Quadratic roots, full range

Average Accuracy: 47.7% → 82.9%

Time: 19.8s

Precision: binary64

Cost: 20620

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{+104}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq -4.2 \cdot 10^{-162}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq -1.5 \cdot 10^{-235}:\\ \;\;\;\;b \cdot \frac{-0.5}{a} - \frac{-0.5}{a} \cdot \mathsf{hypot}\left(\sqrt{a \cdot -4} \cdot \sqrt{c}, b\right)\\ \mathbf{elif}\;b \leq 1020000000:\\ \;\;\;\;\frac{1}{a \cdot \left(\frac{-2}{4 \cdot \left(a \cdot c\right)} \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.7e+104)
   (/ (- b) a)
   (if (<= b -4.2e-162)
     (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* a 2.0))
     (if (<= b -1.5e-235)
       (-
        (* b (/ -0.5 a))
        (* (/ -0.5 a) (hypot (* (sqrt (* a -4.0)) (sqrt c)) b)))
       (if (<= b 1020000000.0)
         (/
          1.0
          (*
           a
           (*
            (/ -2.0 (* 4.0 (* a c)))
            (+ b (hypot b (sqrt (* c (* a -4.0))))))))
         (/ (- c) b))))))

C

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

↓

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.7e+104) {
		tmp = -b / a;
	} else if (b <= -4.2e-162) {
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else if (b <= -1.5e-235) {
		tmp = (b * (-0.5 / a)) - ((-0.5 / a) * hypot((sqrt((a * -4.0)) * sqrt(c)), b));
	} else if (b <= 1020000000.0) {
		tmp = 1.0 / (a * ((-2.0 / (4.0 * (a * c))) * (b + hypot(b, sqrt((c * (a * -4.0)))))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Java

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

↓

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.7e+104) {
		tmp = -b / a;
	} else if (b <= -4.2e-162) {
		tmp = (Math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else if (b <= -1.5e-235) {
		tmp = (b * (-0.5 / a)) - ((-0.5 / a) * Math.hypot((Math.sqrt((a * -4.0)) * Math.sqrt(c)), b));
	} else if (b <= 1020000000.0) {
		tmp = 1.0 / (a * ((-2.0 / (4.0 * (a * c))) * (b + Math.hypot(b, Math.sqrt((c * (a * -4.0)))))));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

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

↓

def code(a, b, c):
	tmp = 0
	if b <= -2.7e+104:
		tmp = -b / a
	elif b <= -4.2e-162:
		tmp = (math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0)
	elif b <= -1.5e-235:
		tmp = (b * (-0.5 / a)) - ((-0.5 / a) * math.hypot((math.sqrt((a * -4.0)) * math.sqrt(c)), b))
	elif b <= 1020000000.0:
		tmp = 1.0 / (a * ((-2.0 / (4.0 * (a * c))) * (b + math.hypot(b, math.sqrt((c * (a * -4.0)))))))
	else:
		tmp = -c / b
	return tmp

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

↓

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.7e+104)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= -4.2e-162)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))) - b) / Float64(a * 2.0));
	elseif (b <= -1.5e-235)
		tmp = Float64(Float64(b * Float64(-0.5 / a)) - Float64(Float64(-0.5 / a) * hypot(Float64(sqrt(Float64(a * -4.0)) * sqrt(c)), b)));
	elseif (b <= 1020000000.0)
		tmp = Float64(1.0 / Float64(a * Float64(Float64(-2.0 / Float64(4.0 * Float64(a * c))) * Float64(b + hypot(b, sqrt(Float64(c * Float64(a * -4.0))))))));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.7e+104)
		tmp = -b / a;
	elseif (b <= -4.2e-162)
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	elseif (b <= -1.5e-235)
		tmp = (b * (-0.5 / a)) - ((-0.5 / a) * hypot((sqrt((a * -4.0)) * sqrt(c)), b));
	elseif (b <= 1020000000.0)
		tmp = 1.0 / (a * ((-2.0 / (4.0 * (a * c))) * (b + hypot(b, sqrt((c * (a * -4.0)))))));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
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]

↓

code[a_, b_, c_] := If[LessEqual[b, -2.7e+104], N[((-b) / a), $MachinePrecision], If[LessEqual[b, -4.2e-162], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -1.5e-235], N[(N[(b * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.5 / a), $MachinePrecision] * N[Sqrt[N[(N[Sqrt[N[(a * -4.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[c], $MachinePrecision]), $MachinePrecision] ^ 2 + b ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1020000000.0], N[(1.0 / N[(a * N[(N[(-2.0 / N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b + N[Sqrt[b ^ 2 + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -2.69999999999999985e104

