quadm (p42, negative)

Percentage Accurate: 52.1% → 85.6%

Time: 22.2s

Alternatives: 13

Speedup: 11.6×

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: 52.1% 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: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-79}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 3.35 \cdot 10^{+137}:\\ \;\;\;\;\frac{b + {\left(\frac{1}{b \cdot b + c \cdot \left(a \cdot -4\right)}\right)}^{-0.5}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.6e-79)
   (- 0.0 (/ c b))
   (if (<= b 3.35e+137)
     (/ (+ b (pow (/ 1.0 (+ (* b b) (* c (* a -4.0)))) -0.5)) (* a -2.0))
     (- 0.0 (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.6e-79) {
		tmp = 0.0 - (c / b);
	} else if (b <= 3.35e+137) {
		tmp = (b + pow((1.0 / ((b * b) + (c * (a * -4.0)))), -0.5)) / (a * -2.0);
	} else {
		tmp = 0.0 - (b / a);
	}
	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 <= (-2.6d-79)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 3.35d+137) then
        tmp = (b + ((1.0d0 / ((b * b) + (c * (a * (-4.0d0))))) ** (-0.5d0))) / (a * (-2.0d0))
    else
        tmp = 0.0d0 - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.6e-79) {
		tmp = 0.0 - (c / b);
	} else if (b <= 3.35e+137) {
		tmp = (b + Math.pow((1.0 / ((b * b) + (c * (a * -4.0)))), -0.5)) / (a * -2.0);
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.6e-79:
		tmp = 0.0 - (c / b)
	elif b <= 3.35e+137:
		tmp = (b + math.pow((1.0 / ((b * b) + (c * (a * -4.0)))), -0.5)) / (a * -2.0)
	else:
		tmp = 0.0 - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.6e-79)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 3.35e+137)
		tmp = Float64(Float64(b + (Float64(1.0 / Float64(Float64(b * b) + Float64(c * Float64(a * -4.0)))) ^ -0.5)) / Float64(a * -2.0));
	else
		tmp = Float64(0.0 - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.6e-79)
		tmp = 0.0 - (c / b);
	elseif (b <= 3.35e+137)
		tmp = (b + ((1.0 / ((b * b) + (c * (a * -4.0)))) ^ -0.5)) / (a * -2.0);
	else
		tmp = 0.0 - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.6e-79], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.35e+137], N[(N[(b + N[Power[N[(1.0 / N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.59999999999999994e-79

   1. Initial program 13.2%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 13.2%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 86.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   7. Simplified 86.3%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

   if -2.59999999999999994e-79 < b < 3.3499999999999999e137

   1. Initial program 84.8%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 84.8%

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

      1. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left({\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)}^{\frac{1}{2}}\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      2. flip3-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left({\left(\frac{{\left(b \cdot b\right)}^{3} + {\left(a \cdot \left(c \cdot -4\right)\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(a \cdot \left(c \cdot -4\right)\right) \cdot \left(a \cdot \left(c \cdot -4\right)\right) - \left(b \cdot b\right) \cdot \left(a \cdot \left(c \cdot -4\right)\right)\right)}\right)}^{\frac{1}{2}}\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left({\left(\frac{1}{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(a \cdot \left(c \cdot -4\right)\right) \cdot \left(a \cdot \left(c \cdot -4\right)\right) - \left(b \cdot b\right) \cdot \left(a \cdot \left(c \cdot -4\right)\right)\right)}{{\left(b \cdot b\right)}^{3} + {\left(a \cdot \left(c \cdot -4\right)\right)}^{3}}}\right)}^{\frac{1}{2}}\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      4. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left({\left({\left(\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(a \cdot \left(c \cdot -4\right)\right) \cdot \left(a \cdot \left(c \cdot -4\right)\right) - \left(b \cdot b\right) \cdot \left(a \cdot \left(c \cdot -4\right)\right)\right)}{{\left(b \cdot b\right)}^{3} + {\left(a \cdot \left(c \cdot -4\right)\right)}^{3}}\right)}^{-1}\right)}^{\frac{1}{2}}\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      5. pow-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left({\left(\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(a \cdot \left(c \cdot -4\right)\right) \cdot \left(a \cdot \left(c \cdot -4\right)\right) - \left(b \cdot b\right) \cdot \left(a \cdot \left(c \cdot -4\right)\right)\right)}{{\left(b \cdot b\right)}^{3} + {\left(a \cdot \left(c \cdot -4\right)\right)}^{3}}\right)}^{\left(-1 \cdot \frac{1}{2}\right)}\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left({\left(\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(a \cdot \left(c \cdot -4\right)\right) \cdot \left(a \cdot \left(c \cdot -4\right)\right) - \left(b \cdot b\right) \cdot \left(a \cdot \left(c \cdot -4\right)\right)\right)}{{\left(b \cdot b\right)}^{3} + {\left(a \cdot \left(c \cdot -4\right)\right)}^{3}}\right)}^{\frac{-1}{2}}\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left({\left(\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(a \cdot \left(c \cdot -4\right)\right) \cdot \left(a \cdot \left(c \cdot -4\right)\right) - \left(b \cdot b\right) \cdot \left(a \cdot \left(c \cdot -4\right)\right)\right)}{{\left(b \cdot b\right)}^{3} + {\left(a \cdot \left(c \cdot -4\right)\right)}^{3}}\right)}^{\left(\frac{1}{-2}\right)}\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{pow.f64}\left(\left(\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(a \cdot \left(c \cdot -4\right)\right) \cdot \left(a \cdot \left(c \cdot -4\right)\right) - \left(b \cdot b\right) \cdot \left(a \cdot \left(c \cdot -4\right)\right)\right)}{{\left(b \cdot b\right)}^{3} + {\left(a \cdot \left(c \cdot -4\right)\right)}^{3}}\right), \left(\frac{1}{-2}\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

   6. Applied egg-rr 84.8%

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

   if 3.3499999999999999e137 < b

   1. Initial program 48.8%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 48.8%

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

   5. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right) \]

      4. /-lowering-/.f64 98.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   7. Simplified 98.3%

