The quadratic formula (r1)

Percentage Accurate: 52.0% → 85.3%

Time: 14.0s

Alternatives: 8

Speedup: 19.1×

Specification

Math

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

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(Float64(4.0 * 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.0% accurate, 1.0× speedup

Math

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

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(Float64(4.0 * 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot 4\right)\\ \mathbf{if}\;b \leq -4 \cdot 10^{+117}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-116}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - t_0} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 3.3 \cdot 10^{+38}:\\ \;\;\;\;\frac{t_0}{b + \mathsf{hypot}\left(b, \sqrt{\left(c \cdot a\right) \cdot -4}\right)} \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a 4.0))))
   (if (<= b -4e+117)
     (- (/ c b) (/ b a))
     (if (<= b 1.15e-116)
       (/ (- (sqrt (- (* b b) t_0)) b) (* a 2.0))
       (if (<= b 3.3e+38)
         (* (/ t_0 (+ b (hypot b (sqrt (* (* c a) -4.0))))) (/ -0.5 a))
         (/ (- c) b))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * 4.0);
	double tmp;
	if (b <= -4e+117) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.15e-116) {
		tmp = (sqrt(((b * b) - t_0)) - b) / (a * 2.0);
	} else if (b <= 3.3e+38) {
		tmp = (t_0 / (b + hypot(b, sqrt(((c * a) * -4.0))))) * (-0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (a * 4.0);
	double tmp;
	if (b <= -4e+117) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.15e-116) {
		tmp = (Math.sqrt(((b * b) - t_0)) - b) / (a * 2.0);
	} else if (b <= 3.3e+38) {
		tmp = (t_0 / (b + Math.hypot(b, Math.sqrt(((c * a) * -4.0))))) * (-0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c * (a * 4.0)
	tmp = 0
	if b <= -4e+117:
		tmp = (c / b) - (b / a)
	elif b <= 1.15e-116:
		tmp = (math.sqrt(((b * b) - t_0)) - b) / (a * 2.0)
	elif b <= 3.3e+38:
		tmp = (t_0 / (b + math.hypot(b, math.sqrt(((c * a) * -4.0))))) * (-0.5 / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * 4.0))
	tmp = 0.0
	if (b <= -4e+117)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.15e-116)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - t_0)) - b) / Float64(a * 2.0));
	elseif (b <= 3.3e+38)
		tmp = Float64(Float64(t_0 / Float64(b + hypot(b, sqrt(Float64(Float64(c * a) * -4.0))))) * Float64(-0.5 / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = c * (a * 4.0);
	tmp = 0.0;
	if (b <= -4e+117)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.15e-116)
		tmp = (sqrt(((b * b) - t_0)) - b) / (a * 2.0);
	elseif (b <= 3.3e+38)
		tmp = (t_0 / (b + hypot(b, sqrt(((c * a) * -4.0))))) * (-0.5 / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4e+117], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.15e-116], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.3e+38], N[(N[(t$95$0 / N[(b + N[Sqrt[b ^ 2 + N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -4.0000000000000002e117

   1. Initial program 47.7%

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

   2. Taylor expanded in b around -inf 96.8%

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

      1. mul-1-neg 96.8%

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

      2. unsub-neg 96.8%

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

   4. Simplified 96.8%

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

   if -4.0000000000000002e117 < b < 1.15000000000000001e-116

   1. Initial program 86.5%

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

   if 1.15000000000000001e-116 < b < 3.2999999999999999e38

   1. Initial program 49.1%

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

      1. frac-2neg 49.1%

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

      2. div-inv 49.1%

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

   3. Applied egg-rr 48.7%

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

      1. *-commutative 48.7%

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

      2. associate-/r* 48.7%

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

      3. metadata-eval 48.7%

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

   5. Simplified 48.7%

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

      1. flip-- 48.2%

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

      2. hypot-udef 48.2%

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

      3. hypot-udef 48.2%

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

      4. add-sqr-sqrt 48.2%

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

      5. add-sqr-sqrt 48.3%

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

   7. Applied egg-rr 48.3%

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

      1. associate--r+ 66.0%

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

      2. +-inverses 66.0%

         \[\leadsto \frac{\color{blue}{0} - c \cdot \left(a \cdot -4\right)}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \cdot \frac{-0.5}{a} \]

