quadp (p42, positive)

Percentage Accurate: 52.6% → 90.5%

Time: 10.4s

Alternatives: 12

Speedup: 2.5×

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: 52.6% 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: 90.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-229}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+108}:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.5e+153)
   (/ (- b) a)
   (if (<= b 3.1e-229)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (if (<= b 2e+108)
       (/ (* c -2.0) (+ b (sqrt (fma c (* a -4.0) (* b b)))))
       (- (/ c b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.5e+153) {
		tmp = -b / a;
	} else if (b <= 3.1e-229) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else if (b <= 2e+108) {
		tmp = (c * -2.0) / (b + sqrt(fma(c, (a * -4.0), (b * b))));
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.5e+153)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 3.1e-229)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	elseif (b <= 2e+108)
		tmp = Float64(Float64(c * -2.0) / Float64(b + sqrt(fma(c, Float64(a * -4.0), Float64(b * b)))));
	else
		tmp = Float64(-Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.5e+153], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 3.1e-229], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e+108], N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(c / b), $MachinePrecision])]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.50000000000000009e153

   1. Initial program 31.7%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 92.9

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

   5. Simplified 92.9%

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

   if -2.50000000000000009e153 < b < 3.1000000000000001e-229

   1. Initial program 89.0%

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

   if 3.1000000000000001e-229 < b < 2.0000000000000001e108

   1. Initial program 33.4%

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

   4. Applied egg-rr 33.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. flip-- N/A

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

      7. lift-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   6. Applied egg-rr 66.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-/.f64 66.0

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

   8. Applied egg-rr 85.0%

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

   if 2.0000000000000001e108 < b

   1. Initial program 4.0%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 94.6

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

   5. Simplified 94.6%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-229}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+108}:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.4 \cdot 10^{+139}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.75 \cdot 10^{-305}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}\right)\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+108}:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.4e+139)
   (/ (- b) a)
   (if (<= b 1.75e-305)
     (* (/ -0.5 a) (- b (sqrt (fma b b (* (* a c) -4.0)))))
     (if (<= b 2e+108)
       (/ (* c -2.0) (+ b (sqrt (fma c (* a -4.0) (* b b)))))
       (- (/ c b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.4e+139) {
		tmp = -b / a;
	} else if (b <= 1.75e-305) {
		tmp = (-0.5 / a) * (b - sqrt(fma(b, b, ((a * c) * -4.0))));
	} else if (b <= 2e+108) {
		tmp = (c * -2.0) / (b + sqrt(fma(c, (a * -4.0), (b * b))));
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.4e+139)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 1.75e-305)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(fma(b, b, Float64(Float64(a * c) * -4.0)))));
	elseif (b <= 2e+108)
		tmp = Float64(Float64(c * -2.0) / Float64(b + sqrt(fma(c, Float64(a * -4.0), Float64(b * b)))));
	else
		tmp = Float64(-Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.4e+139], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 1.75e-305], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(b * b + N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e+108], N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(c / b), $MachinePrecision])]]]

Derivation

1. Split input into 4 regimes

2. if b < -8.3999999999999995e139

   1. Initial program 43.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 94.1

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

   5. Simplified 94.1%

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

   if -8.3999999999999995e139 < b < 1.7499999999999999e-305

   1. Initial program 87.7%

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

   4. Applied egg-rr 87.4%

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

   if 1.7499999999999999e-305 < b < 2.0000000000000001e108

   1. Initial program 44.4%

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

   4. Applied egg-rr 44.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. flip-- N/A

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

      7. lift-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   6. Applied egg-rr 70.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-/.f64 70.9

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

   8. Applied egg-rr 86.5%

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

   if 2.0000000000000001e108 < b

   1. Initial program 4.0%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 94.6

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

   5. Simplified 94.6%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.4 \cdot 10^{+139}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.75 \cdot 10^{-305}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}\right)\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+108}:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.4 \cdot 10^{+139}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq -6.6 \cdot 10^{-279}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}\right)\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+108}:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.4e+139)
   (/ (- b) a)
   (if (<= b -6.6e-279)
     (* (/ -0.5 a) (- b (sqrt (fma b b (* (* a c) -4.0)))))
     (if (<= b 2e+108)
       (* c (/ -2.0 (+ b (sqrt (fma c (* a -4.0) (* b b))))))
       (- (/ c b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.4e+139) {
		tmp = -b / a;
	} else if (b <= -6.6e-279) {
		tmp = (-0.5 / a) * (b - sqrt(fma(b, b, ((a * c) * -4.0))));
	} else if (b <= 2e+108) {
		tmp = c * (-2.0 / (b + sqrt(fma(c, (a * -4.0), (b * b)))));
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.4e+139)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= -6.6e-279)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(fma(b, b, Float64(Float64(a * c) * -4.0)))));
	elseif (b <= 2e+108)
		tmp = Float64(c * Float64(-2.0 / Float64(b + sqrt(fma(c, Float64(a * -4.0), Float64(b * b))))));
	else
		tmp = Float64(-Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.4e+139], N[((-b) / a), $MachinePrecision], If[LessEqual[b, -6.6e-279], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(b * b + N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e+108], N[(c * N[(-2.0 / N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(c / b), $MachinePrecision])]]]