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{b}{a}\right)\right) \]

      3. /-lowering-/.f64 98.3%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(b, a\right)\right) \]

   9. Applied egg-rr 98.3%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-79}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 3.35 \cdot 10^{+137}:\\ \;\;\;\;\frac{b + {\left(\frac{1}{b \cdot b + c \cdot \left(a \cdot -4\right)}\right)}^{-0.5}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-78}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 10^{+137}:\\ \;\;\;\;-0.5 \cdot \left(\frac{b}{a} + \frac{\sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-78)
   (- 0.0 (/ c b))
   (if (<= b 1e+137)
     (* -0.5 (+ (/ b a) (/ (sqrt (+ (* b b) (* -4.0 (* c a)))) a)))
     (- 0.0 (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-78) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1e+137) {
		tmp = -0.5 * ((b / a) + (sqrt(((b * b) + (-4.0 * (c * a)))) / a));
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-78)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 1d+137) then
        tmp = (-0.5d0) * ((b / a) + (sqrt(((b * b) + ((-4.0d0) * (c * a)))) / a))
    else
        tmp = 0.0d0 - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-78) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1e+137) {
		tmp = -0.5 * ((b / a) + (Math.sqrt(((b * b) + (-4.0 * (c * a)))) / a));
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-78:
		tmp = 0.0 - (c / b)
	elif b <= 1e+137:
		tmp = -0.5 * ((b / a) + (math.sqrt(((b * b) + (-4.0 * (c * a)))) / a))
	else:
		tmp = 0.0 - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-78)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 1e+137)
		tmp = Float64(-0.5 * Float64(Float64(b / a) + Float64(sqrt(Float64(Float64(b * b) + Float64(-4.0 * Float64(c * a)))) / a)));
	else
		tmp = Float64(0.0 - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-78)
		tmp = 0.0 - (c / b);
	elseif (b <= 1e+137)
		tmp = -0.5 * ((b / a) + (sqrt(((b * b) + (-4.0 * (c * a)))) / a));
	else
		tmp = 0.0 - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-78], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e+137], N[(-0.5 * N[(N[(b / a), $MachinePrecision] + N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2e-78

   1. Initial program 13.2%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 13.2%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 86.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   7. Simplified 86.3%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

   if -2e-78 < b < 1e137

   1. Initial program 84.8%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 84.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a \cdot -2}\right), \color{blue}{\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{-2 \cdot a}\right), \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{-2}}{a}\right), \left(\color{blue}{b} + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{-2}\right), a\right), \left(\color{blue}{b} + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(c \cdot a\right) \cdot -4\right)\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(c \cdot \left(a \cdot -4\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(a \cdot -4\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 84.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 84.5%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2}}{a} \cdot \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right), \color{blue}{\left(b \cdot \frac{\frac{-1}{2}}{a}\right)}\right) \]

   8. Applied egg-rr 84.8%

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

      1. associate-/r/ N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-out N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{\sqrt{b \cdot b + -4 \cdot \left(a \cdot c\right)}}{a} + \frac{b}{a}\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(\left(\frac{\sqrt{b \cdot b + -4 \cdot \left(a \cdot c\right)}}{a}\right), \color{blue}{\left(\frac{b}{a}\right)}\right)\right) \]

   10. Applied egg-rr 84.8%

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

   if 1e137 < b

   1. Initial program 48.8%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 48.8%

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

   5. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right) \]

      4. /-lowering-/.f64 98.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   7. Simplified 98.3%

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{b}{a}\right)\right) \]

      3. /-lowering-/.f64 98.3%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(b, a\right)\right) \]

   9. Applied egg-rr 98.3%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-78}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 10^{+137}:\\ \;\;\;\;-0.5 \cdot \left(\frac{b}{a} + \frac{\sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-82}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+137}:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-82)
   (- 0.0 (/ c b))
   (if (<= b 5e+137)
     (/ (+ b (sqrt (+ (* b b) (* a (* c -4.0))))) (* a -2.0))
     (- 0.0 (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-82) {
		tmp = 0.0 - (c / b);
	} else if (b <= 5e+137) {
		tmp = (b + sqrt(((b * b) + (a * (c * -4.0))))) / (a * -2.0);
	} else {
		tmp = 0.0 - (b / a);
	}
	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 <= (-4d-82)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 5d+137) then
        tmp = (b + sqrt(((b * b) + (a * (c * (-4.0d0)))))) / (a * (-2.0d0))
    else
        tmp = 0.0d0 - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-82) {
		tmp = 0.0 - (c / b);
	} else if (b <= 5e+137) {
		tmp = (b + Math.sqrt(((b * b) + (a * (c * -4.0))))) / (a * -2.0);
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-82:
		tmp = 0.0 - (c / b)
	elif b <= 5e+137:
		tmp = (b + math.sqrt(((b * b) + (a * (c * -4.0))))) / (a * -2.0)
	else:
		tmp = 0.0 - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-82)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 5e+137)
		tmp = Float64(Float64(b + sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0))))) / Float64(a * -2.0));
	else
		tmp = Float64(0.0 - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e-82)
		tmp = 0.0 - (c / b);
	elseif (b <= 5e+137)
		tmp = (b + sqrt(((b * b) + (a * (c * -4.0))))) / (a * -2.0);
	else
		tmp = 0.0 - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-82], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e+137], N[(N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4e-82

   1. Initial program 13.2%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 13.2%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 86.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   7. Simplified 86.3%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

   if -4e-82 < b < 5.0000000000000002e137

   1. Initial program 84.8%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 84.8%

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

   if 5.0000000000000002e137 < b

   1. Initial program 48.8%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 48.8%

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

   5. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right) \]

      4. /-lowering-/.f64 98.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   7. Simplified 98.3%

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{b}{a}\right)\right) \]

      3. /-lowering-/.f64 98.3%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(b, a\right)\right) \]

