jeff quadratic root 2

Average Error: 31.13% → 12.23%

Time: 21.0s

Precision: binary64

Cost: 43164

Math

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

↓

\[\begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array}\\ t_1 := \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\\ t_2 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-255}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \left(a \cdot \frac{c}{b}\right) - b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, c \cdot \frac{a}{b}, b \cdot -2\right) \cdot \frac{0.5}{a}\right)} + -1\\ \end{array}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+178}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0)
   (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))
   (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0
         (if (>= b 0.0) (/ (* 2.0 c) (- (- b) b)) (/ (* b -2.0) (* 2.0 a))))
        (t_1 (sqrt (- (* b b) (* c (* 4.0 a)))))
        (t_2
         (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_1)) (/ (- t_1 b) (* 2.0 a)))))
   (if (<= t_2 (- INFINITY))
     t_0
     (if (<= t_2 -2e-255)
       t_2
       (if (<= t_2 0.0)
         (if (>= b 0.0)
           (/ (* 2.0 c) (- (- (* 2.0 (* a (/ c b))) b) b))
           (+
            (exp (log1p (* (fma 2.0 (* c (/ a b)) (* b -2.0)) (/ 0.5 a))))
            -1.0))
         (if (<= t_2 2e+178) t_2 t_0))))))

C

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

↓

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (-b - b);
	} else {
		tmp = (b * -2.0) / (2.0 * a);
	}
	double t_0 = tmp;
	double t_1 = sqrt(((b * b) - (c * (4.0 * a))));
	double tmp_1;
	if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-b - t_1);
	} else {
		tmp_1 = (t_1 - b) / (2.0 * a);
	}
	double t_2 = tmp_1;
	double tmp_2;
	if (t_2 <= -((double) INFINITY)) {
		tmp_2 = t_0;
	} else if (t_2 <= -2e-255) {
		tmp_2 = t_2;
	} else if (t_2 <= 0.0) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * c) / (((2.0 * (a * (c / b))) - b) - b);
		} else {
			tmp_3 = exp(log1p((fma(2.0, (c * (a / b)), (b * -2.0)) * (0.5 / a)))) + -1.0;
		}
		tmp_2 = tmp_3;
	} else if (t_2 <= 2e+178) {
		tmp_2 = t_2;
	} else {
		tmp_2 = t_0;
	}
	return tmp_2;
}

Julia

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

↓

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) - b));
	else
		tmp = Float64(Float64(b * -2.0) / Float64(2.0 * a));
	end
	t_0 = tmp
	t_1 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a))))
	tmp_1 = 0.0
	if (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_1));
	else
		tmp_1 = Float64(Float64(t_1 - b) / Float64(2.0 * a));
	end
	t_2 = tmp_1
	tmp_2 = 0.0
	if (t_2 <= Float64(-Inf))
		tmp_2 = t_0;
	elseif (t_2 <= -2e-255)
		tmp_2 = t_2;
	elseif (t_2 <= 0.0)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(2.0 * c) / Float64(Float64(Float64(2.0 * Float64(a * Float64(c / b))) - b) - b));
		else
			tmp_3 = Float64(exp(log1p(Float64(fma(2.0, Float64(c * Float64(a / b)), Float64(b * -2.0)) * Float64(0.5 / a)))) + -1.0);
		end
		tmp_2 = tmp_3;
	elseif (t_2 <= 2e+178)
		tmp_2 = t_2;
	else
		tmp_2 = t_0;
	end
	return tmp_2
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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 = If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - b), $MachinePrecision]), $MachinePrecision], N[(N[(b * -2.0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]}, If[LessEqual[t$95$2, (-Infinity)], t$95$0, If[LessEqual[t$95$2, -2e-255], t$95$2, If[LessEqual[t$95$2, 0.0], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(N[(N[(2.0 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[Log[1 + N[(N[(2.0 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision] + N[(b * -2.0), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]], If[LessEqual[t$95$2, 2e+178], t$95$2, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (if (>=.f64 b 0) (/.f64 (*.f64 2 c) (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))) < -inf.0 or 2.0000000000000001e178 < (if (>=.f64 b 0) (/.f64 (*.f64 2 c) (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)))

