The quadratic formula (r2)

Percentage Accurate: 52.9% → 90.4%

Time: 22.0s

Alternatives: 11

Speedup: 12.9×

Specification

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.9% 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: 90.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{+160}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{-232}:\\ \;\;\;\;\frac{c \cdot -2}{b - \sqrt{a \cdot \left(c \cdot -4\right) + {b}^{2}}}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+126}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.6e+160)
   (/ c (- b))
   (if (<= b 4.6e-232)
     (/ (* c -2.0) (- b (sqrt (+ (* a (* c -4.0)) (pow b 2.0)))))
     (if (<= b 5e+126)
       (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* c a))))) (* a 2.0))
       (/ b (- a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.6e+160) {
		tmp = c / -b;
	} else if (b <= 4.6e-232) {
		tmp = (c * -2.0) / (b - sqrt(((a * (c * -4.0)) + pow(b, 2.0))));
	} else if (b <= 5e+126) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	} else {
		tmp = b / -a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-6.6d+160)) then
        tmp = c / -b
    else if (b <= 4.6d-232) then
        tmp = (c * (-2.0d0)) / (b - sqrt(((a * (c * (-4.0d0))) + (b ** 2.0d0))))
    else if (b <= 5d+126) then
        tmp = (-b - sqrt(((b * b) - (4.0d0 * (c * a))))) / (a * 2.0d0)
    else
        tmp = b / -a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.6e+160) {
		tmp = c / -b;
	} else if (b <= 4.6e-232) {
		tmp = (c * -2.0) / (b - Math.sqrt(((a * (c * -4.0)) + Math.pow(b, 2.0))));
	} else if (b <= 5e+126) {
		tmp = (-b - Math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	} else {
		tmp = b / -a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.6e+160:
		tmp = c / -b
	elif b <= 4.6e-232:
		tmp = (c * -2.0) / (b - math.sqrt(((a * (c * -4.0)) + math.pow(b, 2.0))))
	elif b <= 5e+126:
		tmp = (-b - math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0)
	else:
		tmp = b / -a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.6e+160)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 4.6e-232)
		tmp = Float64(Float64(c * -2.0) / Float64(b - sqrt(Float64(Float64(a * Float64(c * -4.0)) + (b ^ 2.0)))));
	elseif (b <= 5e+126)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a))))) / Float64(a * 2.0));
	else
		tmp = Float64(b / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.6e+160)
		tmp = c / -b;
	elseif (b <= 4.6e-232)
		tmp = (c * -2.0) / (b - sqrt(((a * (c * -4.0)) + (b ^ 2.0))));
	elseif (b <= 5e+126)
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	else
		tmp = b / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.6e+160], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 4.6e-232], N[(N[(c * -2.0), $MachinePrecision] / N[(b - N[Sqrt[N[(N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e+126], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(b / (-a)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -6.5999999999999994e160

   1. Initial program 1.7%

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

      1. div-sub 1.3%

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

      2. sub-neg 1.3%

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

      3. neg-mul-1 1.3%

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

      4. *-commutative 1.3%

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

      5. associate-/l* 1.3%

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

      6. distribute-neg-frac 1.3%

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

      7. neg-mul-1 1.3%

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

      8. *-commutative 1.3%

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

      9. associate-/l* 1.3%

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

      10. distribute-rgt-out 1.7%

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

      11. associate-/r* 1.7%

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

      12. metadata-eval 1.7%

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

      13. sub-neg 1.7%

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

      14. +-commutative 1.7%

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

   3. Simplified 1.8%

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

   5. Taylor expanded in b around -inf 95.6%

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

      1. mul-1-neg 95.6%

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

      2. distribute-neg-frac2 95.6%

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

   7. Simplified 95.6%

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

   if -6.5999999999999994e160 < b < 4.6e-232

   1. Initial program 48.2%

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

      1. div-sub 46.9%

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

      2. sub-neg 46.9%

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

      3. neg-mul-1 46.9%

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

      4. *-commutative 46.9%

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

      5. associate-/l* 46.4%

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

      6. distribute-neg-frac 46.4%

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

      7. neg-mul-1 46.4%

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

      8. *-commutative 46.4%

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

      9. associate-/l* 46.9%

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

      10. distribute-rgt-out 48.1%

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

      11. associate-/r* 48.1%

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

      12. metadata-eval 48.1%

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

      13. sub-neg 48.1%

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

      14. +-commutative 48.1%

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

   3. Simplified 48.1%

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

      1. clear-num 48.1%

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

      2. flip-+ 47.6%

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

      3. frac-times 41.7%

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

      4. add-sqr-sqrt 41.8%

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

      5. *-un-lft-identity 41.8%

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

      6. pow2 41.8%

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

      7. pow2 41.8%

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

      8. div-inv 41.8%

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

      9. metadata-eval 41.8%

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

      10. pow2 41.8%

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

   6. Applied egg-rr 41.8%

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

      1. associate-/r* 47.9%

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

      2. *-commutative 47.9%

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

   8. Simplified 47.9%

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

   9. Taylor expanded in b around 0 86.5%

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

       1. *-commutative 86.5%

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

   11. Simplified 86.5%

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

       1. fma-undefine 86.5%

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

   13. Applied egg-rr 86.5%

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

   if 4.6e-232 < b < 4.99999999999999977e126

   1. Initial program 91.7%

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

   if 4.99999999999999977e126 < b

   1. Initial program 38.6%

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

      1. div-sub 38.6%

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

      2. sub-neg 38.6%

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

      3. neg-mul-1 38.6%

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

      4. *-commutative 38.6%

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

      5. associate-/l* 38.6%

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

      6. distribute-neg-frac 38.6%

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

      7. neg-mul-1 38.6%

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

      8. *-commutative 38.6%

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

      9. associate-/l* 38.5%

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

      10. distribute-rgt-out 38.5%

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

      11. associate-/r* 38.5%

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

      12. metadata-eval 38.5%

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

      13. sub-neg 38.5%

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

      14. +-commutative 38.5%

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

   3. Simplified 38.7%

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

   5. Taylor expanded in a around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{+160}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{-232}:\\ \;\;\;\;\frac{c \cdot -2}{b - \sqrt{a \cdot \left(c \cdot -4\right) + {b}^{2}}}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+126}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.5}{a} \cdot \left(b + \sqrt{-4 \cdot \left(c \cdot a\right)}\right)\\ \mathbf{if}\;b \leq -3.8 \cdot 10^{-119}:\\ \;\;\;\;\frac{c \cdot -2}{b + \left(b - \left(a \cdot \frac{c}{b}\right) \cdot 2\right)}\\ \mathbf{elif}\;b \leq 10^{-88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 1100:\\ \;\;\;\;\frac{b \cdot -2 + \frac{c \cdot a}{b} \cdot 2}{a \cdot 2}\\ \mathbf{elif}\;b \leq 13000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (/ -0.5 a) (+ b (sqrt (* -4.0 (* c a)))))))
   (if (<= b -3.8e-119)
     (/ (* c -2.0) (+ b (- b (* (* a (/ c b)) 2.0))))
     (if (<= b 1e-88)
       t_0
       (if (<= b 1100.0)
         (/ (+ (* b -2.0) (* (/ (* c a) b) 2.0)) (* a 2.0))
         (if (<= b 13000000.0) t_0 (- (/ c b) (/ b a))))))))