   9. Applied egg-rr 98.3%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-82}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+137}:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-80}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{+137}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.35e-80)
   (- 0.0 (/ c b))
   (if (<= b 2.6e+137)
     (* (/ -0.5 a) (+ b (sqrt (+ (* b b) (* c (* a -4.0))))))
     (- 0.0 (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.35e-80) {
		tmp = 0.0 - (c / b);
	} else if (b <= 2.6e+137) {
		tmp = (-0.5 / a) * (b + sqrt(((b * b) + (c * (a * -4.0)))));
	} else {
		tmp = 0.0 - (b / a);
	}
	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 <= (-1.35d-80)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 2.6d+137) then
        tmp = ((-0.5d0) / a) * (b + sqrt(((b * b) + (c * (a * (-4.0d0))))))
    else
        tmp = 0.0d0 - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.35e-80) {
		tmp = 0.0 - (c / b);
	} else if (b <= 2.6e+137) {
		tmp = (-0.5 / a) * (b + Math.sqrt(((b * b) + (c * (a * -4.0)))));
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.35e-80:
		tmp = 0.0 - (c / b)
	elif b <= 2.6e+137:
		tmp = (-0.5 / a) * (b + math.sqrt(((b * b) + (c * (a * -4.0)))))
	else:
		tmp = 0.0 - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.35e-80)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 2.6e+137)
		tmp = Float64(Float64(-0.5 / a) * Float64(b + sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0))))));
	else
		tmp = Float64(0.0 - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.35e-80)
		tmp = 0.0 - (c / b);
	elseif (b <= 2.6e+137)
		tmp = (-0.5 / a) * (b + sqrt(((b * b) + (c * (a * -4.0)))));
	else
		tmp = 0.0 - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.35e-80], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.6e+137], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.3500000000000001e-80

   1. Initial program 13.2%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 13.2%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 86.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   7. Simplified 86.3%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

   if -1.3500000000000001e-80 < b < 2.5999999999999999e137

   1. Initial program 84.8%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 84.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a \cdot -2}\right), \color{blue}{\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{-2 \cdot a}\right), \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{-2}}{a}\right), \left(\color{blue}{b} + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{-2}\right), a\right), \left(\color{blue}{b} + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(c \cdot a\right) \cdot -4\right)\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(c \cdot \left(a \cdot -4\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(a \cdot -4\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 84.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 84.5%

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

   if 2.5999999999999999e137 < b

   1. Initial program 48.8%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 48.8%

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

   5. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right) \]

      4. /-lowering-/.f64 98.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   7. Simplified 98.3%

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{b}{a}\right)\right) \]

      3. /-lowering-/.f64 98.3%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(b, a\right)\right) \]

   9. Applied egg-rr 98.3%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-80}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{+137}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-82}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-102}:\\ \;\;\;\;\frac{b + {\left(\frac{-0.25}{c \cdot a}\right)}^{-0.5}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} + \frac{b \cdot -0.5}{a}\right) + -0.5 \cdot \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.3e-82)
   (- 0.0 (/ c b))
   (if (<= b 2.5e-102)
     (/ (+ b (pow (/ -0.25 (* c a)) -0.5)) (* a -2.0))
     (+ (+ (/ c b) (/ (* b -0.5) a)) (* -0.5 (/ b a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e-82) {
		tmp = 0.0 - (c / b);
	} else if (b <= 2.5e-102) {
		tmp = (b + pow((-0.25 / (c * a)), -0.5)) / (a * -2.0);
	} else {
		tmp = ((c / b) + ((b * -0.5) / a)) + (-0.5 * (b / a));
	}
	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 <= (-2.3d-82)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 2.5d-102) then
        tmp = (b + (((-0.25d0) / (c * a)) ** (-0.5d0))) / (a * (-2.0d0))
    else
        tmp = ((c / b) + ((b * (-0.5d0)) / a)) + ((-0.5d0) * (b / a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e-82) {
		tmp = 0.0 - (c / b);
	} else if (b <= 2.5e-102) {
		tmp = (b + Math.pow((-0.25 / (c * a)), -0.5)) / (a * -2.0);
	} else {
		tmp = ((c / b) + ((b * -0.5) / a)) + (-0.5 * (b / a));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.3e-82:
		tmp = 0.0 - (c / b)
	elif b <= 2.5e-102:
		tmp = (b + math.pow((-0.25 / (c * a)), -0.5)) / (a * -2.0)
	else:
		tmp = ((c / b) + ((b * -0.5) / a)) + (-0.5 * (b / a))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.3e-82)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 2.5e-102)
		tmp = Float64(Float64(b + (Float64(-0.25 / Float64(c * a)) ^ -0.5)) / Float64(a * -2.0));
	else
		tmp = Float64(Float64(Float64(c / b) + Float64(Float64(b * -0.5) / a)) + Float64(-0.5 * Float64(b / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.3e-82)
		tmp = 0.0 - (c / b);
	elseif (b <= 2.5e-102)
		tmp = (b + ((-0.25 / (c * a)) ^ -0.5)) / (a * -2.0);
	else
		tmp = ((c / b) + ((b * -0.5) / a)) + (-0.5 * (b / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.3e-82], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.5e-102], N[(N[(b + N[Power[N[(-0.25 / N[(c * a), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c / b), $MachinePrecision] + N[(N[(b * -0.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.29999999999999997e-82

   1. Initial program 13.2%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 13.2%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 86.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   7. Simplified 86.3%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