   1. Initial program 75.43

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

   2. Taylor expanded in b around inf 77.3

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

   3. Taylor expanded in b around -inf 27.25

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

   4. Simplified 27.25

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \]
      Proof

      [Start] 27.25	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \end{array} \]
      *-commutative [=>] 27.25	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \]

   if -inf.0 < (if (>=.f64 b 0) (/.f64 (*.f64 2 c) (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))) < -2e-255 or 0.0 < (if (>=.f64 b 0) (/.f64 (*.f64 2 c) (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))) < 2.0000000000000001e178

   1. Initial program 4.86

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

   if -2e-255 < (if (>=.f64 b 0) (/.f64 (*.f64 2 c) (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))) < 0.0

   1. Initial program 55.72

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

   2. Taylor expanded in b around -inf 56.12

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

   3. Simplified 56.13

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{2 \cdot a}\\ \end{array} \]
      Proof

      [Start] 56.12	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{c \cdot a}{b} + -2 \cdot b}{2 \cdot a}\\ \end{array} \]
      fma-def [=>] 56.12	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c \cdot a}{b}, -2 \cdot b\right)}{2 \cdot a}\\ \end{array} \]
      associate-/l* [=>] 56.13	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, -2 \cdot b\right)}{2 \cdot a}\\ \end{array} \]
      *-commutative [=>] 56.13	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{2 \cdot a}\\ \end{array} \]

   4. Taylor expanded in b around inf 51.01

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\left(-2 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}} + \left(b + -2 \cdot \frac{c \cdot a}{b}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{2 \cdot a}\\ \end{array} \]

   5. Simplified 35.06

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\left(b + -2 \cdot \left(\frac{c}{b} \cdot a + \frac{c}{\frac{{b}^{3}}{c}} \cdot \left(a \cdot a\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{2 \cdot a}\\ \end{array} \]
      Proof

      [Start] 51.01	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(-2 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}} + \left(b + -2 \cdot \frac{c \cdot a}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{2 \cdot a}\\ \end{array} \]
      +-commutative [=>] 51.01	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\left(\left(b + -2 \cdot \frac{c \cdot a}{b}\right) + -2 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{2 \cdot a}\\ \end{array} \]
      associate-+l+ [=>] 51.01	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\left(b + \left(-2 \cdot \frac{c \cdot a}{b} + -2 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{2 \cdot a}\\ \end{array} \]
      +-commutative [<=] 51.01	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + \color{blue}{\left(-2 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}} + -2 \cdot \frac{c \cdot a}{b}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{2 \cdot a}\\ \end{array} \]
      distribute-lft-out [=>] 51.01	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + \color{blue}{-2 \cdot \left(\frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}} + \frac{c \cdot a}{b}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{2 \cdot a}\\ \end{array} \]
      +-commutative [=>] 51.01	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + -2 \cdot \color{blue}{\left(\frac{c \cdot a}{b} + \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{2 \cdot a}\\ \end{array} \]
      associate-/l* [=>] 50.98	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + -2 \cdot \left(\color{blue}{\frac{c}{\frac{b}{a}}} + \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{2 \cdot a}\\ \end{array} \]
      associate-/r/ [=>] 50.98	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + -2 \cdot \left(\color{blue}{\frac{c}{b} \cdot a} + \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{2 \cdot a}\\ \end{array} \]
      associate-/l* [=>] 48.88	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + -2 \cdot \left(\frac{c}{b} \cdot a + \color{blue}{\frac{{c}^{2}}{\frac{{b}^{3}}{{a}^{2}}}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{2 \cdot a}\\ \end{array} \]
      associate-/r/ [=>] 48.73	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + -2 \cdot \left(\frac{c}{b} \cdot a + \color{blue}{\frac{{c}^{2}}{{b}^{3}} \cdot {a}^{2}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{2 \cdot a}\\ \end{array} \]
      unpow2 [=>] 48.73	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + -2 \cdot \left(\frac{c}{b} \cdot a + \frac{\color{blue}{c \cdot c}}{{b}^{3}} \cdot {a}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{2 \cdot a}\\ \end{array} \]
      associate-/l* [=>] 35.06	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + -2 \cdot \left(\frac{c}{b} \cdot a + \color{blue}{\frac{c}{\frac{{b}^{3}}{c}}} \cdot {a}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{2 \cdot a}\\ \end{array} \]
      unpow2 [=>] 35.06	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + -2 \cdot \left(\frac{c}{b} \cdot a + \frac{c}{\frac{{b}^{3}}{c}} \cdot \color{blue}{\left(a \cdot a\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{2 \cdot a}\\ \end{array} \]

