The quadratic formula (r1)

Percentage Accurate: 38.8% → 65.1%

Time: 24.9s

Precision: binary64

Cost: 20424

• 31.1% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_0 := \frac{c}{\frac{b}{a}}\\ \mathbf{if}\;b \leq -6.2 \cdot 10^{+73}:\\ \;\;\;\;\left(-2 \cdot \sqrt[3]{t_0 \cdot \left(t_0 \cdot t_0\right)} + b \cdot 2\right) \cdot \frac{-0.5}{a}\\ \mathbf{elif}\;b \leq -2.15 \cdot 10^{-201}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - {\left({\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{0.25}\right)}^{2}\right)\\ \mathbf{elif}\;b \leq -4.5 \cdot 10^{-285} \lor \neg \left(b \leq 3.8 \cdot 10^{-89}\right):\\ \;\;\;\;\frac{-c}{b} - a \cdot \left(c \cdot \frac{c}{{b}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - {\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{0.25}\right)}^{2}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ c (/ b a))))
   (if (<= b -6.2e+73)
     (* (+ (* -2.0 (cbrt (* t_0 (* t_0 t_0)))) (* b 2.0)) (/ -0.5 a))
     (if (<= b -2.15e-201)
       (* (/ -0.5 a) (- b (pow (pow (fma a (* c -4.0) (* b b)) 0.25) 2.0)))
       (if (or (<= b -4.5e-285) (not (<= b 3.8e-89)))
         (- (/ (- c) b) (* a (* c (/ c (pow b 3.0)))))
         (* (/ -0.5 a) (- b (pow (pow (* a (* c -4.0)) 0.25) 2.0))))))))

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 t_0 = c / (b / a);
	double tmp;
	if (b <= -6.2e+73) {
		tmp = ((-2.0 * cbrt((t_0 * (t_0 * t_0)))) + (b * 2.0)) * (-0.5 / a);
	} else if (b <= -2.15e-201) {
		tmp = (-0.5 / a) * (b - pow(pow(fma(a, (c * -4.0), (b * b)), 0.25), 2.0));
	} else if ((b <= -4.5e-285) || !(b <= 3.8e-89)) {
		tmp = (-c / b) - (a * (c * (c / pow(b, 3.0))));
	} else {
		tmp = (-0.5 / a) * (b - pow(pow((a * (c * -4.0)), 0.25), 2.0));
	}
	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)
	t_0 = Float64(c / Float64(b / a))
	tmp = 0.0
	if (b <= -6.2e+73)
		tmp = Float64(Float64(Float64(-2.0 * cbrt(Float64(t_0 * Float64(t_0 * t_0)))) + Float64(b * 2.0)) * Float64(-0.5 / a));
	elseif (b <= -2.15e-201)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - ((fma(a, Float64(c * -4.0), Float64(b * b)) ^ 0.25) ^ 2.0)));
	elseif ((b <= -4.5e-285) || !(b <= 3.8e-89))
		tmp = Float64(Float64(Float64(-c) / b) - Float64(a * Float64(c * Float64(c / (b ^ 3.0)))));
	else
		tmp = Float64(Float64(-0.5 / a) * Float64(b - ((Float64(a * Float64(c * -4.0)) ^ 0.25) ^ 2.0)));
	end
	return 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_] := Block[{t$95$0 = N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -6.2e+73], N[(N[(N[(-2.0 * N[Power[N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(b * 2.0), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -2.15e-201], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Power[N[Power[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[b, -4.5e-285], N[Not[LessEqual[b, 3.8e-89]], $MachinePrecision]], N[(N[((-c) / b), $MachinePrecision] - N[(a * N[(c * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Power[N[Power[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	38.8%
Target	52.7%
Herbie	65.1%