   if -2.29999999999999997e-82 < b < 2.50000000000000013e-102

   1. Initial program 76.7%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 76.7%

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

      1. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left({\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)}^{\frac{1}{2}}\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      2. flip3-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left({\left(\frac{{\left(b \cdot b\right)}^{3} + {\left(a \cdot \left(c \cdot -4\right)\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(a \cdot \left(c \cdot -4\right)\right) \cdot \left(a \cdot \left(c \cdot -4\right)\right) - \left(b \cdot b\right) \cdot \left(a \cdot \left(c \cdot -4\right)\right)\right)}\right)}^{\frac{1}{2}}\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left({\left(\frac{1}{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(a \cdot \left(c \cdot -4\right)\right) \cdot \left(a \cdot \left(c \cdot -4\right)\right) - \left(b \cdot b\right) \cdot \left(a \cdot \left(c \cdot -4\right)\right)\right)}{{\left(b \cdot b\right)}^{3} + {\left(a \cdot \left(c \cdot -4\right)\right)}^{3}}}\right)}^{\frac{1}{2}}\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      4. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left({\left({\left(\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(a \cdot \left(c \cdot -4\right)\right) \cdot \left(a \cdot \left(c \cdot -4\right)\right) - \left(b \cdot b\right) \cdot \left(a \cdot \left(c \cdot -4\right)\right)\right)}{{\left(b \cdot b\right)}^{3} + {\left(a \cdot \left(c \cdot -4\right)\right)}^{3}}\right)}^{-1}\right)}^{\frac{1}{2}}\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      5. pow-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left({\left(\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(a \cdot \left(c \cdot -4\right)\right) \cdot \left(a \cdot \left(c \cdot -4\right)\right) - \left(b \cdot b\right) \cdot \left(a \cdot \left(c \cdot -4\right)\right)\right)}{{\left(b \cdot b\right)}^{3} + {\left(a \cdot \left(c \cdot -4\right)\right)}^{3}}\right)}^{\left(-1 \cdot \frac{1}{2}\right)}\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left({\left(\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(a \cdot \left(c \cdot -4\right)\right) \cdot \left(a \cdot \left(c \cdot -4\right)\right) - \left(b \cdot b\right) \cdot \left(a \cdot \left(c \cdot -4\right)\right)\right)}{{\left(b \cdot b\right)}^{3} + {\left(a \cdot \left(c \cdot -4\right)\right)}^{3}}\right)}^{\frac{-1}{2}}\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left({\left(\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(a \cdot \left(c \cdot -4\right)\right) \cdot \left(a \cdot \left(c \cdot -4\right)\right) - \left(b \cdot b\right) \cdot \left(a \cdot \left(c \cdot -4\right)\right)\right)}{{\left(b \cdot b\right)}^{3} + {\left(a \cdot \left(c \cdot -4\right)\right)}^{3}}\right)}^{\left(\frac{1}{-2}\right)}\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{pow.f64}\left(\left(\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(a \cdot \left(c \cdot -4\right)\right) \cdot \left(a \cdot \left(c \cdot -4\right)\right) - \left(b \cdot b\right) \cdot \left(a \cdot \left(c \cdot -4\right)\right)\right)}{{\left(b \cdot b\right)}^{3} + {\left(a \cdot \left(c \cdot -4\right)\right)}^{3}}\right), \left(\frac{1}{-2}\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

   6. Applied egg-rr 76.7%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{pow.f64}\left(\color{blue}{\left(\frac{\frac{-1}{4}}{a \cdot c}\right)}, \frac{-1}{2}\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\frac{-1}{4}, \left(a \cdot c\right)\right), \frac{-1}{2}\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\frac{-1}{4}, \left(c \cdot a\right)\right), \frac{-1}{2}\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      3. *-lowering-*.f64 75.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(c, a\right)\right), \frac{-1}{2}\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

   9. Simplified 75.7%

      \[\leadsto \frac{b + {\color{blue}{\left(\frac{-0.25}{c \cdot a}\right)}}^{-0.5}}{a \cdot -2} \]

   if 2.50000000000000013e-102 < b

   1. Initial program 68.2%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 68.2%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a \cdot -2}\right), \color{blue}{\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{-2 \cdot a}\right), \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{-2}}{a}\right), \left(\color{blue}{b} + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{-2}\right), a\right), \left(\color{blue}{b} + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(c \cdot a\right) \cdot -4\right)\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(c \cdot \left(a \cdot -4\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(a \cdot -4\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 68.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 68.0%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2}}{a} \cdot \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right), \color{blue}{\left(b \cdot \frac{\frac{-1}{2}}{a}\right)}\right) \]

   8. Applied egg-rr 68.2%

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

   9. Taylor expanded in c around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{b}{a} + \frac{c}{b}\right)}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(b, a\right)\right)\right) \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{b}{a}\right), \left(\frac{c}{b}\right)\right), \mathsf{*.f64}\left(\color{blue}{\frac{-1}{2}}, \mathsf{/.f64}\left(b, a\right)\right)\right) \]

       2. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{-1}{2} \cdot b}{a}\right), \left(\frac{c}{b}\right)\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(b, a\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot b\right), a\right), \left(\frac{c}{b}\right)\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(b, a\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, b\right), a\right), \left(\frac{c}{b}\right)\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(b, a\right)\right)\right) \]

       5. /-lowering-/.f64 94.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, b\right), a\right), \mathsf{/.f64}\left(c, b\right)\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(b, a\right)\right)\right) \]

   11. Simplified 94.8%

       \[\leadsto \color{blue}{\left(\frac{-0.5 \cdot b}{a} + \frac{c}{b}\right)} + -0.5 \cdot \frac{b}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-82}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-102}:\\ \;\;\;\;\frac{b + {\left(\frac{-0.25}{c \cdot a}\right)}^{-0.5}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} + \frac{b \cdot -0.5}{a}\right) + -0.5 \cdot \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{-83}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-102}:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} + \frac{b \cdot -0.5}{a}\right) + -0.5 \cdot \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7e-83)
   (- 0.0 (/ c b))
   (if (<= b 1.8e-102)
     (/ (+ b (sqrt (* c (* a -4.0)))) (* a -2.0))
     (+ (+ (/ c b) (/ (* b -0.5) a)) (* -0.5 (/ b a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7e-83) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1.8e-102) {
		tmp = (b + sqrt((c * (a * -4.0)))) / (a * -2.0);
	} else {
		tmp = ((c / b) + ((b * -0.5) / a)) + (-0.5 * (b / a));
	}
	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 <= (-7d-83)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 1.8d-102) then
        tmp = (b + sqrt((c * (a * (-4.0d0))))) / (a * (-2.0d0))
    else
        tmp = ((c / b) + ((b * (-0.5d0)) / a)) + ((-0.5d0) * (b / a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7e-83) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1.8e-102) {
		tmp = (b + Math.sqrt((c * (a * -4.0)))) / (a * -2.0);
	} else {
		tmp = ((c / b) + ((b * -0.5) / a)) + (-0.5 * (b / a));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7e-83:
		tmp = 0.0 - (c / b)
	elif b <= 1.8e-102:
		tmp = (b + math.sqrt((c * (a * -4.0)))) / (a * -2.0)
	else:
		tmp = ((c / b) + ((b * -0.5) / a)) + (-0.5 * (b / a))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7e-83)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 1.8e-102)
		tmp = Float64(Float64(b + sqrt(Float64(c * Float64(a * -4.0)))) / Float64(a * -2.0));
	else
		tmp = Float64(Float64(Float64(c / b) + Float64(Float64(b * -0.5) / a)) + Float64(-0.5 * Float64(b / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7e-83)
		tmp = 0.0 - (c / b);
	elseif (b <= 1.8e-102)
		tmp = (b + sqrt((c * (a * -4.0)))) / (a * -2.0);
	else
		tmp = ((c / b) + ((b * -0.5) / a)) + (-0.5 * (b / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7e-83], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.8e-102], N[(N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c / b), $MachinePrecision] + N[(N[(b * -0.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.00000000000000061e-83

