. Taylor expanded in a around 0 1.9%

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

      1. add-cube-cbrt 1.9%

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

      2. *-un-lft-identity 1.9%

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

      3. times-frac 1.9%

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

   7. Applied egg-rr 86.9%

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

      1. /-rgt-identity 86.9%

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

      2. associate-*r/ 86.9%

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

      3. unpow2 86.9%

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

      4. rem-3cbrt-lft 88.1%

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

   9. Simplified 88.1%

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

   10. Taylor expanded in a around inf 88.2%

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

       1. +-commutative 88.2%

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

       2. mul-1-neg 88.2%

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

       3. unsub-neg 88.2%

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

   12. Simplified 88.2%

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

   if -3.3000000000000003e-5 < b < 4.0000000000000001e-127

   1. Initial program 83.3%

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

      1. *-commutative 83.3%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in b around 0 76.0%

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

      1. *-commutative 76.0%

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

      2. associate-*r* 76.0%

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

   7. Simplified 76.0%

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

      1. add-sqr-sqrt 75.6%

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

      2. pow2 75.6%

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

      3. pow1/2 75.6%

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

      4. sqrt-pow1 75.7%

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

      5. metadata-eval 75.7%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{a \cdot 2} \]

   9. Applied egg-rr 75.7%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{0.25}\right)}^{2}}}{a \cdot 2} \]
   10. Step-by-step derivation

       1. frac-2neg 75.7%

          \[\leadsto \color{blue}{\frac{-\left(\left(-b\right) + {\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{0.25}\right)}^{2}\right)}{-a \cdot 2}} \]

       2. div-inv 75.7%

          \[\leadsto \color{blue}{\left(-\left(\left(-b\right) + {\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{0.25}\right)}^{2}\right)\right) \cdot \frac{1}{-a \cdot 2}} \]

       3. distribute-neg-in 75.7%

          \[\leadsto \color{blue}{\left(\left(-\left(-b\right)\right) + \left(-{\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{0.25}\right)}^{2}\right)\right)} \cdot \frac{1}{-a \cdot 2} \]

       4. add-sqr-sqrt 40.7%

          \[\leadsto \left(\left(-\color{blue}{\sqrt{-b} \cdot \sqrt{-b}}\right) + \left(-{\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{0.25}\right)}^{2}\right)\right) \cdot \frac{1}{-a \cdot 2} \]

       5. sqrt-unprod 75.2%

          \[\leadsto \left(\left(-\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}\right) + \left(-{\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{0.25}\right)}^{2}\right)\right) \cdot \frac{1}{-a \cdot 2} \]

       6. sqr-neg 75.2%

          \[\leadsto \left(\left(-\sqrt{\color{blue}{b \cdot b}}\right) + \left(-{\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{0.25}\right)}^{2}\right)\right) \cdot \frac{1}{-a \cdot 2} \]

       7. sqrt-unprod 34.8%

          \[\leadsto \left(\left(-\color{blue}{\sqrt{b} \cdot \sqrt{b}}\right) + \left(-{\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{0.25}\right)}^{2}\right)\right) \cdot \frac{1}{-a \cdot 2} \]

       8. add-sqr-sqrt 73.2%

          \[\leadsto \left(\left(-\color{blue}{b}\right) + \left(-{\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{0.25}\right)}^{2}\right)\right) \cdot \frac{1}{-a \cdot 2} \]

       9. sub-neg 73.2%

          \[\leadsto \color{blue}{\left(\left(-b\right) - {\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{0.25}\right)}^{2}\right)} \cdot \frac{1}{-a \cdot 2} \]

       10. add-sqr-sqrt 38.4%

           \[\leadsto \left(\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} - {\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{0.25}\right)}^{2}\right) \cdot \frac{1}{-a \cdot 2} \]

       11. sqrt-unprod 73.3%

           \[\leadsto \left(\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} - {\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{0.25}\right)}^{2}\right) \cdot \frac{1}{-a \cdot 2} \]

       12. sqr-neg 73.3%

           \[\leadsto \left(\sqrt{\color{blue}{b \cdot b}} - {\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{0.25}\right)}^{2}\right) \cdot \frac{1}{-a \cdot 2} \]

       13. sqrt-unprod 35.0%

           \[\leadsto \left(\color{blue}{\sqrt{b} \cdot \sqrt{b}} - {\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{0.25}\right)}^{2}\right) \cdot \frac{1}{-a \cdot 2} \]

       14. add-sqr-sqrt 75.7%

           \[\leadsto \left(\color{blue}{b} - {\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{0.25}\right)}^{2}\right) \cdot \frac{1}{-a \cdot 2} \]

       15. pow-pow 76.1%

           \[\leadsto \left(b - \color{blue}{{\left(a \cdot \left(c \cdot -4\right)\right)}^{\left(0.25 \cdot 2\right)}}\right) \cdot \frac{1}{-a \cdot 2} \]

       16. metadata-eval 76.1%

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

       17. pow1/2 76.1%

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

       18. distribute-rgt-neg-in 76.1%

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

       19. metadata-eval 76.1%

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

       20. inv-pow 76.1%

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

   11. Applied egg-rr 76.1%

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

       1. *-commutative 76.1%

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

       2. associate-*r/ 76.1%

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

       3. metadata-eval 76.1%

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

       4. *-commutative 76.1%

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

   13. Simplified 76.1%

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

   if 4.0000000000000001e-127 < b

   1. Initial program 22.4%

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

      1. *-commutative 22.4%

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

   3. Simplified 22.4%

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

   5. Taylor expanded in b around inf 80.0%

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

      1. associate-*r/ 80.0%

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

      2. neg-mul-1 80.0%

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

   7. Simplified 80.0%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-311) (- (/ c b) (/ b a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		tmp = (c / b) - (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 <= (-5d-311)) then
        tmp = (c / b) - (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 <= -5e-311) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-311:
		tmp = (c / b) - (b / a)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-311)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-311)
		tmp = (c / b) - (b / a);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-311], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.00000000000023e-311