Derivation

1. Split input into 4 regimes

2. if b < -8.3999999999999995e139

   1. Initial program 43.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 94.1

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

   5. Simplified 94.1%

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

   if -8.3999999999999995e139 < b < -6.6e-279

   1. Initial program 86.8%

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

   4. Applied egg-rr 86.6%

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

   if -6.6e-279 < b < 2.0000000000000001e108

   1. Initial program 48.1%

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

   4. Applied egg-rr 48.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. flip-- N/A

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

      7. lift-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   6. Applied egg-rr 72.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-/.f64 72.8

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

   8. Applied egg-rr 87.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   10. Applied egg-rr 87.2%

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

   if 2.0000000000000001e108 < b

   1. Initial program 4.0%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 94.6

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

   5. Simplified 94.6%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.4 \cdot 10^{+139}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq -6.6 \cdot 10^{-279}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}\right)\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+108}:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.76 \cdot 10^{-78}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+108}:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.76e-78)
   (/ (- b) a)
   (if (<= b 2e+108)
     (* c (/ -2.0 (+ b (sqrt (fma c (* a -4.0) (* b b))))))
     (- (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.76e-78) {
		tmp = -b / a;
	} else if (b <= 2e+108) {
		tmp = c * (-2.0 / (b + sqrt(fma(c, (a * -4.0), (b * b)))));
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.76e-78)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 2e+108)
		tmp = Float64(c * Float64(-2.0 / Float64(b + sqrt(fma(c, Float64(a * -4.0), Float64(b * b))))));
	else
		tmp = Float64(-Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.76e-78], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 2e+108], N[(c * N[(-2.0 / N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(c / b), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b < -1.7599999999999999e-78

   1. Initial program 77.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 84.8

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

   5. Simplified 84.8%

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

   if -1.7599999999999999e-78 < b < 2.0000000000000001e108

   1. Initial program 55.2%

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

   4. Applied egg-rr 55.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. flip-- N/A

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

      7. lift-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   6. Applied egg-rr 72.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-/.f64 72.0

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

   8. Applied egg-rr 82.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   10. Applied egg-rr 82.4%

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

   if 2.0000000000000001e108 < b

   1. Initial program 4.0%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 94.6

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

   5. Simplified 94.6%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.76 \cdot 10^{-78}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+108}:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.76 \cdot 10^{-78}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-43}:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{a \cdot \left(c \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{\mathsf{fma}\left(b, 2, \frac{\left(a \cdot c\right) \cdot -2}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.76e-78)
   (/ (- b) a)
   (if (<= b 8.6e-43)
     (/ (* c -2.0) (+ b (sqrt (* a (* c -4.0)))))
     (/ (* c -2.0) (fma b 2.0 (/ (* (* a c) -2.0) b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.76e-78) {
		tmp = -b / a;
	} else if (b <= 8.6e-43) {
		tmp = (c * -2.0) / (b + sqrt((a * (c * -4.0))));
	} else {
		tmp = (c * -2.0) / fma(b, 2.0, (((a * c) * -2.0) / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.76e-78)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 8.6e-43)
		tmp = Float64(Float64(c * -2.0) / Float64(b + sqrt(Float64(a * Float64(c * -4.0)))));
	else
		tmp = Float64(Float64(c * -2.0) / fma(b, 2.0, Float64(Float64(Float64(a * c) * -2.0) / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.76e-78], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 8.6e-43], N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -2.0), $MachinePrecision] / N[(b * 2.0 + N[(N[(N[(a * c), $MachinePrecision] * -2.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.7599999999999999e-78

   1. Initial program 77.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 84.8

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

   5. Simplified 84.8%

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

   if -1.7599999999999999e-78 < b < 8.59999999999999927e-43

   1. Initial program 74.0%

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

   4. Applied egg-rr 73.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. flip-- N/A

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

      7. lift-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   6. Applied egg-rr 72.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-/.f64 72.9