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

Derivation

1. Split input into 4 regimes

2. if b < -6.1999999999999999e73

   1. Initial program 47.9%

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

   2. Simplified 47.8%

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

      [Start] 47.9%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      neg-sub0 [=>] 47.9%	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      associate-+l- [=>] 47.9%	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      sub0-neg [=>] 47.9%	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      neg-mul-1 [=>] 47.9%	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      associate-*l/ [<=] 47.8%	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \]
      *-commutative [=>] 47.8%	\[ \color{blue}{\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{2 \cdot a}} \]
      associate-/r* [=>] 47.8%	\[ \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}} \]
      /-rgt-identity [<=] 47.8%	\[ \color{blue}{\frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{-1}{2}}{a} \]
      metadata-eval [<=] 47.8%	\[ \frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{-1}{2}}{a} \]

   3. Taylor expanded in b around -inf 88.8%

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

   4. Applied egg-rr 93.1%

      \[\leadsto \left(-2 \cdot \color{blue}{\sqrt[3]{\left(\frac{c}{\frac{b}{a}} \cdot \frac{c}{\frac{b}{a}}\right) \cdot \frac{c}{\frac{b}{a}}}} + 2 \cdot b\right) \cdot \frac{-0.5}{a} \]
      Step-by-step derivation

      [Start] 88.8%	\[ \left(-2 \cdot \frac{c \cdot a}{b} + 2 \cdot b\right) \cdot \frac{-0.5}{a} \]
      add-cbrt-cube [=>] 93.1%	\[ \left(-2 \cdot \color{blue}{\sqrt[3]{\left(\frac{c \cdot a}{b} \cdot \frac{c \cdot a}{b}\right) \cdot \frac{c \cdot a}{b}}} + 2 \cdot b\right) \cdot \frac{-0.5}{a} \]
      associate-/l* [=>] 93.1%	\[ \left(-2 \cdot \sqrt[3]{\left(\color{blue}{\frac{c}{\frac{b}{a}}} \cdot \frac{c \cdot a}{b}\right) \cdot \frac{c \cdot a}{b}} + 2 \cdot b\right) \cdot \frac{-0.5}{a} \]
      associate-/l* [=>] 93.1%	\[ \left(-2 \cdot \sqrt[3]{\left(\frac{c}{\frac{b}{a}} \cdot \color{blue}{\frac{c}{\frac{b}{a}}}\right) \cdot \frac{c \cdot a}{b}} + 2 \cdot b\right) \cdot \frac{-0.5}{a} \]
      associate-/l* [=>] 93.1%	\[ \left(-2 \cdot \sqrt[3]{\left(\frac{c}{\frac{b}{a}} \cdot \frac{c}{\frac{b}{a}}\right) \cdot \color{blue}{\frac{c}{\frac{b}{a}}}} + 2 \cdot b\right) \cdot \frac{-0.5}{a} \]

   if -6.1999999999999999e73 < b < -2.1499999999999999e-201

   1. Initial program 68.5%

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

   2. Simplified 68.1%

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

      [Start] 68.5%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      neg-sub0 [=>] 68.5%	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      associate-+l- [=>] 68.5%	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      sub0-neg [=>] 68.5%	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      neg-mul-1 [=>] 68.5%	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      associate-*l/ [<=] 68.1%	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \]
      *-commutative [=>] 68.1%	\[ \color{blue}{\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{2 \cdot a}} \]
      associate-/r* [=>] 68.1%	\[ \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}} \]
      /-rgt-identity [<=] 68.1%	\[ \color{blue}{\frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{-1}{2}}{a} \]
      metadata-eval [<=] 68.1%	\[ \frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{-1}{2}}{a} \]