C

double code(double a, double b, double c) {
	double t_0 = (-0.5 / a) * (b + sqrt((-4.0 * (c * a))));
	double tmp;
	if (b <= -3.8e-119) {
		tmp = (c * -2.0) / (b + (b - ((a * (c / b)) * 2.0)));
	} else if (b <= 1e-88) {
		tmp = t_0;
	} else if (b <= 1100.0) {
		tmp = ((b * -2.0) + (((c * a) / b) * 2.0)) / (a * 2.0);
	} else if (b <= 13000000.0) {
		tmp = t_0;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-0.5d0) / a) * (b + sqrt(((-4.0d0) * (c * a))))
    if (b <= (-3.8d-119)) then
        tmp = (c * (-2.0d0)) / (b + (b - ((a * (c / b)) * 2.0d0)))
    else if (b <= 1d-88) then
        tmp = t_0
    else if (b <= 1100.0d0) then
        tmp = ((b * (-2.0d0)) + (((c * a) / b) * 2.0d0)) / (a * 2.0d0)
    else if (b <= 13000000.0d0) then
        tmp = t_0
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (-0.5 / a) * (b + Math.sqrt((-4.0 * (c * a))));
	double tmp;
	if (b <= -3.8e-119) {
		tmp = (c * -2.0) / (b + (b - ((a * (c / b)) * 2.0)));
	} else if (b <= 1e-88) {
		tmp = t_0;
	} else if (b <= 1100.0) {
		tmp = ((b * -2.0) + (((c * a) / b) * 2.0)) / (a * 2.0);
	} else if (b <= 13000000.0) {
		tmp = t_0;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (-0.5 / a) * (b + math.sqrt((-4.0 * (c * a))))
	tmp = 0
	if b <= -3.8e-119:
		tmp = (c * -2.0) / (b + (b - ((a * (c / b)) * 2.0)))
	elif b <= 1e-88:
		tmp = t_0
	elif b <= 1100.0:
		tmp = ((b * -2.0) + (((c * a) / b) * 2.0)) / (a * 2.0)
	elif b <= 13000000.0:
		tmp = t_0
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-0.5 / a) * Float64(b + sqrt(Float64(-4.0 * Float64(c * a)))))
	tmp = 0.0
	if (b <= -3.8e-119)
		tmp = Float64(Float64(c * -2.0) / Float64(b + Float64(b - Float64(Float64(a * Float64(c / b)) * 2.0))));
	elseif (b <= 1e-88)
		tmp = t_0;
	elseif (b <= 1100.0)
		tmp = Float64(Float64(Float64(b * -2.0) + Float64(Float64(Float64(c * a) / b) * 2.0)) / Float64(a * 2.0));
	elseif (b <= 13000000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (-0.5 / a) * (b + sqrt((-4.0 * (c * a))));
	tmp = 0.0;
	if (b <= -3.8e-119)
		tmp = (c * -2.0) / (b + (b - ((a * (c / b)) * 2.0)));
	elseif (b <= 1e-88)
		tmp = t_0;
	elseif (b <= 1100.0)
		tmp = ((b * -2.0) + (((c * a) / b) * 2.0)) / (a * 2.0);
	elseif (b <= 13000000.0)
		tmp = t_0;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.8e-119], N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[(b - N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e-88], t$95$0, If[LessEqual[b, 1100.0], N[(N[(N[(b * -2.0), $MachinePrecision] + N[(N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 13000000.0], t$95$0, N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -3.79999999999999975e-119