   1. Initial program 28.2%

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

   2. Simplified 28.1%

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

      [Start] 28.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      neg-sub0 [=>] 28.2	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      associate-+l- [=>] 28.2	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      sub0-neg [=>] 28.2	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      neg-mul-1 [=>] 28.2	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      associate-*l/ [<=] 28.1	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \]
      *-commutative [=>] 28.1	\[ \color{blue}{\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{2 \cdot a}} \]
      associate-/r* [=>] 28.1	\[ \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}} \]
      /-rgt-identity [<=] 28.1	\[ \color{blue}{\frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{-1}{2}}{a} \]
      metadata-eval [<=] 28.1	\[ \frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{-1}{2}}{a} \]

   3. Taylor expanded in b around -inf 93.7%

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

   4. Simplified 93.7%

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

      [Start] 93.7	\[ -1 \cdot \frac{b}{a} \]
      associate-*r/ [=>] 93.7	\[ \color{blue}{\frac{-1 \cdot b}{a}} \]
      mul-1-neg [=>] 93.7	\[ \frac{\color{blue}{-b}}{a} \]

   if -2.69999999999999985e104 < b < -4.2e-162

   1. Initial program 90.4%

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

   if -4.2e-162 < b < -1.4999999999999999e-235

   1. Initial program 72.2%

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

   2. Simplified 72.1%

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

      [Start] 72.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      neg-sub0 [=>] 72.2	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      associate-+l- [=>] 72.2	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      sub0-neg [=>] 72.2	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      neg-mul-1 [=>] 72.2	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      associate-*l/ [<=] 72.1	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \]
      *-commutative [=>] 72.1	\[ \color{blue}{\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{2 \cdot a}} \]
      associate-/r* [=>] 72.1	\[ \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}} \]
      /-rgt-identity [<=] 72.1	\[ \color{blue}{\frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{-1}{2}}{a} \]
      metadata-eval [<=] 72.1	\[ \frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{-1}{2}}{a} \]

   3. Applied egg-rr 81.9%

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

      [Start] 72.1	\[ \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a} \]
      fma-udef [=>] 72.1	\[ \left(b - \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + b \cdot b}}\right) \cdot \frac{-0.5}{a} \]
      add-sqr-sqrt [=>] 72.1	\[ \left(b - \sqrt{\color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)}} + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      hypot-def [=>] 81.9	\[ \left(b - \color{blue}{\mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)}\right) \cdot \frac{-0.5}{a} \]

   4. Simplified 81.9%

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

      [Start] 81.9	\[ \left(b - \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right) \cdot \frac{-0.5}{a} \]
      associate-*r* [=>] 81.9	\[ \left(b - \mathsf{hypot}\left(\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}, b\right)\right) \cdot \frac{-0.5}{a} \]
      *-commutative [<=] 81.9	\[ \left(b - \mathsf{hypot}\left(\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4}, b\right)\right) \cdot \frac{-0.5}{a} \]
      associate-*l* [=>] 81.9	\[ \left(b - \mathsf{hypot}\left(\sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)}}, b\right)\right) \cdot \frac{-0.5}{a} \]

   5. Applied egg-rr 81.9%

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

      [Start] 81.9	\[ \left(b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)\right) \cdot \frac{-0.5}{a} \]
      *-commutative [=>] 81.9	\[ \color{blue}{\frac{-0.5}{a} \cdot \left(b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)\right)} \]
      sub-neg [=>] 81.9	\[ \frac{-0.5}{a} \cdot \color{blue}{\left(b + \left(-\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)\right)\right)} \]
      distribute-lft-in [=>] 81.9	\[ \color{blue}{\frac{-0.5}{a} \cdot b + \frac{-0.5}{a} \cdot \left(-\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)\right)} \]