      3. rem-square-sqrt 0.0%

         \[\leadsto \frac{0 - c \cdot \left(a \cdot \color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)}\right)}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \cdot \frac{-0.5}{a} \]

      4. unpow2 0.0%

         \[\leadsto \frac{0 - c \cdot \left(a \cdot \color{blue}{{\left(\sqrt{-4}\right)}^{2}}\right)}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)} \cdot \frac{-0.5}{a} \]

      5. neg-sub0 0.0%

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

      6. distribute-lft-neg-in 0.0%

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

      7. neg-mul-1 0.0%

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

      8. *-commutative 0.0%

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

      9. unpow2 0.0%

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

      10. rem-square-sqrt 66.0%

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

      11. associate-*r* 66.0%

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

      12. neg-mul-1 66.0%

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

      13. distribute-lft-neg-in 66.0%

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

      14. distribute-rgt-neg-in 66.0%

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

      15. metadata-eval 66.0%

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

      16. associate-*r* 66.0%

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

      17. *-commutative 66.0%

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

      18. associate-*r* 66.0%

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

   9. Simplified 66.0%

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

   if 3.2999999999999999e38 < b

   1. Initial program 12.2%

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

   2. Taylor expanded in b around inf 90.5%

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

      1. associate-*r/ 90.5%

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

      2. neg-mul-1 90.5%

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

   4. Simplified 90.5%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+117}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-116}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 3.3 \cdot 10^{+38}:\\ \;\;\;\;\frac{c \cdot \left(a \cdot 4\right)}{b + \mathsf{hypot}\left(b, \sqrt{\left(c \cdot a\right) \cdot -4}\right)} \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2: 84.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+64}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-95} \lor \neg \left(b \leq 3 \cdot 10^{-79}\right) \land b \leq 0.135:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.8e+64)
   (- (/ c b) (/ b a))
   (if (or (<= b 4e-95) (and (not (<= b 3e-79)) (<= b 0.135)))
     (* (/ -0.5 a) (- b (sqrt (+ (* b b) (* c (* a -4.0))))))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e+64) {
		tmp = (c / b) - (b / a);
	} else if ((b <= 4e-95) || (!(b <= 3e-79) && (b <= 0.135))) {
		tmp = (-0.5 / a) * (b - sqrt(((b * b) + (c * (a * -4.0)))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.8d+64)) then
        tmp = (c / b) - (b / a)
    else if ((b <= 4d-95) .or. (.not. (b <= 3d-79)) .and. (b <= 0.135d0)) then
        tmp = ((-0.5d0) / a) * (b - sqrt(((b * b) + (c * (a * (-4.0d0))))))
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e+64) {
		tmp = (c / b) - (b / a);
	} else if ((b <= 4e-95) || (!(b <= 3e-79) && (b <= 0.135))) {
		tmp = (-0.5 / a) * (b - Math.sqrt(((b * b) + (c * (a * -4.0)))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.8e+64:
		tmp = (c / b) - (b / a)
	elif (b <= 4e-95) or (not (b <= 3e-79) and (b <= 0.135)):
		tmp = (-0.5 / a) * (b - math.sqrt(((b * b) + (c * (a * -4.0)))))
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.8e+64)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif ((b <= 4e-95) || (!(b <= 3e-79) && (b <= 0.135)))
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0))))));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.8e+64)
		tmp = (c / b) - (b / a);
	elseif ((b <= 4e-95) || (~((b <= 3e-79)) && (b <= 0.135)))
		tmp = (-0.5 / a) * (b - sqrt(((b * b) + (c * (a * -4.0)))));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.8e+64], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[b, 4e-95], And[N[Not[LessEqual[b, 3e-79]], $MachinePrecision], LessEqual[b, 0.135]]], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.80000000000000024e64