   1. Initial program 13.2%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 13.2%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 86.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   7. Simplified 86.3%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

   if -7.00000000000000061e-83 < b < 1.8e-102

   1. Initial program 76.7%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 76.7%

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

   5. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right)}\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\left(-4 \cdot a\right) \cdot c\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(c \cdot \left(-4 \cdot a\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(-4 \cdot a\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(a \cdot -4\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      5. *-lowering-*.f64 75.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

   7. Simplified 75.6%

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

   if 1.8e-102 < b

   1. Initial program 68.2%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 68.2%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a \cdot -2}\right), \color{blue}{\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{-2 \cdot a}\right), \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{-2}}{a}\right), \left(\color{blue}{b} + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{-2}\right), a\right), \left(\color{blue}{b} + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(c \cdot a\right) \cdot -4\right)\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(c \cdot \left(a \cdot -4\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(a \cdot -4\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 68.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 68.0%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2}}{a} \cdot \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right), \color{blue}{\left(b \cdot \frac{\frac{-1}{2}}{a}\right)}\right) \]

   8. Applied egg-rr 68.2%

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

   9. Taylor expanded in c around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{b}{a} + \frac{c}{b}\right)}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(b, a\right)\right)\right) \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{b}{a}\right), \left(\frac{c}{b}\right)\right), \mathsf{*.f64}\left(\color{blue}{\frac{-1}{2}}, \mathsf{/.f64}\left(b, a\right)\right)\right) \]

       2. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{-1}{2} \cdot b}{a}\right), \left(\frac{c}{b}\right)\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(b, a\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot b\right), a\right), \left(\frac{c}{b}\right)\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(b, a\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, b\right), a\right), \left(\frac{c}{b}\right)\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(b, a\right)\right)\right) \]

       5. /-lowering-/.f64 94.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, b\right), a\right), \mathsf{/.f64}\left(c, b\right)\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(b, a\right)\right)\right) \]

   11. Simplified 94.8%

       \[\leadsto \color{blue}{\left(\frac{-0.5 \cdot b}{a} + \frac{c}{b}\right)} + -0.5 \cdot \frac{b}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{-83}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-102}:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} + \frac{b \cdot -0.5}{a}\right) + -0.5 \cdot \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{-77}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-103}:\\ \;\;\;\;\frac{-0.5}{\frac{a}{b + \sqrt{c \cdot \left(a \cdot -4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} + \frac{b \cdot -0.5}{a}\right) + -0.5 \cdot \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.5e-77)
   (- 0.0 (/ c b))
   (if (<= b 1.35e-103)
     (/ -0.5 (/ a (+ b (sqrt (* c (* a -4.0))))))
     (+ (+ (/ c b) (/ (* b -0.5) a)) (* -0.5 (/ b a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.5e-77) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1.35e-103) {
		tmp = -0.5 / (a / (b + sqrt((c * (a * -4.0)))));
	} else {
		tmp = ((c / b) + ((b * -0.5) / a)) + (-0.5 * (b / a));
	}
	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 <= (-3.5d-77)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 1.35d-103) then
        tmp = (-0.5d0) / (a / (b + sqrt((c * (a * (-4.0d0))))))
    else
        tmp = ((c / b) + ((b * (-0.5d0)) / a)) + ((-0.5d0) * (b / a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.5e-77) {
		tmp = 0.0 - (c / b);
	} else if (b <= 1.35e-103) {
		tmp = -0.5 / (a / (b + Math.sqrt((c * (a * -4.0)))));
	} else {
		tmp = ((c / b) + ((b * -0.5) / a)) + (-0.5 * (b / a));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.5e-77:
		tmp = 0.0 - (c / b)
	elif b <= 1.35e-103:
		tmp = -0.5 / (a / (b + math.sqrt((c * (a * -4.0)))))
	else:
		tmp = ((c / b) + ((b * -0.5) / a)) + (-0.5 * (b / a))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.5e-77)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 1.35e-103)
		tmp = Float64(-0.5 / Float64(a / Float64(b + sqrt(Float64(c * Float64(a * -4.0))))));
	else
		tmp = Float64(Float64(Float64(c / b) + Float64(Float64(b * -0.5) / a)) + Float64(-0.5 * Float64(b / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.5e-77)
		tmp = 0.0 - (c / b);
	elseif (b <= 1.35e-103)
		tmp = -0.5 / (a / (b + sqrt((c * (a * -4.0)))));
	else
		tmp = ((c / b) + ((b * -0.5) / a)) + (-0.5 * (b / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.5e-77], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e-103], N[(-0.5 / N[(a / N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c / b), $MachinePrecision] + N[(N[(b * -0.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.50000000000000013e-77

   1. Initial program 13.2%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 13.2%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 86.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   7. Simplified 86.3%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

   if -3.50000000000000013e-77 < b < 1.35000000000000005e-103

   1. Initial program 76.7%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 76.7%

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

      1. associate-/r* N/A

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

      2. clear-num N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{-2}\right), \color{blue}{\left(\frac{a}{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}\right)}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \left(\frac{\color{blue}{a}}{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \color{blue}{\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(c \cdot a\right) \cdot -4\right)\right)\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(c \cdot \left(a \cdot -4\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(a \cdot -4\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 76.5%