   3. Applied egg-rr 70.0%

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

      [Start] 68.1%	\[ \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a} \]
      fma-udef [=>] 68.1%	\[ \left(b - \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + b \cdot b}}\right) \cdot \frac{-0.5}{a} \]
      *-commutative [=>] 68.1%	\[ \left(b - \sqrt{a \cdot \color{blue}{\left(-4 \cdot c\right)} + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      associate-*r* [=>] 68.1%	\[ \left(b - \sqrt{\color{blue}{\left(a \cdot -4\right) \cdot c} + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      metadata-eval [<=] 68.1%	\[ \left(b - \sqrt{\left(a \cdot \color{blue}{\left(-4\right)}\right) \cdot c + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      distribute-rgt-neg-in [<=] 68.1%	\[ \left(b - \sqrt{\color{blue}{\left(-a \cdot 4\right)} \cdot c + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      *-commutative [<=] 68.1%	\[ \left(b - \sqrt{\left(-\color{blue}{4 \cdot a}\right) \cdot c + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      distribute-lft-neg-in [<=] 68.1%	\[ \left(b - \sqrt{\color{blue}{\left(-\left(4 \cdot a\right) \cdot c\right)} + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      +-commutative [=>] 68.1%	\[ \left(b - \sqrt{\color{blue}{b \cdot b + \left(-\left(4 \cdot a\right) \cdot c\right)}}\right) \cdot \frac{-0.5}{a} \]
      sub-neg [<=] 68.1%	\[ \left(b - \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot \frac{-0.5}{a} \]
      add-sqr-sqrt [=>] 68.2%	\[ \left(b - \color{blue}{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\right) \cdot \frac{-0.5}{a} \]
      pow2 [=>] 68.2%	\[ \left(b - \color{blue}{{\left(\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}^{2}}\right) \cdot \frac{-0.5}{a} \]

   if -2.1499999999999999e-201 < b < -4.5000000000000002e-285 or 3.8000000000000001e-89 < b

   1. Initial program 16.8%

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

   2. Simplified 16.8%

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

      [Start] 16.8%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      neg-sub0 [=>] 16.8%	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      associate-+l- [=>] 16.8%	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      sub0-neg [=>] 16.8%	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      neg-mul-1 [=>] 16.8%	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      associate-*l/ [<=] 16.7%	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \]
      *-commutative [=>] 16.7%	\[ \color{blue}{\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{2 \cdot a}} \]
      associate-/r* [=>] 16.7%	\[ \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}} \]
      /-rgt-identity [<=] 16.7%	\[ \color{blue}{\frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{-1}{2}}{a} \]
      metadata-eval [<=] 16.7%	\[ \frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{-1}{2}}{a} \]

   3. Applied egg-rr 10.7%

      \[\leadsto \left(b - \color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{1.5}\right)}^{0.3333333333333333}}\right) \cdot \frac{-0.5}{a} \]
      Step-by-step derivation

      [Start] 16.8%	\[ \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a} \]
      fma-udef [=>] 16.7%	\[ \left(b - \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + b \cdot b}}\right) \cdot \frac{-0.5}{a} \]
      *-commutative [=>] 16.7%	\[ \left(b - \sqrt{a \cdot \color{blue}{\left(-4 \cdot c\right)} + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      associate-*r* [=>] 16.7%	\[ \left(b - \sqrt{\color{blue}{\left(a \cdot -4\right) \cdot c} + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      metadata-eval [<=] 16.7%	\[ \left(b - \sqrt{\left(a \cdot \color{blue}{\left(-4\right)}\right) \cdot c + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      distribute-rgt-neg-in [<=] 16.7%	\[ \left(b - \sqrt{\color{blue}{\left(-a \cdot 4\right)} \cdot c + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      *-commutative [<=] 16.7%	\[ \left(b - \sqrt{\left(-\color{blue}{4 \cdot a}\right) \cdot c + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      distribute-lft-neg-in [<=] 16.7%	\[ \left(b - \sqrt{\color{blue}{\left(-\left(4 \cdot a\right) \cdot c\right)} + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      +-commutative [=>] 16.7%	\[ \left(b - \sqrt{\color{blue}{b \cdot b + \left(-\left(4 \cdot a\right) \cdot c\right)}}\right) \cdot \frac{-0.5}{a} \]
      sub-neg [<=] 16.7%	\[ \left(b - \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot \frac{-0.5}{a} \]
      add-cbrt-cube [=>] 11.3%	\[ \left(b - \color{blue}{\sqrt[3]{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\right) \cdot \frac{-0.5}{a} \]
      pow3 [=>] 11.4%	\[ \left(b - \sqrt[3]{\color{blue}{{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}^{3}}}\right) \cdot \frac{-0.5}{a} \]
      pow1/3 [=>] 9.8%	\[ \left(b - \color{blue}{{\left({\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}^{3}\right)}^{0.3333333333333333}}\right) \cdot \frac{-0.5}{a} \]