   1. Initial program 55.4%

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

   2. Taylor expanded in b around -inf 97.3%

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

      1. mul-1-neg 97.3%

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

      2. unsub-neg 97.3%

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

   4. Simplified 97.3%

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

   if -2.80000000000000024e64 < b < 3.99999999999999996e-95 or 3e-79 < b < 0.13500000000000001

   1. Initial program 81.4%

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

      1. frac-2neg 81.4%

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

      2. div-inv 81.2%

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

   3. Applied egg-rr 73.0%

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

      1. *-commutative 73.0%

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

      2. associate-/r* 73.0%

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

      3. metadata-eval 73.0%

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

   5. Simplified 73.0%

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

      1. hypot-udef 73.0%

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

      2. add-sqr-sqrt 81.2%

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

   7. Applied egg-rr 81.2%

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

   if 3.99999999999999996e-95 < b < 3e-79 or 0.13500000000000001 < b

   1. Initial program 16.5%

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

   2. Taylor expanded in b around inf 83.2%

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

      1. associate-*r/ 83.2%

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

      2. neg-mul-1 83.2%

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

   4. Simplified 83.2%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+64}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-95} \lor \neg \left(b \leq 3 \cdot 10^{-79}\right) \land b \leq 0.135:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.06 \cdot 10^{+117}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-94} \lor \neg \left(b \leq 6.3 \cdot 10^{-80}\right) \land b \leq 0.135:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.06e+117)
   (- (/ c b) (/ b a))
   (if (or (<= b 1.7e-94) (and (not (<= b 6.3e-80)) (<= b 0.135)))
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.06e+117) {
		tmp = (c / b) - (b / a);
	} else if ((b <= 1.7e-94) || (!(b <= 6.3e-80) && (b <= 0.135))) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.06d+117)) then
        tmp = (c / b) - (b / a)
    else if ((b <= 1.7d-94) .or. (.not. (b <= 6.3d-80)) .and. (b <= 0.135d0)) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.06e+117) {
		tmp = (c / b) - (b / a);
	} else if ((b <= 1.7e-94) || (!(b <= 6.3e-80) && (b <= 0.135))) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.06e+117:
		tmp = (c / b) - (b / a)
	elif (b <= 1.7e-94) or (not (b <= 6.3e-80) and (b <= 0.135)):
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.06e+117)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif ((b <= 1.7e-94) || (!(b <= 6.3e-80) && (b <= 0.135)))
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.06e+117)
		tmp = (c / b) - (b / a);
	elseif ((b <= 1.7e-94) || (~((b <= 6.3e-80)) && (b <= 0.135)))
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.06e+117], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[b, 1.7e-94], And[N[Not[LessEqual[b, 6.3e-80]], $MachinePrecision], LessEqual[b, 0.135]]], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.06e117

   1. Initial program 47.7%

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

   2. Taylor expanded in b around -inf 96.8%

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

      1. mul-1-neg 96.8%

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

      2. unsub-neg 96.8%

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

   4. Simplified 96.8%

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

   if -1.06e117 < b < 1.6999999999999999e-94 or 6.29999999999999966e-80 < b < 0.13500000000000001

   1. Initial program 83.0%

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

   if 1.6999999999999999e-94 < b < 6.29999999999999966e-80 or 0.13500000000000001 < b