   6. Applied egg-rr 48.0%

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

      [Start] 81.9	\[ \frac{-0.5}{a} \cdot b + \frac{-0.5}{a} \cdot \left(-\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)\right) \]
      *-commutative [=>] 81.9	\[ \frac{-0.5}{a} \cdot b + \frac{-0.5}{a} \cdot \left(-\mathsf{hypot}\left(\sqrt{\color{blue}{\left(a \cdot -4\right) \cdot c}}, b\right)\right) \]
      sqrt-prod [=>] 48.0	\[ \frac{-0.5}{a} \cdot b + \frac{-0.5}{a} \cdot \left(-\mathsf{hypot}\left(\color{blue}{\sqrt{a \cdot -4} \cdot \sqrt{c}}, b\right)\right) \]

   if -1.4999999999999999e-235 < b < 1.02e9

   1. Initial program 61.9%

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

   2. Simplified 61.9%

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

      [Start] 61.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      neg-sub0 [=>] 61.9	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      associate-+l- [=>] 61.9	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      sub0-neg [=>] 61.9	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      neg-mul-1 [=>] 61.9	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      associate-*l/ [<=] 61.8	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \]
      *-commutative [=>] 61.8	\[ \color{blue}{\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{2 \cdot a}} \]
      associate-/r* [=>] 61.8	\[ \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}} \]
      /-rgt-identity [<=] 61.8	\[ \color{blue}{\frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{-1}{2}}{a} \]
      metadata-eval [<=] 61.8	\[ \frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{-1}{2}}{a} \]

   3. Applied egg-rr 61.2%

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

      [Start] 61.9	\[ \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a} \]
      associate-*r/ [=>] 62.0	\[ \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot -0.5}{a}} \]
      clear-num [=>] 61.9	\[ \color{blue}{\frac{1}{\frac{a}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot -0.5}}} \]
      fma-udef [=>] 61.9	\[ \frac{1}{\frac{a}{\left(b - \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + b \cdot b}}\right) \cdot -0.5}} \]
      +-commutative [=>] 61.9	\[ \frac{1}{\frac{a}{\left(b - \sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -4\right)}}\right) \cdot -0.5}} \]
      add-sqr-sqrt [=>] 61.1	\[ \frac{1}{\frac{a}{\left(b - \sqrt{b \cdot b + \color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)}}}\right) \cdot -0.5}} \]
      hypot-def [=>] 61.2	\[ \frac{1}{\frac{a}{\left(b - \color{blue}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\right) \cdot -0.5}} \]

   4. Applied egg-rr 61.2%

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

      [Start] 61.2	\[ \frac{1}{\frac{a}{\left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right) \cdot -0.5}} \]
      clear-num [=>] 61.3	\[ \frac{1}{\color{blue}{\frac{1}{\frac{\left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right) \cdot -0.5}{a}}}} \]
      associate-/r/ [=>] 61.2	\[ \frac{1}{\color{blue}{\frac{1}{\left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right) \cdot -0.5} \cdot a}} \]
      *-commutative [=>] 61.2	\[ \frac{1}{\frac{1}{\color{blue}{-0.5 \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)}} \cdot a} \]
      associate-/r* [=>] 61.2	\[ \frac{1}{\color{blue}{\frac{\frac{1}{-0.5}}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}} \cdot a} \]
      metadata-eval [=>] 61.2	\[ \frac{1}{\frac{\color{blue}{-2}}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)} \cdot a} \]
      hypot-udef [=>] 61.0	\[ \frac{1}{\frac{-2}{b - \color{blue}{\sqrt{b \cdot b + \sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)}}}} \cdot a} \]
      add-sqr-sqrt [<=] 61.8	\[ \frac{1}{\frac{-2}{b - \sqrt{b \cdot b + \color{blue}{a \cdot \left(c \cdot -4\right)}}} \cdot a} \]
      +-commutative [=>] 61.8	\[ \frac{1}{\frac{-2}{b - \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + b \cdot b}}} \cdot a} \]
      add-sqr-sqrt [=>] 61.0	\[ \frac{1}{\frac{-2}{b - \sqrt{\color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)}} + b \cdot b}} \cdot a} \]
      hypot-def [=>] 61.2	\[ \frac{1}{\frac{-2}{b - \color{blue}{\mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)}} \cdot a} \]

