jeff quadratic root 1

Average Error: 19.5 → 6.6

Time: 20.3s

Precision: binary64

Cost: 38052

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\\ t_1 := \frac{\left(-b\right) - t_0}{a \cdot 2}\\ t_2 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t_0 - b}\\ \end{array}\\ t_3 := \left(-b\right) - b\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{t_3}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;t_2 \leq -5 \cdot 10^{-254}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(2, a \cdot \frac{c}{b}, b \cdot -2\right)}\\ \end{array}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+248}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t_3}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* b b) (* c (* a -4.0)))))
        (t_1 (/ (- (- b) t_0) (* a 2.0)))
        (t_2 (if (>= b 0.0) t_1 (/ (* c 2.0) (- t_0 b))))
        (t_3 (- (- b) b)))
   (if (<= t_2 (- INFINITY))
     (if (>= b 0.0) (/ t_3 (* a 2.0)) (/ b a))
     (if (<= t_2 -5e-254)
       t_2
       (if (<= t_2 0.0)
         (if (>= b 0.0) t_1 (/ (* c 2.0) (fma 2.0 (* a (/ c b)) (* b -2.0))))
         (if (<= t_2 5e+248)
           t_2
           (if (>= b 0.0) (- (/ c b) (/ b a)) (/ (* c 2.0) t_3))))))))

C

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

↓

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) + (c * (a * -4.0))));
	double t_1 = (-b - t_0) / (a * 2.0);
	double tmp;
	if (b >= 0.0) {
		tmp = t_1;
	} else {
		tmp = (c * 2.0) / (t_0 - b);
	}
	double t_2 = tmp;
	double t_3 = -b - b;
	double tmp_2;
	if (t_2 <= -((double) INFINITY)) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_3 / (a * 2.0);
		} else {
			tmp_3 = b / a;
		}
		tmp_2 = tmp_3;
	} else if (t_2 <= -5e-254) {
		tmp_2 = t_2;
	} else if (t_2 <= 0.0) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = t_1;
		} else {
			tmp_4 = (c * 2.0) / fma(2.0, (a * (c / b)), (b * -2.0));
		}
		tmp_2 = tmp_4;
	} else if (t_2 <= 5e+248) {
		tmp_2 = t_2;
	} else if (b >= 0.0) {
		tmp_2 = (c / b) - (b / a);
	} else {
		tmp_2 = (c * 2.0) / t_3;
	}
	return tmp_2;
}

Julia

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

↓

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0))))
	t_1 = Float64(Float64(Float64(-b) - t_0) / Float64(a * 2.0))
	tmp = 0.0
	if (b >= 0.0)
		tmp = t_1;
	else
		tmp = Float64(Float64(c * 2.0) / Float64(t_0 - b));
	end
	t_2 = tmp
	t_3 = Float64(Float64(-b) - b)
	tmp_2 = 0.0
	if (t_2 <= Float64(-Inf))
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(t_3 / Float64(a * 2.0));
		else
			tmp_3 = Float64(b / a);
		end
		tmp_2 = tmp_3;
	elseif (t_2 <= -5e-254)
		tmp_2 = t_2;
	elseif (t_2 <= 0.0)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = t_1;
		else
			tmp_4 = Float64(Float64(c * 2.0) / fma(2.0, Float64(a * Float64(c / b)), Float64(b * -2.0)));
		end
		tmp_2 = tmp_4;
	elseif (t_2 <= 5e+248)
		tmp_2 = t_2;
	elseif (b >= 0.0)
		tmp_2 = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp_2 = Float64(Float64(c * 2.0) / t_3);
	end
	return tmp_2
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], 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], 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]]

↓

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[((-b) - t$95$0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[GreaterEqual[b, 0.0], t$95$1, N[(N[(c * 2.0), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]]}, Block[{t$95$3 = N[((-b) - b), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], If[GreaterEqual[b, 0.0], N[(t$95$3 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(b / a), $MachinePrecision]], If[LessEqual[t$95$2, -5e-254], t$95$2, If[LessEqual[t$95$2, 0.0], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(c * 2.0), $MachinePrecision] / N[(2.0 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(b * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[t$95$2, 5e+248], t$95$2, If[GreaterEqual[b, 0.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 64.0