   1. Initial program 16.5%

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

   2. Taylor expanded in b around inf 83.2%

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

      1. associate-*r/ 83.2%

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

      2. neg-mul-1 83.2%

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

   4. Simplified 83.2%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.06 \cdot 10^{+117}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-94} \lor \neg \left(b \leq 6.3 \cdot 10^{-80}\right) \land b \leq 0.135:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 81.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-94}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot a\right) \cdot -4} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.4e-78)
   (- (/ c b) (/ b a))
   (if (<= b 2e-94) (/ (- (sqrt (* (* c a) -4.0)) b) (* a 2.0)) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.4e-78) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2e-94) {
		tmp = (sqrt(((c * a) * -4.0)) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.4d-78)) then
        tmp = (c / b) - (b / a)
    else if (b <= 2d-94) then
        tmp = (sqrt(((c * a) * (-4.0d0))) - b) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.4e-78) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2e-94) {
		tmp = (Math.sqrt(((c * a) * -4.0)) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.4e-78:
		tmp = (c / b) - (b / a)
	elif b <= 2e-94:
		tmp = (math.sqrt(((c * a) * -4.0)) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.4e-78)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 2e-94)
		tmp = Float64(Float64(sqrt(Float64(Float64(c * a) * -4.0)) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.4e-78)
		tmp = (c / b) - (b / a);
	elseif (b <= 2e-94)
		tmp = (sqrt(((c * a) * -4.0)) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.4e-78], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e-94], N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.4e-78

   1. Initial program 68.0%

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

   2. Taylor expanded in b around -inf 93.4%

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

      1. mul-1-neg 93.4%

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

      2. unsub-neg 93.4%

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

   4. Simplified 93.4%

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

   if -2.4e-78 < b < 1.9999999999999999e-94

   1. Initial program 80.2%

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

   2. Taylor expanded in b around 0 71.7%

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

      1. *-commutative 71.7%

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

      2. associate-*r* 71.8%

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

   4. Simplified 71.8%

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

      1. expm1-log1p-u 51.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{c \cdot \left(a \cdot -4\right)}}{2 \cdot a}\right)\right)} \]

      2. expm1-udef 16.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{c \cdot \left(a \cdot -4\right)}}{2 \cdot a}\right)} - 1} \]

      3. neg-mul-1 16.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{-1 \cdot b} + \sqrt{c \cdot \left(a \cdot -4\right)}}{2 \cdot a}\right)} - 1 \]

      4. metadata-eval 16.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{-2}{2}} \cdot b + \sqrt{c \cdot \left(a \cdot -4\right)}}{2 \cdot a}\right)} - 1 \]

      5. fma-def 16.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\mathsf{fma}\left(\frac{-2}{2}, b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}{2 \cdot a}\right)} - 1 \]

      6. metadata-eval 16.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(\color{blue}{-1}, b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}{2 \cdot a}\right)} - 1 \]

      7. *-commutative 16.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}{\color{blue}{a \cdot 2}}\right)} - 1 \]

   6. Applied egg-rr 16.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}{a \cdot 2}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 51.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}{a \cdot 2}\right)\right)} \]

      2. expm1-log1p 71.8%

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

      3. rem-log-exp 5.8%

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

      4. fma-udef 5.8%

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

      5. neg-mul-1 5.8%

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

      6. prod-exp 5.8%

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

      7. *-commutative 5.8%

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

      8. prod-exp 5.8%

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

      9. rem-log-exp 71.8%

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

      10. unsub-neg 71.8%

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

      11. associate-*r* 71.7%

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

      12. *-commutative 71.7%

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

   8. Simplified 71.7%

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

   if 1.9999999999999999e-94 < b

   1. Initial program 23.4%

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

   2. Taylor expanded in b around inf 74.3%

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

      1. associate-*r/ 74.3%

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

      2. neg-mul-1 74.3%

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

   4. Simplified 74.3%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-94}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot a\right) \cdot -4} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5: 81.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{-71}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.2e-71)
   (- (/ c b) (/ b a))
   (if (<= b 7e-95) (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0)) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.2e-71) {
		tmp = (c / b) - (b / a);
	} else if (b <= 7e-95) {
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-7.2d-71)) then
        tmp = (c / b) - (b / a)
    else if (b <= 7d-95) then
        tmp = (sqrt((c * (a * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.2e-71) {
		tmp = (c / b) - (b / a);
	} else if (b <= 7e-95) {
		tmp = (Math.sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.2e-71:
		tmp = (c / b) - (b / a)
	elif b <= 7e-95:
		tmp = (math.sqrt((c * (a * -4.0))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.2e-71)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 7e-95)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.2e-71)
		tmp = (c / b) - (b / a);
	elseif (b <= 7e-95)
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.2e-71], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7e-95], N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.2e-71