   1. Initial program 18.4%

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

      1. div-sub 17.1%

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

      2. sub-neg 17.1%

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

      3. neg-mul-1 17.1%

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

      4. *-commutative 17.1%

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

      5. associate-/l* 16.5%

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

      6. distribute-neg-frac 16.5%

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

      7. neg-mul-1 16.5%

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

      8. *-commutative 16.5%

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

      9. associate-/l* 17.0%

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

      10. distribute-rgt-out 18.4%

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

      11. associate-/r* 18.4%

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

      12. metadata-eval 18.4%

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

      13. sub-neg 18.4%

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

      14. +-commutative 18.4%

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

   3. Simplified 18.4%

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

      1. clear-num 18.4%

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

      2. flip-+ 17.5%

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

      3. frac-times 13.9%

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

      4. add-sqr-sqrt 13.9%

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

      5. *-un-lft-identity 13.9%

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

      6. pow2 13.9%

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

      7. pow2 13.9%

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

      8. div-inv 13.9%

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

      9. metadata-eval 13.9%

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

      10. pow2 13.9%

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

   6. Applied egg-rr 13.9%

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

      1. associate-/r* 17.6%

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

      2. *-commutative 17.6%

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

   8. Simplified 17.6%

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

   9. Taylor expanded in b around 0 70.8%

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

       1. *-commutative 70.8%

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

   11. Simplified 70.8%

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

   12. Taylor expanded in b around -inf 83.4%

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

       1. neg-mul-1 83.4%

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

       2. +-commutative 83.4%

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

       3. unsub-neg 83.4%

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

       4. *-commutative 83.4%

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

       5. associate-/l* 87.5%

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

   14. Simplified 87.5%

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

   if -3.79999999999999975e-119 < b < 9.99999999999999934e-89 or 1100 < b < 1.3e7

   1. Initial program 81.2%

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

      1. div-sub 81.2%

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

      2. sub-neg 81.2%

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

      3. neg-mul-1 81.2%

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

      4. *-commutative 81.2%

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

      5. associate-/l* 81.2%

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

      6. distribute-neg-frac 81.2%

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

      7. neg-mul-1 81.2%

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

      8. *-commutative 81.2%

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

      9. associate-/l* 81.0%

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

      10. distribute-rgt-out 81.0%

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

      11. associate-/r* 81.0%

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

      12. metadata-eval 81.0%

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

      13. sub-neg 81.0%

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

      14. +-commutative 81.0%

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

   3. Simplified 81.0%

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

   5. Taylor expanded in a around inf 78.3%

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

      1. *-commutative 78.3%

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

   7. Simplified 78.3%

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

   if 9.99999999999999934e-89 < b < 1100

   1. Initial program 90.6%

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

      1. *-commutative 90.6%

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

      2. *-commutative 90.6%

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

      3. sqr-neg 90.6%

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

      4. *-commutative 90.6%

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

      5. sqr-neg 90.6%

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

      6. *-commutative 90.6%

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

      7. associate-*r* 90.6%

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

   3. Simplified 90.6%

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

   5. Taylor expanded in b around inf 64.5%

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

   if 1.3e7 < b

   1. Initial program 56.9%

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

      1. div-sub 56.9%

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

      2. sub-neg 56.9%

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

      3. neg-mul-1 56.9%

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

      4. *-commutative 56.9%

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

      5. associate-/l* 56.8%

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

      6. distribute-neg-frac 56.8%

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

      7. neg-mul-1 56.8%

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

      8. *-commutative 56.8%

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

      9. associate-/l* 56.7%

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

      10. distribute-rgt-out 56.7%

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

      11. associate-/r* 56.7%

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

      12. metadata-eval 56.7%

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

      13. sub-neg 56.7%

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

      14. +-commutative 56.7%

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

   3. Simplified 56.9%

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

   5. Taylor expanded in a around 0 95.8%

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

      1. +-commutative 95.8%

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

      2. mul-1-neg 95.8%

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

      3. unsub-neg 95.8%

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

   7. Simplified 95.8%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-119}:\\ \;\;\;\;\frac{c \cdot -2}{b + \left(b - \left(a \cdot \frac{c}{b}\right) \cdot 2\right)}\\ \mathbf{elif}\;b \leq 10^{-88}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{-4 \cdot \left(c \cdot a\right)}\right)\\ \mathbf{elif}\;b \leq 1100:\\ \;\;\;\;\frac{b \cdot -2 + \frac{c \cdot a}{b} \cdot 2}{a \cdot 2}\\ \mathbf{elif}\;b \leq 13000000:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{-4 \cdot \left(c \cdot a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{-94}:\\ \;\;\;\;\frac{c \cdot -2}{b + \left(b - \left(a \cdot \frac{c}{b}\right) \cdot 2\right)}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-127}:\\ \;\;\;\;\frac{c \cdot -2}{b - \sqrt{a \cdot \left(c \cdot -4\right)}}\\ \mathbf{elif}\;b \leq 1100:\\ \;\;\;\;\frac{b \cdot -2 + \frac{c \cdot a}{b} \cdot 2}{a \cdot 2}\\ \mathbf{elif}\;b \leq 40000000:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{-4 \cdot \left(c \cdot a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.65e-94)
   (/ (* c -2.0) (+ b (- b (* (* a (/ c b)) 2.0))))
   (if (<= b 7e-127)
     (/ (* c -2.0) (- b (sqrt (* a (* c -4.0)))))
     (if (<= b 1100.0)
       (/ (+ (* b -2.0) (* (/ (* c a) b) 2.0)) (* a 2.0))
       (if (<= b 40000000.0)
         (* (/ -0.5 a) (+ b (sqrt (* -4.0 (* c a)))))
         (- (/ c b) (/ b a)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.65e-94) {
		tmp = (c * -2.0) / (b + (b - ((a * (c / b)) * 2.0)));
	} else if (b <= 7e-127) {
		tmp = (c * -2.0) / (b - sqrt((a * (c * -4.0))));
	} else if (b <= 1100.0) {
		tmp = ((b * -2.0) + (((c * a) / b) * 2.0)) / (a * 2.0);
	} else if (b <= 40000000.0) {
		tmp = (-0.5 / a) * (b + sqrt((-4.0 * (c * a))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.65d-94)) then
        tmp = (c * (-2.0d0)) / (b + (b - ((a * (c / b)) * 2.0d0)))
    else if (b <= 7d-127) then
        tmp = (c * (-2.0d0)) / (b - sqrt((a * (c * (-4.0d0)))))
    else if (b <= 1100.0d0) then
        tmp = ((b * (-2.0d0)) + (((c * a) / b) * 2.0d0)) / (a * 2.0d0)
    else if (b <= 40000000.0d0) then
        tmp = ((-0.5d0) / a) * (b + sqrt(((-4.0d0) * (c * a))))
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.65e-94) {
		tmp = (c * -2.0) / (b + (b - ((a * (c / b)) * 2.0)));
	} else if (b <= 7e-127) {
		tmp = (c * -2.0) / (b - Math.sqrt((a * (c * -4.0))));
	} else if (b <= 1100.0) {
		tmp = ((b * -2.0) + (((c * a) / b) * 2.0)) / (a * 2.0);
	} else if (b <= 40000000.0) {
		tmp = (-0.5 / a) * (b + Math.sqrt((-4.0 * (c * a))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.65e-94:
		tmp = (c * -2.0) / (b + (b - ((a * (c / b)) * 2.0)))
	elif b <= 7e-127:
		tmp = (c * -2.0) / (b - math.sqrt((a * (c * -4.0))))
	elif b <= 1100.0:
		tmp = ((b * -2.0) + (((c * a) / b) * 2.0)) / (a * 2.0)
	elif b <= 40000000.0:
		tmp = (-0.5 / a) * (b + math.sqrt((-4.0 * (c * a))))
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.65e-94)
		tmp = Float64(Float64(c * -2.0) / Float64(b + Float64(b - Float64(Float64(a * Float64(c / b)) * 2.0))));
	elseif (b <= 7e-127)
		tmp = Float64(Float64(c * -2.0) / Float64(b - sqrt(Float64(a * Float64(c * -4.0)))));
	elseif (b <= 1100.0)
		tmp = Float64(Float64(Float64(b * -2.0) + Float64(Float64(Float64(c * a) / b) * 2.0)) / Float64(a * 2.0));
	elseif (b <= 40000000.0)
		tmp = Float64(Float64(-0.5 / a) * Float64(b + sqrt(Float64(-4.0 * Float64(c * a)))));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.65e-94)
		tmp = (c * -2.0) / (b + (b - ((a * (c / b)) * 2.0)));
	elseif (b <= 7e-127)
		tmp = (c * -2.0) / (b - sqrt((a * (c * -4.0))));
	elseif (b <= 1100.0)
		tmp = ((b * -2.0) + (((c * a) / b) * 2.0)) / (a * 2.0);
	elseif (b <= 40000000.0)
		tmp = (-0.5 / a) * (b + sqrt((-4.0 * (c * a))));
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.65e-94], N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[(b - N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7e-127], N[(N[(c * -2.0), $MachinePrecision] / N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1100.0], N[(N[(N[(b * -2.0), $MachinePrecision] + N[(N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 40000000.0], N[(N[(-0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -1.6500000000000001e-94