          \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 76.5%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right)}\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\left(c \cdot a\right) \cdot -4\right)\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(c \cdot \left(a \cdot -4\right)\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(c \cdot \left(-4 \cdot a\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(-4 \cdot a\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \left(a \cdot -4\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 75.5%

         \[\leadsto \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right)\right) \]

   9. Simplified 75.5%

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

   if 1.35000000000000005e-103 < b

   1. Initial program 68.2%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 68.2%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a \cdot -2}\right), \color{blue}{\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{-2 \cdot a}\right), \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{-2}}{a}\right), \left(\color{blue}{b} + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{-2}\right), a\right), \left(\color{blue}{b} + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(c \cdot a\right) \cdot -4\right)\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(c \cdot \left(a \cdot -4\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(a \cdot -4\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 68.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 68.0%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2}}{a} \cdot \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right), \color{blue}{\left(b \cdot \frac{\frac{-1}{2}}{a}\right)}\right) \]

   8. Applied egg-rr 68.2%

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

   9. Taylor expanded in c around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{b}{a} + \frac{c}{b}\right)}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(b, a\right)\right)\right) \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{b}{a}\right), \left(\frac{c}{b}\right)\right), \mathsf{*.f64}\left(\color{blue}{\frac{-1}{2}}, \mathsf{/.f64}\left(b, a\right)\right)\right) \]

       2. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{-1}{2} \cdot b}{a}\right), \left(\frac{c}{b}\right)\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(b, a\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot b\right), a\right), \left(\frac{c}{b}\right)\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(b, a\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, b\right), a\right), \left(\frac{c}{b}\right)\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(b, a\right)\right)\right) \]

       5. /-lowering-/.f64 94.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, b\right), a\right), \mathsf{/.f64}\left(c, b\right)\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(b, a\right)\right)\right) \]

   11. Simplified 94.8%

       \[\leadsto \color{blue}{\left(\frac{-0.5 \cdot b}{a} + \frac{c}{b}\right)} + -0.5 \cdot \frac{b}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{-77}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-103}:\\ \;\;\;\;\frac{-0.5}{\frac{a}{b + \sqrt{c \cdot \left(a \cdot -4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} + \frac{b \cdot -0.5}{a}\right) + -0.5 \cdot \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{-80}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{-113}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} + \frac{b \cdot -0.5}{a}\right) + -0.5 \cdot \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.1e-80)
   (- 0.0 (/ c b))
   (if (<= b 3.7e-113)
     (* (/ -0.5 a) (+ b (sqrt (* c (* a -4.0)))))
     (+ (+ (/ c b) (/ (* b -0.5) a)) (* -0.5 (/ b a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.1e-80) {
		tmp = 0.0 - (c / b);
	} else if (b <= 3.7e-113) {
		tmp = (-0.5 / a) * (b + sqrt((c * (a * -4.0))));
	} else {
		tmp = ((c / b) + ((b * -0.5) / a)) + (-0.5 * (b / a));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.1d-80)) then
        tmp = 0.0d0 - (c / b)
    else if (b <= 3.7d-113) then
        tmp = ((-0.5d0) / a) * (b + sqrt((c * (a * (-4.0d0)))))
    else
        tmp = ((c / b) + ((b * (-0.5d0)) / a)) + ((-0.5d0) * (b / a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.1e-80) {
		tmp = 0.0 - (c / b);
	} else if (b <= 3.7e-113) {
		tmp = (-0.5 / a) * (b + Math.sqrt((c * (a * -4.0))));
	} else {
		tmp = ((c / b) + ((b * -0.5) / a)) + (-0.5 * (b / a));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.1e-80:
		tmp = 0.0 - (c / b)
	elif b <= 3.7e-113:
		tmp = (-0.5 / a) * (b + math.sqrt((c * (a * -4.0))))
	else:
		tmp = ((c / b) + ((b * -0.5) / a)) + (-0.5 * (b / a))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.1e-80)
		tmp = Float64(0.0 - Float64(c / b));
	elseif (b <= 3.7e-113)
		tmp = Float64(Float64(-0.5 / a) * Float64(b + sqrt(Float64(c * Float64(a * -4.0)))));
	else
		tmp = Float64(Float64(Float64(c / b) + Float64(Float64(b * -0.5) / a)) + Float64(-0.5 * Float64(b / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.1e-80)
		tmp = 0.0 - (c / b);
	elseif (b <= 3.7e-113)
		tmp = (-0.5 / a) * (b + sqrt((c * (a * -4.0))));
	else
		tmp = ((c / b) + ((b * -0.5) / a)) + (-0.5 * (b / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.1e-80], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.7e-113], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c / b), $MachinePrecision] + N[(N[(b * -0.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.10000000000000001e-80

   1. Initial program 13.2%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 13.2%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 86.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   7. Simplified 86.3%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

   if -2.10000000000000001e-80 < b < 3.6999999999999998e-113

   1. Initial program 76.7%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 76.7%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a \cdot -2}\right), \color{blue}{\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{-2 \cdot a}\right), \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{-2}}{a}\right), \left(\color{blue}{b} + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{-2}\right), a\right), \left(\color{blue}{b} + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(c \cdot a\right) \cdot -4\right)\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(c \cdot \left(a \cdot -4\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(a \cdot -4\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 76.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 76.5%

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

   7. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right)}\right)\right)\right) \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\left(-4 \cdot a\right) \cdot c\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(-4 \cdot a\right), c\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(a \cdot -4\right), c\right)\right)\right)\right) \]

      4. *-lowering-*.f64 75.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, -4\right), c\right)\right)\right)\right) \]