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

   8. Applied egg-rr 79.6%

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

   9. Taylor expanded in c around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 74.6

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

   11. Simplified 74.6%

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

   if 8.59999999999999927e-43 < b

   1. Initial program 12.9%

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

   4. Applied egg-rr 12.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. flip-- N/A

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

      7. lift-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   6. Applied egg-rr 60.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-/.f64 60.3

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

   8. Applied egg-rr 68.3%

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

   9. Taylor expanded in c around 0

      \[\leadsto \frac{c \cdot -2}{\color{blue}{-2 \cdot \frac{a \cdot c}{b} + 2 \cdot b}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-/l* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. *-commutative N/A

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

       9. associate-*r* N/A

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

       10. associate-/l* N/A

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

       11. *-commutative N/A

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

       12. associate-*r/ N/A

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

       13. lower-/.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. lower-*.f64 86.2

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

   11. Simplified 86.2%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.76 \cdot 10^{-78}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-43}:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{a \cdot \left(c \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{\mathsf{fma}\left(b, 2, \frac{\left(a \cdot c\right) \cdot -2}{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.76 \cdot 10^{-78}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-43}:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{a \cdot \left(c \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{b + \mathsf{fma}\left(-2, \frac{a \cdot c}{b}, b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.76e-78)
   (/ (- b) a)
   (if (<= b 8.6e-43)
     (/ (* c -2.0) (+ b (sqrt (* a (* c -4.0)))))
     (/ (* c -2.0) (+ b (fma -2.0 (/ (* a c) b) b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.76e-78) {
		tmp = -b / a;
	} else if (b <= 8.6e-43) {
		tmp = (c * -2.0) / (b + sqrt((a * (c * -4.0))));
	} else {
		tmp = (c * -2.0) / (b + fma(-2.0, ((a * c) / b), b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.76e-78)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 8.6e-43)
		tmp = Float64(Float64(c * -2.0) / Float64(b + sqrt(Float64(a * Float64(c * -4.0)))));
	else
		tmp = Float64(Float64(c * -2.0) / Float64(b + fma(-2.0, Float64(Float64(a * c) / b), b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.76e-78], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 8.6e-43], N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[(-2.0 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.7599999999999999e-78

   1. Initial program 77.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 84.8

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

   5. Simplified 84.8%

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

   if -1.7599999999999999e-78 < b < 8.59999999999999927e-43

   1. Initial program 74.0%

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

   4. Applied egg-rr 73.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. flip-- N/A

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

      7. lift-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   6. Applied egg-rr 72.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-/.f64 72.9

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

   8. Applied egg-rr 79.6%

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

   9. Taylor expanded in c around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 74.6

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

   11. Simplified 74.6%

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

   if 8.59999999999999927e-43 < b

   1. Initial program 12.9%

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

   4. Applied egg-rr 12.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. flip-- N/A

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

      7. lift-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   6. Applied egg-rr 60.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-/.f64 60.3

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

   8. Applied egg-rr 68.3%

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

   9. Taylor expanded in c around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-*.f64 86.2

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

   11. Simplified 86.2%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.76 \cdot 10^{-78}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-43}:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{a \cdot \left(c \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{b + \mathsf{fma}\left(-2, \frac{a \cdot c}{b}, b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.76 \cdot 10^{-78}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-43}:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{a \cdot \left(c \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.76e-78)
   (/ (- b) a)
   (if (<= b 8.6e-43)
     (/ (* c -2.0) (+ b (sqrt (* a (* c -4.0)))))
     (- (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.76e-78) {
		tmp = -b / a;
	} else if (b <= 8.6e-43) {
		tmp = (c * -2.0) / (b + sqrt((a * (c * -4.0))));
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.76d-78)) then
        tmp = -b / a
    else if (b <= 8.6d-43) then
        tmp = (c * (-2.0d0)) / (b + sqrt((a * (c * (-4.0d0)))))
    else
        tmp = -(c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.76e-78) {
		tmp = -b / a;
	} else if (b <= 8.6e-43) {
		tmp = (c * -2.0) / (b + Math.sqrt((a * (c * -4.0))));
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.76e-78:
		tmp = -b / a
	elif b <= 8.6e-43:
		tmp = (c * -2.0) / (b + math.sqrt((a * (c * -4.0))))
	else:
		tmp = -(c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.76e-78)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 8.6e-43)
		tmp = Float64(Float64(c * -2.0) / Float64(b + sqrt(Float64(a * Float64(c * -4.0)))));
	else
		tmp = Float64(-Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.76e-78)
		tmp = -b / a;
	elseif (b <= 8.6e-43)
		tmp = (c * -2.0) / (b + sqrt((a * (c * -4.0))));
	else
		tmp = -(c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.76e-78], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 8.6e-43], N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(c / b), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b < -1.7599999999999999e-78