   4. Simplified 13.0%

      \[\leadsto \left(b - \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\right)}^{1.5}}}\right) \cdot \frac{-0.5}{a} \]
      Step-by-step derivation

      [Start] 10.7%	\[ \left(b - {\left({\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right) \cdot \frac{-0.5}{a} \]
      unpow1/3 [=>] 12.2%	\[ \left(b - \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{1.5}}}\right) \cdot \frac{-0.5}{a} \]
      fma-def [<=] 12.1%	\[ \left(b - \sqrt[3]{{\color{blue}{\left(a \cdot \left(c \cdot -4\right) + b \cdot b\right)}}^{1.5}}\right) \cdot \frac{-0.5}{a} \]
      +-commutative [=>] 12.1%	\[ \left(b - \sqrt[3]{{\color{blue}{\left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)}}^{1.5}}\right) \cdot \frac{-0.5}{a} \]
      fma-def [=>] 13.0%	\[ \left(b - \sqrt[3]{{\color{blue}{\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\right)}}^{1.5}}\right) \cdot \frac{-0.5}{a} \]

   5. Taylor expanded in b around inf 61.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b}} \]

   6. Simplified 73.5%

      \[\leadsto \color{blue}{-\mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)} \]
      Step-by-step derivation

      [Start] 61.7%	\[ -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b} \]
      mul-1-neg [=>] 61.7%	\[ \color{blue}{\left(-\frac{{c}^{2} \cdot a}{{b}^{3}}\right)} + -1 \cdot \frac{c}{b} \]
      associate-*l/ [<=] 66.2%	\[ \left(-\color{blue}{\frac{{c}^{2}}{{b}^{3}} \cdot a}\right) + -1 \cdot \frac{c}{b} \]
      unpow2 [=>] 66.2%	\[ \left(-\frac{\color{blue}{c \cdot c}}{{b}^{3}} \cdot a\right) + -1 \cdot \frac{c}{b} \]
      mul-1-neg [=>] 66.2%	\[ \left(-\frac{c \cdot c}{{b}^{3}} \cdot a\right) + \color{blue}{\left(-\frac{c}{b}\right)} \]
      distribute-neg-out [=>] 66.2%	\[ \color{blue}{-\left(\frac{c \cdot c}{{b}^{3}} \cdot a + \frac{c}{b}\right)} \]
      fma-def [=>] 66.2%	\[ -\color{blue}{\mathsf{fma}\left(\frac{c \cdot c}{{b}^{3}}, a, \frac{c}{b}\right)} \]
      associate-/l* [=>] 73.5%	\[ -\mathsf{fma}\left(\color{blue}{\frac{c}{\frac{{b}^{3}}{c}}}, a, \frac{c}{b}\right) \]

   7. Applied egg-rr 73.5%

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

      [Start] 73.5%	\[ -\mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right) \]
      fma-udef [=>] 73.5%	\[ -\color{blue}{\left(\frac{c}{\frac{{b}^{3}}{c}} \cdot a + \frac{c}{b}\right)} \]