   9. Simplified 75.4%

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

   if 3.6999999999999998e-113 < b

   1. Initial program 68.2%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 68.2%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a \cdot -2}\right), \color{blue}{\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{-2 \cdot a}\right), \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{-2}}{a}\right), \left(\color{blue}{b} + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{-2}\right), a\right), \left(\color{blue}{b} + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(c \cdot a\right) \cdot -4\right)\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(c \cdot \left(a \cdot -4\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(a \cdot -4\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 68.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 68.0%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2}}{a} \cdot \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right), \color{blue}{\left(b \cdot \frac{\frac{-1}{2}}{a}\right)}\right) \]

   8. Applied egg-rr 68.2%

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

   9. Taylor expanded in c around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{b}{a} + \frac{c}{b}\right)}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(b, a\right)\right)\right) \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{b}{a}\right), \left(\frac{c}{b}\right)\right), \mathsf{*.f64}\left(\color{blue}{\frac{-1}{2}}, \mathsf{/.f64}\left(b, a\right)\right)\right) \]

       2. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{-1}{2} \cdot b}{a}\right), \left(\frac{c}{b}\right)\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(b, a\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot b\right), a\right), \left(\frac{c}{b}\right)\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(b, a\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, b\right), a\right), \left(\frac{c}{b}\right)\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(b, a\right)\right)\right) \]

       5. /-lowering-/.f64 94.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, b\right), a\right), \mathsf{/.f64}\left(c, b\right)\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(b, a\right)\right)\right) \]

   11. Simplified 94.8%

       \[\leadsto \color{blue}{\left(\frac{-0.5 \cdot b}{a} + \frac{c}{b}\right)} + -0.5 \cdot \frac{b}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{-80}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{-113}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} + \frac{b \cdot -0.5}{a}\right) + -0.5 \cdot \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.4% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} + \frac{b \cdot -0.5}{a}\right) + -0.5 \cdot \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = 0.0 - (c / b);
	} else {
		tmp = ((c / b) + ((b * -0.5) / a)) + (-0.5 * (b / a));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = 0.0d0 - (c / b)
    else
        tmp = ((c / b) + ((b * (-0.5d0)) / a)) + ((-0.5d0) * (b / a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = 0.0 - (c / b);
	} else {
		tmp = ((c / b) + ((b * -0.5) / a)) + (-0.5 * (b / a));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = 0.0 - (c / b)
	else:
		tmp = ((c / b) + ((b * -0.5) / a)) + (-0.5 * (b / a))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c / b), $MachinePrecision] + N[(N[(b * -0.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 32.6%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 32.6%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 64.5%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   7. Simplified 64.5%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

   if -1.999999999999994e-310 < b

   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. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 68.4%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{a \cdot -2}\right), \color{blue}{\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{-2 \cdot a}\right), \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{-2}}{a}\right), \left(\color{blue}{b} + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{-2}\right), a\right), \left(\color{blue}{b} + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(c \cdot -4\right)\right)\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(a \cdot c\right) \cdot -4\right)\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(c \cdot a\right) \cdot -4\right)\right)\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(c \cdot \left(a \cdot -4\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(a \cdot -4\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 68.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -4\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 68.2%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2}}{a} \cdot \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right), \color{blue}{\left(b \cdot \frac{\frac{-1}{2}}{a}\right)}\right) \]

   8. Applied egg-rr 68.4%

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

   9. Taylor expanded in c around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{b}{a} + \frac{c}{b}\right)}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(b, a\right)\right)\right) \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{b}{a}\right), \left(\frac{c}{b}\right)\right), \mathsf{*.f64}\left(\color{blue}{\frac{-1}{2}}, \mathsf{/.f64}\left(b, a\right)\right)\right) \]

       2. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{-1}{2} \cdot b}{a}\right), \left(\frac{c}{b}\right)\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(b, a\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot b\right), a\right), \left(\frac{c}{b}\right)\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(b, a\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, b\right), a\right), \left(\frac{c}{b}\right)\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(b, a\right)\right)\right) \]

       5. /-lowering-/.f64 81.1%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, b\right), a\right), \mathsf{/.f64}\left(c, b\right)\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(b, a\right)\right)\right) \]

   11. Simplified 81.1%

       \[\leadsto \color{blue}{\left(\frac{-0.5 \cdot b}{a} + \frac{c}{b}\right)} + -0.5 \cdot \frac{b}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} + \frac{b \cdot -0.5}{a}\right) + -0.5 \cdot \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.5% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (- 0.0 (/ c b)) (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = 0.0 - (c / b);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = 0.0d0 - (c / b)
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = 0.0 - (c / b);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = 0.0 - (c / b)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 32.6%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 32.6%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 64.5%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   7. Simplified 64.5%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

   if -1.999999999999994e-310 < b

   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. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 68.4%

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

   5. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{c}{b} + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right) \]

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{c}{b}\right), \color{blue}{\left(\frac{b}{a}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(c, b\right), \left(\frac{\color{blue}{b}}{a}\right)\right) \]

      6. /-lowering-/.f64 81.1%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(c, b\right), \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   7. Simplified 81.1%

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

Alternative 11: 68.3% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (- 0.0 (/ c b)) (- 0.0 (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = 0.0 - (c / b);
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = 0.0d0 - (c / b)
    else
        tmp = 0.0d0 - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = 0.0 - (c / b);
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = 0.0 - (c / b)
	else:
		tmp = 0.0 - (b / a)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 32.6%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 32.6%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      4. /-lowering-/.f64 64.5%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   7. Simplified 64.5%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

   if -1.999999999999994e-310 < b

   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. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 68.4%

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

   5. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right) \]

      4. /-lowering-/.f64 80.4%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   7. Simplified 80.4%

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{b}{a}\right)\right) \]

      3. /-lowering-/.f64 80.4%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(b, a\right)\right) \]

   9. Applied egg-rr 80.4%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;0 - \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 44.1% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{-299}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 2.4e-299) 0.0 (- 0.0 (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.4e-299) {
		tmp = 0.0;
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.4d-299) then
        tmp = 0.0d0
    else
        tmp = 0.0d0 - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.4e-299) {
		tmp = 0.0;
	} else {
		tmp = 0.0 - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.4e-299:
		tmp = 0.0
	else:
		tmp = 0.0 - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.4e-299)
		tmp = 0.0;
	else
		tmp = Float64(0.0 - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.4e-299)
		tmp = 0.0;
	else
		tmp = 0.0 - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.4e-299], 0.0, N[(0.0 - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.40000000000000019e-299