   1. Initial program 17.9%

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

      1. div-sub 16.5%

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

      2. sub-neg 16.5%

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

      3. neg-mul-1 16.5%

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

      4. *-commutative 16.5%

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

      5. associate-/l* 15.9%

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

      6. distribute-neg-frac 15.9%

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

      7. neg-mul-1 15.9%

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

      8. *-commutative 15.9%

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

      9. associate-/l* 16.5%

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

      10. distribute-rgt-out 17.9%

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

      11. associate-/r* 17.9%

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

      12. metadata-eval 17.9%

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

      13. sub-neg 17.9%

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

      14. +-commutative 17.9%

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

   3. Simplified 17.9%

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

      1. clear-num 17.9%

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

      2. flip-+ 17.0%

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

      3. frac-times 14.2%

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

      4. add-sqr-sqrt 14.3%

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

      5. *-un-lft-identity 14.3%

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

      6. pow2 14.3%

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

      7. pow2 14.3%

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

      8. div-inv 14.3%

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

      9. metadata-eval 14.3%

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

      10. pow2 14.3%

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

   6. Applied egg-rr 14.3%

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

      1. associate-/r* 17.1%

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

      2. *-commutative 17.1%

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

   8. Simplified 17.1%

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

   9. Taylor expanded in b around 0 70.9%

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

       1. *-commutative 70.9%

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

   11. Simplified 70.9%

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

   12. Taylor expanded in b around -inf 84.9%

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

       1. neg-mul-1 84.9%

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

       2. +-commutative 84.9%

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

       3. unsub-neg 84.9%

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

       4. *-commutative 84.9%

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

       5. associate-/l* 89.1%

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

   14. Simplified 89.1%

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

   if -1.6500000000000001e-94 < b < 6.99999999999999979e-127

   1. Initial program 75.6%

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

      1. div-sub 75.6%

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

      2. sub-neg 75.6%

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

      3. neg-mul-1 75.6%

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

      4. *-commutative 75.6%

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

      5. associate-/l* 75.6%

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

      6. distribute-neg-frac 75.6%

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

      7. neg-mul-1 75.6%

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

      8. *-commutative 75.6%

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

      9. associate-/l* 75.5%

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

      10. distribute-rgt-out 75.5%

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

      11. associate-/r* 75.5%

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

      12. metadata-eval 75.5%

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

      13. sub-neg 75.5%

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

      14. +-commutative 75.5%

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

   3. Simplified 75.5%

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

      1. clear-num 75.5%

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

      2. flip-+ 74.5%

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

      3. frac-times 64.4%

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

      4. add-sqr-sqrt 64.5%

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

      5. *-un-lft-identity 64.5%

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

      6. pow2 64.5%

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

      7. pow2 64.5%

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

      8. div-inv 64.5%

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

      9. metadata-eval 64.5%

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

      10. pow2 64.5%

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

   6. Applied egg-rr 64.5%

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

      1. associate-/r* 74.8%

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

      2. *-commutative 74.8%

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

   8. Simplified 74.8%

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

   9. Taylor expanded in b around 0 78.4%

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

       1. *-commutative 78.4%

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

   11. Simplified 78.4%

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

   12. Taylor expanded in a around inf 76.3%

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

       1. *-commutative 76.3%

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

       2. *-commutative 76.3%

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

       3. *-commutative 76.3%

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

       4. associate-*r* 76.3%

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

   14. Simplified 76.3%

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

   if 6.99999999999999979e-127 < b < 1100

   1. Initial program 92.1%

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

      1. *-commutative 92.1%

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

      2. *-commutative 92.1%

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

      3. sqr-neg 92.1%

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

      4. *-commutative 92.1%

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

      5. sqr-neg 92.1%

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

      6. *-commutative 92.1%

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

      7. associate-*r* 92.1%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in b around inf 62.6%

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

   if 1100 < b < 4e7

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. sub-neg 100.0%

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

      3. neg-mul-1 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-/l* 100.0%

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

      6. distribute-neg-frac 100.0%

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

      7. neg-mul-1 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-/l* 99.7%

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

      10. distribute-rgt-out 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

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

      13. sub-neg 99.7%

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

      14. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in a around inf 99.7%

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

      1. *-commutative 99.7%

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

   7. Simplified 99.7%

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

   if 4e7 < b

   1. Initial program 56.9%

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

      1. div-sub 56.9%

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

      2. sub-neg 56.9%

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

      3. neg-mul-1 56.9%

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

      4. *-commutative 56.9%

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

      5. associate-/l* 56.8%

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

      6. distribute-neg-frac 56.8%

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

      7. neg-mul-1 56.8%

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

      8. *-commutative 56.8%

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

      9. associate-/l* 56.7%

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

      10. distribute-rgt-out 56.7%

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

      11. associate-/r* 56.7%

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

      12. metadata-eval 56.7%

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

      13. sub-neg 56.7%

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

      14. +-commutative 56.7%

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

   3. Simplified 56.9%

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

   5. Taylor expanded in a around 0 95.8%

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

      1. +-commutative 95.8%

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

      2. mul-1-neg 95.8%

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

      3. unsub-neg 95.8%

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

   7. Simplified 95.8%

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{-94}:\\ \;\;\;\;\frac{c \cdot -2}{b + \left(b - \left(a \cdot \frac{c}{b}\right) \cdot 2\right)}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-127}:\\ \;\;\;\;\frac{c \cdot -2}{b - \sqrt{a \cdot \left(c \cdot -4\right)}}\\ \mathbf{elif}\;b \leq 1100:\\ \;\;\;\;\frac{b \cdot -2 + \frac{c \cdot a}{b} \cdot 2}{a \cdot 2}\\ \mathbf{elif}\;b \leq 40000000:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{-4 \cdot \left(c \cdot a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \frac{{\left(a \cdot \left(c \cdot -4\right)\right)}^{0.5}}{a}\\ \mathbf{if}\;b \leq -1.65 \cdot 10^{-94}:\\ \;\;\;\;\frac{c \cdot -2}{b + \left(b - \left(a \cdot \frac{c}{b}\right) \cdot 2\right)}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-130}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 820:\\ \;\;\;\;\frac{b \cdot -2 + \frac{c \cdot a}{b} \cdot 2}{a \cdot 2}\\ \mathbf{elif}\;b \leq 11000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* -0.5 (/ (pow (* a (* c -4.0)) 0.5) a))))
   (if (<= b -1.65e-94)
     (/ (* c -2.0) (+ b (- b (* (* a (/ c b)) 2.0))))
     (if (<= b 8.5e-130)
       t_0
       (if (<= b 820.0)
         (/ (+ (* b -2.0) (* (/ (* c a) b) 2.0)) (* a 2.0))
         (if (<= b 11000000.0) t_0 (- (/ c b) (/ b a))))))))