   6. Taylor expanded in c around 0 20.62

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + -2 \cdot \color{blue}{\frac{c \cdot a}{b}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{2 \cdot a}\\ \end{array} \]

   7. Simplified 16.93

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + -2 \cdot \color{blue}{\left(a \cdot \frac{c}{b}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{2 \cdot a}\\ \end{array} \]
      Proof

      [Start] 20.62	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + -2 \cdot \frac{c \cdot a}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{2 \cdot a}\\ \end{array} \]
      *-commutative [=>] 20.62	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + -2 \cdot \frac{\color{blue}{a \cdot c}}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{2 \cdot a}\\ \end{array} \]
      associate-*r/ [<=] 16.93	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + -2 \cdot \color{blue}{\left(a \cdot \frac{c}{b}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{2 \cdot a}\\ \end{array} \]

   8. Applied egg-rr 16.93

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + -2 \cdot \left(a \cdot \frac{c}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, c \cdot \frac{a}{b}, b \cdot -2\right) \cdot \frac{0.5}{a}\right)} - 1\\ \end{array} \]
3. Recombined 3 regimes into one program.

4. Final simplification 12.23

   \[\leadsto \begin{array}{l} \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array} \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array} \leq -2 \cdot 10^{-255}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array} \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \left(a \cdot \frac{c}{b}\right) - b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(\mathsf{fma}\left(2, c \cdot \frac{a}{b}, b \cdot -2\right) \cdot \frac{0.5}{a}\right)} + -1\\ \end{array}\\ \mathbf{elif}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array} \leq 2 \cdot 10^{+178}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.22%
Cost	38052

\[\begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array}\\ t_1 := \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\\ t_2 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-255}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \left(a \cdot \frac{c}{b}\right) - b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2 + \left(c \cdot a\right) \cdot \frac{2}{b}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+178}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	12.22%
Cost	38052

\[\begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{2 \cdot a}\\ \end{array}\\ t_1 := \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\\ t_2 := \frac{t_1 - b}{2 \cdot a}\\ t_3 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array}\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_3 \leq -5 \cdot 10^{-241}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(\frac{2 \cdot c}{\frac{b}{a}} - b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array}\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{+178}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	20.12%
Cost	7760

\[\begin{array}{l} t_0 := \frac{2 \cdot c}{\left(-b\right) - b}\\ t_1 := \sqrt{c \cdot \left(a \cdot -4\right)}\\ \mathbf{if}\;b \leq -3 \cdot 10^{-72}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq -6 \cdot 10^{-307}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.85 \cdot 10^{-70}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \left(a \cdot \frac{c}{b}\right) - b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2 + \left(c \cdot a\right) \cdot \frac{2}{b}}{2 \cdot a}\\ \end{array} \]

Alternative 4
Error	20.14%
Cost	7760

\[\begin{array}{l} t_0 := \frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{2 \cdot a}\\ t_1 := \frac{2 \cdot c}{\left(-b\right) - b}\\ t_2 := \sqrt{c \cdot \left(a \cdot -4\right)}\\ \mathbf{if}\;b \leq -4.9 \cdot 10^{-73}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq -6 \cdot 10^{-307}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-70}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t_2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(\frac{2 \cdot c}{\frac{b}{a}} - b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	15.14%
Cost	7760

\[\begin{array}{l} t_0 := \frac{2 \cdot c}{\left(-b\right) - b}\\ t_1 := \frac{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{2 \cdot a}\\ \mathbf{if}\;b \leq -1 \cdot 10^{+137}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq -6 \cdot 10^{-307}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-70}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(\frac{2 \cdot c}{\frac{b}{a}} - b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	27.78%
Cost	7368

\[\begin{array}{l} t_0 := \frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{if}\;b \leq -8.4 \cdot 10^{-73}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{2 \cdot a}\\ \end{array} \]

Alternative 7
Error	34.44%
Cost	644

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

Reproduce

herbie shell --seed 2023121 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))