   1. Initial program 77.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 84.8

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

   5. Simplified 84.8%

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

   if -1.7599999999999999e-78 < b < 8.59999999999999927e-43

   1. Initial program 74.0%

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

   4. Applied egg-rr 73.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. flip-- N/A

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

      7. lift-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   6. Applied egg-rr 72.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-/.f64 72.9

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

   8. Applied egg-rr 79.6%

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

   9. Taylor expanded in c around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 74.6

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

   11. Simplified 74.6%

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

   if 8.59999999999999927e-43 < b

   1. Initial program 12.9%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 85.7

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

   5. Simplified 85.7%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.76 \cdot 10^{-78}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-43}:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{a \cdot \left(c \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.76 \cdot 10^{-78}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\left(a \cdot c\right) \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.76e-78)
   (/ (- b) a)
   (if (<= b 5.2e-51)
     (* (/ -0.5 a) (- b (sqrt (* (* a c) -4.0))))
     (- (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.76e-78) {
		tmp = -b / a;
	} else if (b <= 5.2e-51) {
		tmp = (-0.5 / a) * (b - sqrt(((a * c) * -4.0)));
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.76d-78)) then
        tmp = -b / a
    else if (b <= 5.2d-51) then
        tmp = ((-0.5d0) / a) * (b - sqrt(((a * c) * (-4.0d0))))
    else
        tmp = -(c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.76e-78) {
		tmp = -b / a;
	} else if (b <= 5.2e-51) {
		tmp = (-0.5 / a) * (b - Math.sqrt(((a * c) * -4.0)));
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.76e-78:
		tmp = -b / a
	elif b <= 5.2e-51:
		tmp = (-0.5 / a) * (b - math.sqrt(((a * c) * -4.0)))
	else:
		tmp = -(c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.76e-78)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 5.2e-51)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(Float64(Float64(a * c) * -4.0))));
	else
		tmp = Float64(-Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.76e-78)
		tmp = -b / a;
	elseif (b <= 5.2e-51)
		tmp = (-0.5 / a) * (b - sqrt(((a * c) * -4.0)));
	else
		tmp = -(c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.76e-78], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 5.2e-51], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(c / b), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b < -1.7599999999999999e-78

   1. Initial program 77.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 84.8

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

   5. Simplified 84.8%

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

   if -1.7599999999999999e-78 < b < 5.2e-51

   1. Initial program 74.6%

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

   4. Applied egg-rr 74.4%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 74.3

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

   7. Simplified 74.3%

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

   if 5.2e-51 < b

   1. Initial program 13.7%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 85.1

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

   5. Simplified 85.1%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.76 \cdot 10^{-78}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\left(a \cdot c\right) \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-272}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.1e-272) (/ (- b) a) (- (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-272) {
		tmp = -b / a;
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 5.1d-272) then
        tmp = -b / a
    else
        tmp = -(c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.1e-272) {
		tmp = -b / a;
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.1e-272:
		tmp = -b / a
	else:
		tmp = -(c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.1e-272)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(-Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.1e-272)
		tmp = -b / a;
	else
		tmp = -(c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.1e-272], N[((-b) / a), $MachinePrecision], (-N[(c / b), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if b < 5.0999999999999998e-272

   1. Initial program 77.9%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 55.3

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

   5. Simplified 55.3%

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

   if 5.0999999999999998e-272 < b

   1. Initial program 24.8%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 71.8

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

   5. Simplified 71.8%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{-272}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1000000:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 1000000.0) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1000000.0) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1000000.0d0) then
        tmp = -b / a
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1000000.0) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1000000.0:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1000000.0)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1000000.0)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1000000.0], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1e6

   1. Initial program 70.5%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 41.7

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

   5. Simplified 41.7%

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

   if 1e6 < b

   1. Initial program 11.5%

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

   4. Applied egg-rr 5.4%

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

   5. Taylor expanded in b around -inf

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

      1. lower-/.f64 23.3

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

   7. Simplified 23.3%

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

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1000000:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 10.3% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

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

4. Applied egg-rr 32.6%

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

5. Taylor expanded in b around -inf

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

   1. lower-/.f64 9.7

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

7. Simplified 9.7%

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

Alternative 12: 2.6% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

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

4. Applied egg-rr 32.6%

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

5. Taylor expanded in a around 0

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

   1. lower-/.f64 2.7

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

7. Simplified 2.7%

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

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024215 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

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

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