C

double code(double a, double b, double c) {
	double t_0 = -0.5 * (pow((a * (c * -4.0)), 0.5) / a);
	double tmp;
	if (b <= -1.65e-94) {
		tmp = (c * -2.0) / (b + (b - ((a * (c / b)) * 2.0)));
	} else if (b <= 8.5e-130) {
		tmp = t_0;
	} else if (b <= 820.0) {
		tmp = ((b * -2.0) + (((c * a) / b) * 2.0)) / (a * 2.0);
	} else if (b <= 11000000.0) {
		tmp = t_0;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-0.5d0) * (((a * (c * (-4.0d0))) ** 0.5d0) / a)
    if (b <= (-1.65d-94)) then
        tmp = (c * (-2.0d0)) / (b + (b - ((a * (c / b)) * 2.0d0)))
    else if (b <= 8.5d-130) then
        tmp = t_0
    else if (b <= 820.0d0) then
        tmp = ((b * (-2.0d0)) + (((c * a) / b) * 2.0d0)) / (a * 2.0d0)
    else if (b <= 11000000.0d0) then
        tmp = t_0
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -0.5 * (Math.pow((a * (c * -4.0)), 0.5) / a);
	double tmp;
	if (b <= -1.65e-94) {
		tmp = (c * -2.0) / (b + (b - ((a * (c / b)) * 2.0)));
	} else if (b <= 8.5e-130) {
		tmp = t_0;
	} else if (b <= 820.0) {
		tmp = ((b * -2.0) + (((c * a) / b) * 2.0)) / (a * 2.0);
	} else if (b <= 11000000.0) {
		tmp = t_0;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = -0.5 * (math.pow((a * (c * -4.0)), 0.5) / a)
	tmp = 0
	if b <= -1.65e-94:
		tmp = (c * -2.0) / (b + (b - ((a * (c / b)) * 2.0)))
	elif b <= 8.5e-130:
		tmp = t_0
	elif b <= 820.0:
		tmp = ((b * -2.0) + (((c * a) / b) * 2.0)) / (a * 2.0)
	elif b <= 11000000.0:
		tmp = t_0
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(-0.5 * Float64((Float64(a * Float64(c * -4.0)) ^ 0.5) / a))
	tmp = 0.0
	if (b <= -1.65e-94)
		tmp = Float64(Float64(c * -2.0) / Float64(b + Float64(b - Float64(Float64(a * Float64(c / b)) * 2.0))));
	elseif (b <= 8.5e-130)
		tmp = t_0;
	elseif (b <= 820.0)
		tmp = Float64(Float64(Float64(b * -2.0) + Float64(Float64(Float64(c * a) / b) * 2.0)) / Float64(a * 2.0));
	elseif (b <= 11000000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = -0.5 * (((a * (c * -4.0)) ^ 0.5) / a);
	tmp = 0.0;
	if (b <= -1.65e-94)
		tmp = (c * -2.0) / (b + (b - ((a * (c / b)) * 2.0)));
	elseif (b <= 8.5e-130)
		tmp = t_0;
	elseif (b <= 820.0)
		tmp = ((b * -2.0) + (((c * a) / b) * 2.0)) / (a * 2.0);
	elseif (b <= 11000000.0)
		tmp = t_0;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-0.5 * N[(N[Power[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.65e-94], N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[(b - N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.5e-130], t$95$0, If[LessEqual[b, 820.0], N[(N[(N[(b * -2.0), $MachinePrecision] + N[(N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 11000000.0], t$95$0, N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.6500000000000001e-94

   1. Initial program 17.9%

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

      1. div-sub 16.5%

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

      2. sub-neg 16.5%

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

      3. neg-mul-1 16.5%

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

      4. *-commutative 16.5%

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

      5. associate-/l* 15.9%

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

      6. distribute-neg-frac 15.9%

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

      7. neg-mul-1 15.9%

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

      8. *-commutative 15.9%

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

      9. associate-/l* 16.5%

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

      10. distribute-rgt-out 17.9%

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

      11. associate-/r* 17.9%

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

      12. metadata-eval 17.9%

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

      13. sub-neg 17.9%

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

      14. +-commutative 17.9%

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

   3. Simplified 17.9%

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

      1. clear-num 17.9%

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

      2. flip-+ 17.0%

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

      3. frac-times 14.2%

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

      4. add-sqr-sqrt 14.3%

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

      5. *-un-lft-identity 14.3%

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

      6. pow2 14.3%

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

      7. pow2 14.3%

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

      8. div-inv 14.3%

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

      9. metadata-eval 14.3%

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

      10. pow2 14.3%

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

   6. Applied egg-rr 14.3%

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

      1. associate-/r* 17.1%

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

      2. *-commutative 17.1%

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

   8. Simplified 17.1%

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

   9. Taylor expanded in b around 0 70.9%

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

       1. *-commutative 70.9%

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

   11. Simplified 70.9%

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

   12. Taylor expanded in b around -inf 84.9%

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

       1. neg-mul-1 84.9%

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

       2. +-commutative 84.9%

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

       3. unsub-neg 84.9%

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

       4. *-commutative 84.9%

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

       5. associate-/l* 89.1%

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

   14. Simplified 89.1%

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

   if -1.6500000000000001e-94 < b < 8.50000000000000033e-130 or 820 < b < 1.1e7

   1. Initial program 78.9%

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

      1. div-sub 78.9%

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

      2. sub-neg 78.9%

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

      3. neg-mul-1 78.9%

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

      4. *-commutative 78.9%

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

      5. associate-/l* 78.9%

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

      6. distribute-neg-frac 78.9%

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

      7. neg-mul-1 78.9%

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

      8. *-commutative 78.9%

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

      9. associate-/l* 78.8%

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

      10. distribute-rgt-out 78.8%

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

      11. associate-/r* 78.8%

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

      12. metadata-eval 78.8%

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

      13. sub-neg 78.8%

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

      14. +-commutative 78.8%

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

   3. Simplified 78.8%

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

      1. add-sqr-sqrt 78.5%

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

      2. pow2 78.5%

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

      3. pow1/2 78.5%

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

      4. sqrt-pow1 78.7%

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

      5. pow2 78.7%

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

      6. metadata-eval 78.7%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + {\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{\color{blue}{0.25}}\right)}^{2}\right) \]

   6. Applied egg-rr 78.7%

      \[\leadsto \frac{-0.5}{a} \cdot \left(b + \color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right) \]

   7. Taylor expanded in c around inf 53.8%

      \[\leadsto \frac{-0.5}{a} \cdot \color{blue}{\left(b + {\left(e^{0.25 \cdot \left(\log \left(-4 \cdot a\right) + -1 \cdot \log \left(\frac{1}{c}\right)\right)}\right)}^{2}\right)} \]
   8. Step-by-step derivation

      1. unpow2 53.8%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + \color{blue}{e^{0.25 \cdot \left(\log \left(-4 \cdot a\right) + -1 \cdot \log \left(\frac{1}{c}\right)\right)} \cdot e^{0.25 \cdot \left(\log \left(-4 \cdot a\right) + -1 \cdot \log \left(\frac{1}{c}\right)\right)}}\right) \]

      2. exp-prod 52.7%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + \color{blue}{{\left(e^{0.25}\right)}^{\left(\log \left(-4 \cdot a\right) + -1 \cdot \log \left(\frac{1}{c}\right)\right)}} \cdot e^{0.25 \cdot \left(\log \left(-4 \cdot a\right) + -1 \cdot \log \left(\frac{1}{c}\right)\right)}\right) \]

      3. exp-prod 51.7%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + {\left(e^{0.25}\right)}^{\left(\log \left(-4 \cdot a\right) + -1 \cdot \log \left(\frac{1}{c}\right)\right)} \cdot \color{blue}{{\left(e^{0.25}\right)}^{\left(\log \left(-4 \cdot a\right) + -1 \cdot \log \left(\frac{1}{c}\right)\right)}}\right) \]