   5. Applied egg-rr 60.6%

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

      [Start] 61.2	\[ \frac{1}{\frac{-2}{b - \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)} \cdot a} \]
      flip-- [=>] 60.8	\[ \frac{1}{\frac{-2}{\color{blue}{\frac{b \cdot b - \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right) \cdot \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)}{b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)}}} \cdot a} \]
      associate-/r/ [=>] 60.6	\[ \frac{1}{\color{blue}{\left(\frac{-2}{b \cdot b - \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right) \cdot \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)} \cdot \left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right)\right)} \cdot a} \]
      hypot-udef [=>] 60.6	\[ \frac{1}{\left(\frac{-2}{b \cdot b - \color{blue}{\sqrt{\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)} + b \cdot b}} \cdot \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)} \cdot \left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right)\right) \cdot a} \]
      hypot-udef [=>] 60.6	\[ \frac{1}{\left(\frac{-2}{b \cdot b - \sqrt{\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)} + b \cdot b} \cdot \color{blue}{\sqrt{\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)} + b \cdot b}}} \cdot \left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right)\right) \cdot a} \]
      add-sqr-sqrt [<=] 60.6	\[ \frac{1}{\left(\frac{-2}{b \cdot b - \color{blue}{\left(\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)} + b \cdot b\right)}} \cdot \left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right)\right) \cdot a} \]
      add-sqr-sqrt [<=] 60.6	\[ \frac{1}{\left(\frac{-2}{b \cdot b - \left(\color{blue}{a \cdot \left(c \cdot -4\right)} + b \cdot b\right)} \cdot \left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right)\right) \cdot a} \]
      +-commutative [=>] 60.6	\[ \frac{1}{\left(\frac{-2}{b \cdot b - \color{blue}{\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)}} \cdot \left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right)\right) \cdot a} \]
      fma-def [=>] 60.6	\[ \frac{1}{\left(\frac{-2}{b \cdot b - \color{blue}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}} \cdot \left(b + \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -4\right)}, b\right)\right)\right) \cdot a} \]
      hypot-udef [=>] 60.7	\[ \frac{1}{\left(\frac{-2}{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} \cdot \left(b + \color{blue}{\sqrt{\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)} + b \cdot b}}\right)\right) \cdot a} \]

   6. Simplified 66.5%

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

      [Start] 60.6	\[ \frac{1}{\left(\frac{-2}{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)\right)\right) \cdot a} \]

   if 1.02e9 < b

   1. Initial program 12.0%

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

   2. Simplified 12.0%

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

      [Start] 12.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      neg-sub0 [=>] 12.0	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      associate-+l- [=>] 12.0	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      sub0-neg [=>] 12.0	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      neg-mul-1 [=>] 12.0	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      associate-*l/ [<=] 12.0	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \]
      *-commutative [=>] 12.0	\[ \color{blue}{\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{2 \cdot a}} \]
      associate-/r* [=>] 12.0	\[ \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}} \]
      /-rgt-identity [<=] 12.0	\[ \color{blue}{\frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{-1}{2}}{a} \]
      metadata-eval [<=] 12.0	\[ \frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{-1}{2}}{a} \]

   3. Taylor expanded in b around inf 91.5%

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

   4. Simplified 91.5%

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

      [Start] 91.5	\[ -1 \cdot \frac{c}{b} \]
      associate-*r/ [=>] 91.5	\[ \color{blue}{\frac{-1 \cdot c}{b}} \]
      mul-1-neg [=>] 91.5	\[ \frac{\color{blue}{-c}}{b} \]

3. Recombined 5 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{+104}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq -4.2 \cdot 10^{-162}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq -1.5 \cdot 10^{-235}:\\ \;\;\;\;b \cdot \frac{-0.5}{a} - \frac{-0.5}{a} \cdot \mathsf{hypot}\left(\sqrt{a \cdot -4} \cdot \sqrt{c}, b\right)\\ \mathbf{elif}\;b \leq 1020000000:\\ \;\;\;\;\frac{1}{a \cdot \left(\frac{-2}{4 \cdot \left(a \cdot c\right)} \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	82.9%
Cost	20364

\[\begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{+103}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq -4.2 \cdot 10^{-162}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq -1.7 \cdot 10^{-235}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \mathsf{hypot}\left(\sqrt{a \cdot -4} \cdot \sqrt{c}, b\right)\right)\\ \mathbf{elif}\;b \leq 1100000:\\ \;\;\;\;\frac{1}{a \cdot \left(\frac{-2}{4 \cdot \left(a \cdot c\right)} \cdot \left(b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 2
Accuracy	83.9%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -8.6 \cdot 10^{+103}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-38}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 3
Accuracy	78.1%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-40}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{-36}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4
Accuracy	37.8%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-308}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Accuracy	63.9%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq 2.3 \cdot 10^{-255}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6
Accuracy	11.8%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023151 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))