   8. Applied egg-rr 73.5%

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

      [Start] 73.5%	\[ -\left(\frac{c}{\frac{{b}^{3}}{c}} \cdot a + \frac{c}{b}\right) \]
      associate-/r/ [=>] 73.5%	\[ -\left(\color{blue}{\left(\frac{c}{{b}^{3}} \cdot c\right)} \cdot a + \frac{c}{b}\right) \]

   if -4.5000000000000002e-285 < b < 3.8000000000000001e-89

   1. Initial program 32.9%

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

   2. Simplified 32.9%

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

      [Start] 32.9%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      neg-sub0 [=>] 32.9%	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      associate-+l- [=>] 32.9%	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      sub0-neg [=>] 32.9%	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      neg-mul-1 [=>] 32.9%	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]
      associate-*l/ [<=] 32.9%	\[ \color{blue}{\frac{-1}{2 \cdot a} \cdot \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \]
      *-commutative [=>] 32.9%	\[ \color{blue}{\left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{-1}{2 \cdot a}} \]
      associate-/r* [=>] 32.9%	\[ \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{a}} \]
      /-rgt-identity [<=] 32.9%	\[ \color{blue}{\frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{1}} \cdot \frac{\frac{-1}{2}}{a} \]
      metadata-eval [<=] 32.9%	\[ \frac{b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{--1}} \cdot \frac{\frac{-1}{2}}{a} \]

   3. Applied egg-rr 41.8%

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

      [Start] 32.9%	\[ \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a} \]
      fma-udef [=>] 32.9%	\[ \left(b - \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + b \cdot b}}\right) \cdot \frac{-0.5}{a} \]
      *-commutative [=>] 32.9%	\[ \left(b - \sqrt{a \cdot \color{blue}{\left(-4 \cdot c\right)} + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      associate-*r* [=>] 32.9%	\[ \left(b - \sqrt{\color{blue}{\left(a \cdot -4\right) \cdot c} + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      metadata-eval [<=] 32.9%	\[ \left(b - \sqrt{\left(a \cdot \color{blue}{\left(-4\right)}\right) \cdot c + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      distribute-rgt-neg-in [<=] 32.9%	\[ \left(b - \sqrt{\color{blue}{\left(-a \cdot 4\right)} \cdot c + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      *-commutative [<=] 32.9%	\[ \left(b - \sqrt{\left(-\color{blue}{4 \cdot a}\right) \cdot c + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      distribute-lft-neg-in [<=] 32.9%	\[ \left(b - \sqrt{\color{blue}{\left(-\left(4 \cdot a\right) \cdot c\right)} + b \cdot b}\right) \cdot \frac{-0.5}{a} \]
      +-commutative [=>] 32.9%	\[ \left(b - \sqrt{\color{blue}{b \cdot b + \left(-\left(4 \cdot a\right) \cdot c\right)}}\right) \cdot \frac{-0.5}{a} \]
      sub-neg [<=] 32.9%	\[ \left(b - \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot \frac{-0.5}{a} \]
      add-sqr-sqrt [=>] 32.7%	\[ \left(b - \color{blue}{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\right) \cdot \frac{-0.5}{a} \]
      pow2 [=>] 32.7%	\[ \left(b - \color{blue}{{\left(\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}^{2}}\right) \cdot \frac{-0.5}{a} \]

   4. Taylor expanded in b around 0 41.7%

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

   5. Simplified 41.8%

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

      [Start] 41.7%	\[ \left(b - {\left({\left(-4 \cdot \left(c \cdot a\right)\right)}^{0.25}\right)}^{2}\right) \cdot \frac{-0.5}{a} \]
      associate-*r* [=>] 41.8%	\[ \left(b - {\left({\color{blue}{\left(\left(-4 \cdot c\right) \cdot a\right)}}^{0.25}\right)}^{2}\right) \cdot \frac{-0.5}{a} \]
      *-commutative [<=] 41.8%	\[ \left(b - {\left({\left(\color{blue}{\left(c \cdot -4\right)} \cdot a\right)}^{0.25}\right)}^{2}\right) \cdot \frac{-0.5}{a} \]
      *-commutative [<=] 41.8%	\[ \left(b - {\left({\color{blue}{\left(a \cdot \left(c \cdot -4\right)\right)}}^{0.25}\right)}^{2}\right) \cdot \frac{-0.5}{a} \]