      4. pow-sqr 51.7%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + \color{blue}{{\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log \left(-4 \cdot a\right) + -1 \cdot \log \left(\frac{1}{c}\right)\right)\right)}}\right) \]

      5. *-commutative 51.7%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + {\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log \color{blue}{\left(a \cdot -4\right)} + -1 \cdot \log \left(\frac{1}{c}\right)\right)\right)}\right) \]

      6. mul-1-neg 51.7%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + {\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log \left(a \cdot -4\right) + \color{blue}{\left(-\log \left(\frac{1}{c}\right)\right)}\right)\right)}\right) \]

      7. log-rec 51.7%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + {\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log \left(a \cdot -4\right) + \left(-\color{blue}{\left(-\log c\right)}\right)\right)\right)}\right) \]

      8. remove-double-neg 51.7%

         \[\leadsto \frac{-0.5}{a} \cdot \left(b + {\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log \left(a \cdot -4\right) + \color{blue}{\log c}\right)\right)}\right) \]

   9. Simplified 51.7%

      \[\leadsto \frac{-0.5}{a} \cdot \color{blue}{\left(b + {\left(e^{0.25}\right)}^{\left(2 \cdot \left(\log \left(a \cdot -4\right) + \log c\right)\right)}\right)} \]

   10. Taylor expanded in b around 0 54.0%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{e^{0.5 \cdot \left(\log c + \log \left(-4 \cdot a\right)\right)}}{a}} \]
   11. Step-by-step derivation

       1. *-commutative 54.0%

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

       2. *-commutative 54.0%

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

       3. log-prod 73.4%

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

       4. associate-*r* 73.4%

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

       5. *-commutative 73.4%

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

       6. associate-*r* 73.4%

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

       7. exp-to-pow 78.4%

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

   12. Simplified 78.4%

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

   if 8.50000000000000033e-130 < b < 820

   1. Initial program 88.7%

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

      1. *-commutative 88.7%

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

      2. *-commutative 88.7%

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

      3. sqr-neg 88.7%

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

      4. *-commutative 88.7%

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

      5. sqr-neg 88.7%

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

      6. *-commutative 88.7%

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

      7. associate-*r* 88.7%

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

   3. Simplified 88.7%

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

   5. Taylor expanded in b around inf 60.3%

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

   if 1.1e7 < b

   1. Initial program 56.9%

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

      1. div-sub 56.9%

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

      2. sub-neg 56.9%

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

      3. neg-mul-1 56.9%

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

      4. *-commutative 56.9%

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

      5. associate-/l* 56.8%

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

      6. distribute-neg-frac 56.8%

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

      7. neg-mul-1 56.8%

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

      8. *-commutative 56.8%

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

      9. associate-/l* 56.7%

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

      10. distribute-rgt-out 56.7%

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

      11. associate-/r* 56.7%

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

      12. metadata-eval 56.7%

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

      13. sub-neg 56.7%

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

      14. +-commutative 56.7%

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

   3. Simplified 56.9%

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

   5. Taylor expanded in a around 0 95.8%

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

      1. +-commutative 95.8%

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

      2. mul-1-neg 95.8%

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

      3. unsub-neg 95.8%

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

   7. Simplified 95.8%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{-94}:\\ \;\;\;\;\frac{c \cdot -2}{b + \left(b - \left(a \cdot \frac{c}{b}\right) \cdot 2\right)}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-130}:\\ \;\;\;\;-0.5 \cdot \frac{{\left(a \cdot \left(c \cdot -4\right)\right)}^{0.5}}{a}\\ \mathbf{elif}\;b \leq 820:\\ \;\;\;\;\frac{b \cdot -2 + \frac{c \cdot a}{b} \cdot 2}{a \cdot 2}\\ \mathbf{elif}\;b \leq 11000000:\\ \;\;\;\;-0.5 \cdot \frac{{\left(a \cdot \left(c \cdot -4\right)\right)}^{0.5}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{-93}:\\ \;\;\;\;\frac{c \cdot -2}{b + \left(b - \left(a \cdot \frac{c}{b}\right) \cdot 2\right)}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+129}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.7e-93)
   (/ (* c -2.0) (+ b (- b (* (* a (/ c b)) 2.0))))
   (if (<= b 4e+129)
     (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* c a))))) (* a 2.0))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.7e-93) {
		tmp = (c * -2.0) / (b + (b - ((a * (c / b)) * 2.0)));
	} else if (b <= 4e+129) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.7d-93)) then
        tmp = (c * (-2.0d0)) / (b + (b - ((a * (c / b)) * 2.0d0)))
    else if (b <= 4d+129) then
        tmp = (-b - sqrt(((b * b) - (4.0d0 * (c * a))))) / (a * 2.0d0)
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.7e-93) {
		tmp = (c * -2.0) / (b + (b - ((a * (c / b)) * 2.0)));
	} else if (b <= 4e+129) {
		tmp = (-b - Math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.7e-93:
		tmp = (c * -2.0) / (b + (b - ((a * (c / b)) * 2.0)))
	elif b <= 4e+129:
		tmp = (-b - math.sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0)
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.7e-93)
		tmp = Float64(Float64(c * -2.0) / Float64(b + Float64(b - Float64(Float64(a * Float64(c / b)) * 2.0))));
	elseif (b <= 4e+129)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a))))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.7e-93)
		tmp = (c * -2.0) / (b + (b - ((a * (c / b)) * 2.0)));
	elseif (b <= 4e+129)
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (a * 2.0);
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.7e-93], N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[(b - N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4e+129], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.70000000000000001e-93