   1. Initial program 33.0%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 33.0%

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

   5. Taylor expanded in b around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \color{blue}{\left(-1 \cdot \left(b \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)}\right), \mathsf{*.f64}\left(a, -2\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left(\mathsf{neg}\left(b \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left(b \cdot \left(\mathsf{neg}\left(\left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(b, \left(\mathsf{neg}\left(\left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      4. neg-mul-1 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(b, \left(-1 \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      5. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(b, \left(1 \cdot -1 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}}\right) \cdot -1\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(b, \left(-1 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}}\right) \cdot -1\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(-1, \left(\left(-2 \cdot \frac{a \cdot c}{{b}^{2}}\right) \cdot -1\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(-2 \cdot \frac{a \cdot c}{{b}^{2}}\right), -1\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, \left(\frac{a \cdot c}{{b}^{2}}\right)\right), -1\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, \left(\frac{a \cdot c}{b \cdot b}\right)\right), -1\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      11. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, \left(\frac{\frac{a \cdot c}{b}}{b}\right)\right), -1\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(\left(\frac{a \cdot c}{b}\right), b\right)\right), -1\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(a \cdot c\right), b\right), b\right)\right), -1\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(c \cdot a\right), b\right), b\right)\right), -1\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

      15. *-lowering-*.f64 18.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, a\right), b\right), b\right)\right), -1\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

   7. Simplified 18.3%

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

   8. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot \left(0 \cdot b\right)}{a} \]

      4. mul0-lft N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot 0}{a} \]

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 17.6%

         \[\leadsto \mathsf{/.f64}\left(0, \color{blue}{a}\right) \]

   10. Simplified 17.6%

       \[\leadsto \color{blue}{\frac{0}{a}} \]
   11. Step-by-step derivation

       1. div0 17.6%

          \[\leadsto 0 \]

   12. Applied egg-rr 17.6%

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

   if 2.40000000000000019e-299 < b

   1. Initial program 68.1%

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

      1. sub-neg N/A

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

      2. distribute-neg-out N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

   3. Simplified 68.1%

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

   5. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{b}{a}\right)}\right) \]

      4. /-lowering-/.f64 81.1%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(b, \color{blue}{a}\right)\right) \]

   7. Simplified 81.1%

      \[\leadsto \color{blue}{0 - \frac{b}{a}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{b}{a}\right)\right) \]

      3. /-lowering-/.f64 81.1%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(b, a\right)\right) \]

   9. Applied egg-rr 81.1%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{-299}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 11.1% 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 47.3%

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

   1. sub-neg N/A

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

   2. distribute-neg-out N/A

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

   3. distribute-frac-neg N/A

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

   4. distribute-neg-frac2 N/A

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

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot a\right)\right)}\right) \]

3. Simplified 47.3%

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

5. Taylor expanded in b around -inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \color{blue}{\left(-1 \cdot \left(b \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)}\right), \mathsf{*.f64}\left(a, -2\right)\right) \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left(\mathsf{neg}\left(b \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

   2. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left(b \cdot \left(\mathsf{neg}\left(\left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(b, \left(\mathsf{neg}\left(\left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

   4. neg-mul-1 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(b, \left(-1 \cdot \left(1 + -2 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

   5. distribute-rgt-in N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(b, \left(1 \cdot -1 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}}\right) \cdot -1\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(b, \left(-1 + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}}\right) \cdot -1\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(-1, \left(\left(-2 \cdot \frac{a \cdot c}{{b}^{2}}\right) \cdot -1\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(-2 \cdot \frac{a \cdot c}{{b}^{2}}\right), -1\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, \left(\frac{a \cdot c}{{b}^{2}}\right)\right), -1\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, \left(\frac{a \cdot c}{b \cdot b}\right)\right), -1\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

   11. associate-/r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, \left(\frac{\frac{a \cdot c}{b}}{b}\right)\right), -1\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(\left(\frac{a \cdot c}{b}\right), b\right)\right), -1\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(a \cdot c\right), b\right), b\right)\right), -1\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(c \cdot a\right), b\right), b\right)\right), -1\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

   15. *-lowering-*.f64 11.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, a\right), b\right), b\right)\right), -1\right)\right)\right)\right), \mathsf{*.f64}\left(a, -2\right)\right) \]

7. Simplified 11.7%

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

8. Taylor expanded in c around 0

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

   1. associate-*r/ N/A

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

   2. distribute-rgt1-in N/A

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

   3. metadata-eval N/A

      \[\leadsto \frac{\frac{-1}{2} \cdot \left(0 \cdot b\right)}{a} \]

   4. mul0-lft N/A

      \[\leadsto \frac{\frac{-1}{2} \cdot 0}{a} \]

   5. metadata-eval N/A

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

   6. /-lowering-/.f64 11.4%

      \[\leadsto \mathsf{/.f64}\left(0, \color{blue}{a}\right) \]

10. Simplified 11.4%

    \[\leadsto \color{blue}{\frac{0}{a}} \]
11. Step-by-step derivation

    1. div0 11.4%

       \[\leadsto 0 \]

12. Applied egg-rr 11.4%

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

Developer Target 1: 99.7% accurate, 0.1× speedup

Math

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

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) (/ c (- t_2 (/ b 2.0))) (/ (+ (/ b 2.0) t_2) (- a)))))

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 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	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 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	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 = c / (t_2 - (b / 2.0))
	else:
		tmp_1 = ((b / 2.0) + t_2) / -a
	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(c / Float64(t_2 - Float64(b / 2.0)));
	else
		tmp_1 = Float64(Float64(Float64(b / 2.0) + t_2) / Float64(-a));
	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 = c / (t_2 - (b / 2.0));
	else
		tmp_2 = ((b / 2.0) + t_2) / -a;
	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[(c / N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision] / (-a)), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024191 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :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) (/ c (- sqtD (/ b 2))) (/ (+ (/ b 2) sqtD) (- a)))))

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