3. Recombined 4 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{+73}:\\ \;\;\;\;\left(-2 \cdot \sqrt[3]{\frac{c}{\frac{b}{a}} \cdot \left(\frac{c}{\frac{b}{a}} \cdot \frac{c}{\frac{b}{a}}\right)} + b \cdot 2\right) \cdot \frac{-0.5}{a}\\ \mathbf{elif}\;b \leq -2.15 \cdot 10^{-201}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - {\left({\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{0.25}\right)}^{2}\right)\\ \mathbf{elif}\;b \leq -4.5 \cdot 10^{-285} \lor \neg \left(b \leq 3.8 \cdot 10^{-89}\right):\\ \;\;\;\;\frac{-c}{b} - a \cdot \left(c \cdot \frac{c}{{b}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - {\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{0.25}\right)}^{2}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	65.1%
Cost	20424

\[\begin{array}{l} t_0 := \frac{c}{\frac{b}{a}}\\ \mathbf{if}\;b \leq -6.2 \cdot 10^{+73}:\\ \;\;\;\;\left(-2 \cdot \sqrt[3]{t_0 \cdot \left(t_0 \cdot t_0\right)} + b \cdot 2\right) \cdot \frac{-0.5}{a}\\ \mathbf{elif}\;b \leq -2.15 \cdot 10^{-201}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - {\left({\left(\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}^{0.25}\right)}^{2}\right)\\ \mathbf{elif}\;b \leq -4.5 \cdot 10^{-285} \lor \neg \left(b \leq 3.8 \cdot 10^{-89}\right):\\ \;\;\;\;\frac{-c}{b} - a \cdot \left(c \cdot \frac{c}{{b}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - {\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{0.25}\right)}^{2}\right)\\ \end{array} \]

Alternative 2
Accuracy	64.4%
Cost	14161

\[\begin{array}{l} t_0 := \frac{c}{\frac{b}{a}}\\ \mathbf{if}\;b \leq -1.76 \cdot 10^{+74}:\\ \;\;\;\;\left(-2 \cdot \sqrt[3]{t_0 \cdot \left(t_0 \cdot t_0\right)} + b \cdot 2\right) \cdot \frac{-0.5}{a}\\ \mathbf{elif}\;b \leq -2.3 \cdot 10^{-200}:\\ \;\;\;\;\frac{-0.5 \cdot \left(b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right)}{a}\\ \mathbf{elif}\;b \leq -1.3 \cdot 10^{-289} \lor \neg \left(b \leq 2.4 \cdot 10^{-88}\right):\\ \;\;\;\;\frac{-c}{b} - a \cdot \left(c \cdot \frac{c}{{b}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - {\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{0.25}\right)}^{2}\right)\\ \end{array} \]

Alternative 3
Accuracy	60.8%
Cost	13969

\[\begin{array}{l} t_0 := \frac{c}{\frac{b}{a}}\\ \mathbf{if}\;b \leq -1.15 \cdot 10^{+74}:\\ \;\;\;\;\left(-2 \cdot \sqrt[3]{t_0 \cdot \left(t_0 \cdot t_0\right)} + b \cdot 2\right) \cdot \frac{-0.5}{a}\\ \mathbf{elif}\;b \leq -1.35 \cdot 10^{-201}:\\ \;\;\;\;\frac{-0.5 \cdot \left(b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right)}{a}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-236} \lor \neg \left(b \leq 2.9 \cdot 10^{-141}\right):\\ \;\;\;\;\frac{-c}{b} - a \cdot \left(c \cdot \frac{c}{{b}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\frac{{b}^{3}}{c}}, a, \frac{c}{b}\right)\\ \end{array} \]