   1. Initial program 17.2%

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

      1. div-sub 15.8%

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

      2. sub-neg 15.8%

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

      3. neg-mul-1 15.8%

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

      4. *-commutative 15.8%

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

      5. associate-/l* 15.3%

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

      6. distribute-neg-frac 15.3%

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

      7. neg-mul-1 15.3%

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

      8. *-commutative 15.3%

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

      9. associate-/l* 15.8%

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

      10. distribute-rgt-out 17.2%

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

      11. associate-/r* 17.2%

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

      12. metadata-eval 17.2%

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

      13. sub-neg 17.2%

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

      14. +-commutative 17.2%

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

   3. Simplified 17.2%

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

      1. clear-num 17.2%

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

      2. flip-+ 16.3%

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

      3. frac-times 14.3%

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

      4. add-sqr-sqrt 14.3%

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

      5. *-un-lft-identity 14.3%

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

      6. pow2 14.3%

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

      7. pow2 14.3%

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

      8. div-inv 14.3%

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

      9. metadata-eval 14.3%

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

      10. pow2 14.3%

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

   6. Applied egg-rr 14.3%

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

      1. associate-/r* 16.4%

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

      2. *-commutative 16.4%

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

   8. Simplified 16.4%

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

   9. Taylor expanded in b around 0 70.6%

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

       1. *-commutative 70.6%

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

   11. Simplified 70.6%

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

   12. Taylor expanded in b around -inf 85.2%

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

       1. neg-mul-1 85.2%

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

       2. +-commutative 85.2%

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

       3. unsub-neg 85.2%

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

       4. *-commutative 85.2%

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

       5. associate-/l* 89.4%

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

   14. Simplified 89.4%

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

   if -1.70000000000000001e-93 < b < 4e129

   1. Initial program 85.0%

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

   if 4e129 < b

   1. Initial program 38.6%

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

      1. div-sub 38.6%

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

      2. sub-neg 38.6%

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

      3. neg-mul-1 38.6%

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

      4. *-commutative 38.6%

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

      5. associate-/l* 38.6%

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

      6. distribute-neg-frac 38.6%

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

      7. neg-mul-1 38.6%

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

      8. *-commutative 38.6%

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

      9. associate-/l* 38.5%

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

      10. distribute-rgt-out 38.5%

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

      11. associate-/r* 38.5%

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

      12. metadata-eval 38.5%

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

      13. sub-neg 38.5%

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

      14. +-commutative 38.5%

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

   3. Simplified 38.7%

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

   5. Taylor expanded in a around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{-93}:\\ \;\;\;\;\frac{c \cdot -2}{b + \left(b - \left(a \cdot \frac{c}{b}\right) \cdot 2\right)}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+129}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.6% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-292}:\\ \;\;\;\;\frac{c \cdot -2}{b + \left(b - \left(a \cdot \frac{c}{b}\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5e-292)
   (/ (* c -2.0) (+ b (- b (* (* a (/ c b)) 2.0))))
   (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5e-292) {
		tmp = (c * -2.0) / (b + (b - ((a * (c / b)) * 2.0)));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 5d-292) then
        tmp = (c * (-2.0d0)) / (b + (b - ((a * (c / b)) * 2.0d0)))
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5e-292) {
		tmp = (c * -2.0) / (b + (b - ((a * (c / b)) * 2.0)));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5e-292:
		tmp = (c * -2.0) / (b + (b - ((a * (c / b)) * 2.0)))
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5e-292)
		tmp = Float64(Float64(c * -2.0) / Float64(b + Float64(b - Float64(Float64(a * Float64(c / b)) * 2.0))));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5e-292)
		tmp = (c * -2.0) / (b + (b - ((a * (c / b)) * 2.0)));
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5e-292], N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[(b - N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.99999999999999981e-292

   1. Initial program 31.4%

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

      1. div-sub 30.4%

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

      2. sub-neg 30.4%

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

      3. neg-mul-1 30.4%

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

      4. *-commutative 30.4%

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

      5. associate-/l* 30.0%

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

      6. distribute-neg-frac 30.0%

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

      7. neg-mul-1 30.0%

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

      8. *-commutative 30.0%

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

      9. associate-/l* 30.3%

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

      10. distribute-rgt-out 31.3%

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

      11. associate-/r* 31.3%

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

      12. metadata-eval 31.3%

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

      13. sub-neg 31.3%

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

      14. +-commutative 31.3%

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

   3. Simplified 31.4%

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

      1. clear-num 31.4%

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

      2. flip-+ 30.5%

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

      3. frac-times 26.9%

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

      4. add-sqr-sqrt 26.9%

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

      5. *-un-lft-identity 26.9%

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

      6. pow2 26.9%

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

      7. pow2 26.9%

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

      8. div-inv 26.9%

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

      9. metadata-eval 26.9%

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

      10. pow2 26.9%

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

   6. Applied egg-rr 26.9%

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

      1. associate-/r* 30.6%

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

      2. *-commutative 30.6%

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

   8. Simplified 30.6%

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

   9. Taylor expanded in b around 0 72.2%

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

       1. *-commutative 72.2%

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

   11. Simplified 72.2%

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

   12. Taylor expanded in b around -inf 66.2%

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

       1. neg-mul-1 66.2%

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

       2. +-commutative 66.2%

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

       3. unsub-neg 66.2%

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

       4. *-commutative 66.2%

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

       5. associate-/l* 69.3%

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

   14. Simplified 69.3%

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

   if 4.99999999999999981e-292 < b

   1. Initial program 70.1%

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

      1. div-sub 70.0%

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

      2. sub-neg 70.0%

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

      3. neg-mul-1 70.0%

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

      4. *-commutative 70.0%

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

      5. associate-/l* 70.0%

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

      6. distribute-neg-frac 70.0%

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

      7. neg-mul-1 70.0%

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

      8. *-commutative 70.0%

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

      9. associate-/l* 69.8%

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

      10. distribute-rgt-out 69.9%

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

      11. associate-/r* 69.9%

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

      12. metadata-eval 69.9%

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

      13. sub-neg 69.9%

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

      14. +-commutative 69.9%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in a around 0 70.7%

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

      1. +-commutative 70.7%

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

      2. mul-1-neg 70.7%

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

      3. unsub-neg 70.7%

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

   7. Simplified 70.7%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-292}:\\ \;\;\;\;\frac{c \cdot -2}{b + \left(b - \left(a \cdot \frac{c}{b}\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.6% accurate, 9.7× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (/ c (- b)) (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = c / -b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = c / -b
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = c / -b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = c / -b
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(c / Float64(-b));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = c / -b;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(c / (-b)), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 31.6%

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

      1. div-sub 30.6%

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

      2. sub-neg 30.6%

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

      3. neg-mul-1 30.6%

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

      4. *-commutative 30.6%

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

      5. associate-/l* 30.2%

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

      6. distribute-neg-frac 30.2%

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

      7. neg-mul-1 30.2%

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

      8. *-commutative 30.2%

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

      9. associate-/l* 30.5%

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

      10. distribute-rgt-out 31.6%

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

      11. associate-/r* 31.6%

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

      12. metadata-eval 31.6%

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

      13. sub-neg 31.6%

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

      14. +-commutative 31.6%

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

   3. Simplified 31.6%

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

   5. Taylor expanded in b around -inf 69.3%

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

      1. mul-1-neg 69.3%

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

      2. distribute-neg-frac2 69.3%

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

   7. Simplified 69.3%

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

   if -4.999999999999985e-310 < b

   1. Initial program 69.5%

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

      1. div-sub 69.5%

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

      2. sub-neg 69.5%

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

      3. neg-mul-1 69.5%

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

      4. *-commutative 69.5%

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

      5. associate-/l* 69.4%

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

      6. distribute-neg-frac 69.4%

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

      7. neg-mul-1 69.4%

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

      8. *-commutative 69.4%

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

      9. associate-/l* 69.3%

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

      10. distribute-rgt-out 69.3%

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

      11. associate-/r* 69.3%

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

      12. metadata-eval 69.3%

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

      13. sub-neg 69.3%

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

      14. +-commutative 69.3%

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

   3. Simplified 69.4%

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

   5. Taylor expanded in a around 0 70.2%

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

      1. +-commutative 70.2%

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

      2. mul-1-neg 70.2%

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

      3. unsub-neg 70.2%

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

   7. Simplified 70.2%

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

4. Final simplification 69.8%

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

Alternative 8: 66.4% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-308}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.1e-308) (/ c (- b)) (/ b (- a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.1e-308) {
		tmp = c / -b;
	} else {
		tmp = b / -a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.1d-308)) then
        tmp = c / -b
    else
        tmp = b / -a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.1e-308) {
		tmp = c / -b;
	} else {
		tmp = b / -a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.1e-308:
		tmp = c / -b
	else:
		tmp = b / -a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.1e-308)
		tmp = Float64(c / Float64(-b));
	else
		tmp = Float64(b / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.1e-308)
		tmp = c / -b;
	else
		tmp = b / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.1e-308], N[(c / (-b)), $MachinePrecision], N[(b / (-a)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.1000000000000001e-308