   1. Initial program 66.7%

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

      1. *-commutative 66.7%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in a around 0 1.9%

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

      1. add-cube-cbrt 1.9%

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

      2. *-un-lft-identity 1.9%

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

      3. times-frac 1.9%

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

   7. Applied egg-rr 66.3%

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

      1. /-rgt-identity 66.3%

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

      2. associate-*r/ 66.4%

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

      3. unpow2 66.4%

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

      4. rem-3cbrt-lft 67.3%

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

   9. Simplified 67.3%

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

   10. Taylor expanded in a around inf 67.4%

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

       1. +-commutative 67.4%

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

       2. mul-1-neg 67.4%

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

       3. unsub-neg 67.4%

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

   12. Simplified 67.4%

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

   if -5.00000000000023e-311 < b

   1. Initial program 35.4%

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

      1. *-commutative 35.4%

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

   3. Simplified 35.4%

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

   5. Taylor expanded in b around inf 64.6%

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

      1. associate-*r/ 64.6%

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

      2. neg-mul-1 64.6%

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

   7. Simplified 64.6%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.2% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.35 \cdot 10^{-145}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 2.35e-145) (/ b (- a)) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.35e-145) {
		tmp = b / -a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.35d-145) then
        tmp = b / -a
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.35e-145) {
		tmp = b / -a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.35e-145:
		tmp = b / -a
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.35e-145)
		tmp = Float64(b / Float64(-a));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.35e-145)
		tmp = b / -a;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.35e-145], N[(b / (-a)), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < 2.3500000000000001e-145

   1. Initial program 68.4%

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

      1. *-commutative 68.4%

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

   3. Simplified 68.4%

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

   5. Taylor expanded in b around -inf 55.4%

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

      1. associate-*r/ 55.4%

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

      2. mul-1-neg 55.4%

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

   7. Simplified 55.4%

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

   if 2.3500000000000001e-145 < b

   1. Initial program 24.6%

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

      1. *-commutative 24.6%

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

   3. Simplified 24.6%

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

   6. Applied egg-rr 24.9%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{0.5}{a} + \left(-0.5 \cdot \frac{b}{a}\right)} \]
   7. Step-by-step derivation

      1. fma-define 19.5%

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

      2. distribute-lft-neg-in 19.5%

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

      3. metadata-eval 19.5%

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

   8. Simplified 19.5%

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

   9. Taylor expanded in a around 0 23.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot b + 0.5 \cdot b}{a}} \]
   10. Step-by-step derivation

       1. distribute-rgt-out 23.0%

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

       2. metadata-eval 23.0%

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

       3. mul0-rgt 23.0%

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

       4. div0 23.0%

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

   11. Simplified 23.0%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.35 \cdot 10^{-145}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.1% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{-293}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.4e-293) (/ b (- a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.4e-293) {
		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 <= 1.4d-293) 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 <= 1.4e-293) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.4e-293:
		tmp = b / -a
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.4e-293)
		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 <= 1.4e-293)
		tmp = b / -a;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.4e-293], N[(b / (-a)), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.40000000000000013e-293

   1. Initial program 67.0%

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

      1. *-commutative 67.0%

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

   3. Simplified 67.0%

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

   5. Taylor expanded in b around -inf 66.7%

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

      1. associate-*r/ 66.7%

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

      2. mul-1-neg 66.7%

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

   7. Simplified 66.7%

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

   if 1.40000000000000013e-293 < b

   1. Initial program 34.9%

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

      1. *-commutative 34.9%

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

   3. Simplified 34.9%

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

   5. Taylor expanded in b around inf 65.1%

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

      1. associate-*r/ 65.1%

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

      2. neg-mul-1 65.1%

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

   7. Simplified 65.1%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{-293}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 11.3% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 50.5%

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

   1. *-commutative 50.5%

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

3. Simplified 50.5%

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

6. Applied egg-rr 48.3%

   \[\leadsto \color{blue}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{0.5}{a} + \left(-0.5 \cdot \frac{b}{a}\right)} \]
7. Step-by-step derivation

   1. fma-define 46.2%

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

   2. distribute-lft-neg-in 46.2%

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

   3. metadata-eval 46.2%

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

8. Simplified 46.2%

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

9. Taylor expanded in a around 0 11.1%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot b + 0.5 \cdot b}{a}} \]
10. Step-by-step derivation

    1. distribute-rgt-out 11.1%

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

    2. metadata-eval 11.1%

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

    3. mul0-rgt 11.1%

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

    4. div0 11.1%

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

11. Simplified 11.1%

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

12. Final simplification 11.1%

    \[\leadsto 0 \]
13. Add Preprocessing

Developer target: 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 2024095 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (if (< b 0.0) (/ (- (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot (/ b 2.0) (* (sqrt (fabs a)) (sqrt (fabs c))))) (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot (/ b 2.0) (* (sqrt (fabs a)) (sqrt (fabs c))))))))

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