Alternative 4
Accuracy	61.0%
Cost	8260

\[\begin{array}{l} t_0 := \frac{c}{\frac{b}{a}}\\ \mathbf{if}\;b \leq -3.2 \cdot 10^{+74}:\\ \;\;\;\;\left(-2 \cdot \sqrt[3]{t_0 \cdot \left(t_0 \cdot t_0\right)} + b \cdot 2\right) \cdot \frac{-0.5}{a}\\ \mathbf{elif}\;b \leq -1.4 \cdot 10^{-200}:\\ \;\;\;\;\frac{-0.5 \cdot \left(b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \left(c \cdot \frac{c}{{b}^{3}}\right)\\ \end{array} \]

Alternative 5
Accuracy	57.4%
Cost	7896

\[\begin{array}{l} t_0 := \frac{-0.5}{a} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{if}\;b \leq -3 \cdot 10^{+37}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \left(b - 2 \cdot \left(c \cdot \frac{a}{b}\right)\right)\right)\\ \mathbf{elif}\;b \leq -2.9 \cdot 10^{-68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -1.35 \cdot 10^{-143}:\\ \;\;\;\;\frac{2 \cdot \frac{c \cdot a}{b} + b \cdot -2}{a \cdot 2}\\ \mathbf{elif}\;b \leq -9 \cdot 10^{-199}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -1.8 \cdot 10^{-286}:\\ \;\;\;\;a \cdot \frac{c \cdot \left(-c\right)}{{b}^{3}}\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-135}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]

Alternative 6
Accuracy	56.7%
Cost	7760

\[\begin{array}{l} t_0 := \frac{-0.5}{a} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{if}\;b \leq -3 \cdot 10^{+37}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \left(b - 2 \cdot \left(c \cdot \frac{a}{b}\right)\right)\right)\\ \mathbf{elif}\;b \leq -2.1 \cdot 10^{-68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -1.26 \cdot 10^{-143}:\\ \;\;\;\;\frac{2 \cdot \frac{c \cdot a}{b} + b \cdot -2}{a \cdot 2}\\ \mathbf{elif}\;b \leq -1 \cdot 10^{-200}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \left(c \cdot \frac{c}{{b}^{3}}\right)\\ \end{array} \]

Alternative 7
Accuracy	61.6%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{+66}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -5.5 \cdot 10^{-201}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \left(c \cdot \frac{c}{{b}^{3}}\right)\\ \end{array} \]

Alternative 8
Accuracy	61.7%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{+74}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq -3.5 \cdot 10^{-200}:\\ \;\;\;\;\frac{-0.5 \cdot \left(b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - a \cdot \left(c \cdot \frac{c}{{b}^{3}}\right)\\ \end{array} \]

Alternative 9
Accuracy	55.9%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{-201}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \left(b - 2 \cdot \left(c \cdot \frac{a}{b}\right)\right)\right)\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-155}:\\ \;\;\;\;a \cdot \frac{c \cdot \left(-c\right)}{{b}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]

Alternative 10
Accuracy	55.7%
Cost	1100

\[\begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{-201}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \left(b - 2 \cdot \left(c \cdot \frac{a}{b}\right)\right)\right)\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-158}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{-c}{\frac{b}{a}}}}\\ \mathbf{elif}\;b \leq 2.25 \cdot 10^{-141}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(-2 \cdot \frac{c}{\frac{b}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]

Alternative 11
Accuracy	54.8%
Cost	904

\[\begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{-198}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.85 \cdot 10^{-154}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{-c}{\frac{b}{a}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]

Alternative 12
Accuracy	54.8%
Cost	904

\[\begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{-201}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{c \cdot a}{b} - b}}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-154}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{-c}{\frac{b}{a}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]

Alternative 13
Accuracy	53.9%
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq 8.5 \cdot 10^{-178}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 14
Accuracy	35.0%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-114}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 15
Accuracy	53.4%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-199}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 16
Accuracy	12.2%
Cost	192

\[\frac{c}{b} \]

Reproduce

herbie shell --seed 2023255 
(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)))