   1. Initial program 31.6%

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

      1. div-sub 30.6%

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

      2. sub-neg 30.6%

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

      3. neg-mul-1 30.6%

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

      4. *-commutative 30.6%

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

      5. associate-/l* 30.2%

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

      6. distribute-neg-frac 30.2%

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

      7. neg-mul-1 30.2%

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

      8. *-commutative 30.2%

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

      9. associate-/l* 30.5%

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

      10. distribute-rgt-out 31.6%

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

      11. associate-/r* 31.6%

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

      12. metadata-eval 31.6%

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

      13. sub-neg 31.6%

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

      14. +-commutative 31.6%

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

   3. Simplified 31.6%

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

   5. Taylor expanded in b around -inf 69.3%

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

      1. mul-1-neg 69.3%

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

      2. distribute-neg-frac2 69.3%

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

   7. Simplified 69.3%

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

   if -1.1000000000000001e-308 < b

   1. Initial program 69.5%

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

      1. div-sub 69.5%

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

      2. sub-neg 69.5%

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

      3. neg-mul-1 69.5%

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

      4. *-commutative 69.5%

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

      5. associate-/l* 69.4%

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

      6. distribute-neg-frac 69.4%

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

      7. neg-mul-1 69.4%

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

      8. *-commutative 69.4%

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

      9. associate-/l* 69.3%

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

      10. distribute-rgt-out 69.3%

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

      11. associate-/r* 69.3%

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

      12. metadata-eval 69.3%

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

      13. sub-neg 69.3%

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

      14. +-commutative 69.3%

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

   3. Simplified 69.4%

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

   5. Taylor expanded in a around 0 69.4%

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

      1. associate-*r/ 69.4%

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

      2. mul-1-neg 69.4%

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

   7. Simplified 69.4%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-308}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.3% accurate, 29.0× 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 / Float64(-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 50.6%

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

   1. div-sub 50.0%

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

   2. sub-neg 50.0%

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

   3. neg-mul-1 50.0%

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

   4. *-commutative 50.0%

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

   5. associate-/l* 49.8%

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

   6. distribute-neg-frac 49.8%

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

   7. neg-mul-1 49.8%

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

   8. *-commutative 49.8%

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

   9. associate-/l* 49.9%

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

   10. distribute-rgt-out 50.5%

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

   11. associate-/r* 50.5%

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

   12. metadata-eval 50.5%

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

   13. sub-neg 50.5%

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

   14. +-commutative 50.5%

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

3. Simplified 50.5%

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

5. Taylor expanded in b around -inf 35.8%

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

   1. mul-1-neg 35.8%

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

   2. distribute-neg-frac2 35.8%

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

7. Simplified 35.8%

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

8. Final simplification 35.8%

   \[\leadsto \frac{c}{-b} \]
9. Add Preprocessing

Alternative 10: 2.6% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.6%

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

   1. div-sub 50.0%

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

   2. sub-neg 50.0%

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

   3. neg-mul-1 50.0%

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

   4. *-commutative 50.0%

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

   5. associate-/l* 49.8%

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

   6. distribute-neg-frac 49.8%

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

   7. neg-mul-1 49.8%

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

   8. *-commutative 49.8%

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

   9. associate-/l* 49.9%

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

   10. distribute-rgt-out 50.5%

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

   11. associate-/r* 50.5%

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

   12. metadata-eval 50.5%

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

   13. sub-neg 50.5%

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

   14. +-commutative 50.5%

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

3. Simplified 50.5%

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

6. Applied egg-rr 30.3%

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

7. Taylor expanded in b around -inf 2.6%

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

8. Final simplification 2.6%

   \[\leadsto \frac{b}{a} \]
9. Add Preprocessing

Alternative 11: 10.6% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.6%

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

   1. div-sub 50.0%

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

   2. sub-neg 50.0%

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

   3. neg-mul-1 50.0%

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

   4. *-commutative 50.0%

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

   5. associate-/l* 49.8%

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

   6. distribute-neg-frac 49.8%

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

   7. neg-mul-1 49.8%

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

   8. *-commutative 49.8%

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

   9. associate-/l* 49.9%

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

   10. distribute-rgt-out 50.5%

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

   11. associate-/r* 50.5%

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

   12. metadata-eval 50.5%

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

   13. sub-neg 50.5%

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

   14. +-commutative 50.5%

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

3. Simplified 50.5%

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

5. Taylor expanded in a around 0 35.3%

   \[\leadsto \frac{-0.5}{a} \cdot \left(b + \color{blue}{\left(b + -2 \cdot \frac{a \cdot c}{b}\right)}\right) \]
6. Step-by-step derivation

   1. associate-/l* 36.1%

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

7. Simplified 36.1%

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

8. Taylor expanded in a around inf 12.5%

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

9. Final simplification 12.5%

   \[\leadsto \frac{c}{b} \]
10. Add Preprocessing

Developer target: 70.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + t\_0}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - t\_0}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* 4.0 (* a c))))))
   (if (< b 0.0)
     (/ c (* a (/ (+ (- b) t_0) (* 2.0 a))))
     (/ (- (- b) t_0) (* 2.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (4.0 * (a * c))));
	double tmp;
	if (b < 0.0) {
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-b - t_0) / (2.0 * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - (4.0d0 * (a * c))))
    if (b < 0.0d0) then
        tmp = c / (a * ((-b + t_0) / (2.0d0 * a)))
    else
        tmp = (-b - t_0) / (2.0d0 * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (4.0 * (a * c))));
	double tmp;
	if (b < 0.0) {
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-b - t_0) / (2.0 * a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (4.0 * (a * c))))
	tmp = 0
	if b < 0.0:
		tmp = c / (a * ((-b + t_0) / (2.0 * a)))
	else:
		tmp = (-b - t_0) / (2.0 * a)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))
	tmp = 0.0
	if (b < 0.0)
		tmp = Float64(c / Float64(a * Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a))));
	else
		tmp = Float64(Float64(Float64(-b) - t_0) / Float64(2.0 * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - (4.0 * (a * c))));
	tmp = 0.0;
	if (b < 0.0)
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	else
		tmp = (-b - t_0) / (2.0 * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[b, 0.0], N[(c / N[(a * N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024040 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

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