   1. Initial program 68.0%

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

   2. Taylor expanded in b around -inf 93.4%

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

      1. mul-1-neg 93.4%

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

      2. unsub-neg 93.4%

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

   4. Simplified 93.4%

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

   if -7.2e-71 < b < 6.9999999999999994e-95

   1. Initial program 80.2%

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

   2. Taylor expanded in b around 0 71.7%

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

      1. *-commutative 71.7%

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

      2. associate-*r* 71.8%

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

   4. Simplified 71.8%

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

      1. pow1/2 71.8%

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

      2. pow-to-exp 67.2%

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

   6. Applied egg-rr 67.2%

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

      1. *-un-lft-identity 67.2%

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

      2. +-commutative 67.2%

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

      3. exp-to-pow 71.8%

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

      4. *-commutative 71.8%

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

   8. Applied egg-rr 71.8%

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

      1. *-lft-identity 71.8%

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

      2. unsub-neg 71.8%

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

      3. unpow1/2 71.8%

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

   10. Simplified 71.8%

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

   if 6.9999999999999994e-95 < b

   1. Initial program 23.4%

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

   2. Taylor expanded in b around inf 74.3%

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

      1. associate-*r/ 74.3%

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

      2. neg-mul-1 74.3%

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

   4. Simplified 74.3%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{-71}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6: 68.2% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 72.7%

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

   2. Taylor expanded in b around -inf 76.4%

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

      1. mul-1-neg 76.4%

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

      2. unsub-neg 76.4%

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

   4. Simplified 76.4%

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

   if -1.999999999999994e-310 < b

   1. Initial program 36.2%

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

   2. Taylor expanded in b around inf 58.1%

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

      1. associate-*r/ 58.1%

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

      2. neg-mul-1 58.1%

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

   4. Simplified 58.1%

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

4. Final simplification 67.2%

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

Alternative 7: 68.0% accurate, 19.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 72.7%

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

   2. Taylor expanded in b around -inf 76.3%

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

      1. associate-*r/ 76.3%

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

      2. mul-1-neg 76.3%

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

   4. Simplified 76.3%

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

   if -1.999999999999994e-310 < b

   1. Initial program 36.2%

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

   2. Taylor expanded in b around inf 58.1%

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

      1. associate-*r/ 58.1%

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

      2. neg-mul-1 58.1%

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

   4. Simplified 58.1%

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

4. Final simplification 67.2%

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

Alternative 8: 35.6% accurate, 29.0× 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(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 54.5%

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

2. Taylor expanded in b around -inf 39.6%

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

   1. associate-*r/ 39.6%

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

   2. mul-1-neg 39.6%

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

4. Simplified 39.6%

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

5. Final simplification 39.6%

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

Developer target: 70.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* (* 4.0 a) c)))))
   (if (< b 0.0)
     (/ (+ (- b) t_0) (* 2.0 a))
     (/ c (* 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 = (-b + t_0) / (2.0 * a);
	} else {
		tmp = c / (a * ((-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 = (-b + t_0) / (2.0d0 * a)
    else
        tmp = c / (a * ((-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 = (-b + t_0) / (2.0 * a);
	} else {
		tmp = c / (a * ((-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 = (-b + t_0) / (2.0 * a)
	else:
		tmp = c / (a * ((-b - t_0) / (2.0 * a)))
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	tmp = 0.0
	if (b < 0.0)
		tmp = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
	else
		tmp = Float64(c / Float64(a * 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 = (-b + t_0) / (2.0 * a);
	else
		tmp = c / (a * ((-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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[b, 0.0], N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(c / N[(a * N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023279 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

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

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