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

   2. Taylor expanded in b around inf 16.5

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

   3. Taylor expanded in b around -inf 16.5

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

   4. Simplified 16.5

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

      [Start] 16.5	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{c \cdot a}{b} + -2 \cdot b}\\ \end{array} \]
      fma-def [=>] 16.5	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c \cdot a}{b}, -2 \cdot b\right)}}\\ \end{array} \]
      associate-/l* [=>] 16.5	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, -2 \cdot b\right)}}\\ \end{array} \]
      associate-/r/ [=>] 16.5	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\mathsf{fma}\left(2, \frac{c}{b} \cdot a, -2 \cdot b\right)}}\\ \end{array} \]
      *-commutative [=>] 16.5	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{b} \cdot a, b \cdot -2\right)}\\ \end{array} \]

   5. Taylor expanded in c around inf 16.5

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

   if -inf.0 < (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (/.f64 (*.f64 2 c) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))) < -5.0000000000000003e-254 or -0.0 < (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (/.f64 (*.f64 2 c) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))) < 4.9999999999999996e248

   1. Initial program 2.7

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

   if -5.0000000000000003e-254 < (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (/.f64 (*.f64 2 c) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))) < -0.0

   1. Initial program 35.7

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

   2. Taylor expanded in b around -inf 11.9

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

   3. Simplified 9.8

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

      [Start] 11.9	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{c \cdot a}{b} + -2 \cdot b}\\ \end{array} \]
      fma-def [=>] 11.9	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c \cdot a}{b}, -2 \cdot b\right)}}\\ \end{array} \]
      associate-/l* [=>] 9.8	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, -2 \cdot b\right)}}\\ \end{array} \]
      associate-/r/ [=>] 9.8	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\mathsf{fma}\left(2, \frac{c}{b} \cdot a, -2 \cdot b\right)}}\\ \end{array} \]
      *-commutative [=>] 9.8	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{b} \cdot a, b \cdot -2\right)}\\ \end{array} \]

   if 4.9999999999999996e248 < (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (/.f64 (*.f64 2 c) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))

   1. Initial program 55.4

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

   2. Taylor expanded in b around -inf 51.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + -1 \cdot b}}\\ \end{array} \]

   3. Taylor expanded in b around inf 16.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + -1 \cdot b}\\ \end{array} \]

   4. Simplified 16.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c}{b} - \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + -1 \cdot b}\\ \end{array} \]
      Proof

      [Start] 16.3	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} + -1 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + -1 \cdot b}\\ \end{array} \]
      mul-1-neg [=>] 16.3	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + -1 \cdot b}\\ \end{array} \]
      unsub-neg [=>] 16.3	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c}{b} - \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + -1 \cdot b}\\ \end{array} \]

3. Recombined 4 regimes into one program.

4. Final simplification 6.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array} \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array} \leq -5 \cdot 10^{-254}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array} \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(2, a \cdot \frac{c}{b}, b \cdot -2\right)}\\ \end{array}\\ \mathbf{elif}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array} \leq 5 \cdot 10^{+248}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.5
Cost	38052

\[\begin{array}{l} t_0 := \left(-b\right) - b\\ t_1 := \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\\ t_2 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t_1 - b}\\ \end{array}\\ t_3 := \frac{t_0}{a \cdot 2}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-285}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{-2 \cdot \left(c \cdot \frac{a}{b} - b\right)}\\ \end{array}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+248}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t_0}\\ \end{array} \]

Alternative 2
Error	10.4
Cost	7888

\[\begin{array}{l} t_0 := \left(-b\right) - b\\ t_1 := \frac{t_0}{a \cdot 2}\\ \mathbf{if}\;b \leq -3.2 \cdot 10^{-108}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{-2 \cdot \left(c \cdot \frac{a}{b} - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq -8.9 \cdot 10^{-305}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{c \cdot \left(a \cdot -4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.25 \cdot 10^{+38}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t_0}\\ \end{array} \]

Alternative 3
Error	7.0
Cost	7888

\[\begin{array}{l} t_0 := \left(-b\right) - b\\ t_1 := \frac{t_0}{a \cdot 2}\\ \mathbf{if}\;b \leq -4 \cdot 10^{+153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{-2 \cdot \left(c \cdot \frac{a}{b} - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq -8.9 \cdot 10^{-305}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.25 \cdot 10^{+38}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t_0}\\ \end{array} \]

Alternative 4
Error	13.5
Cost	7696

\[\begin{array}{l} t_0 := \left(-b\right) - b\\ t_1 := \frac{c \cdot 2}{t_0}\\ t_2 := \frac{t_0}{a \cdot 2}\\ t_3 := \sqrt{c \cdot \left(a \cdot -4\right)}\\ \mathbf{if}\;b \leq -1.4 \cdot 10^{-108}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{-2 \cdot \left(c \cdot \frac{a}{b} - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq -8.9 \cdot 10^{-305}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t_3 - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-40}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_3}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	14.2
Cost	7632

\[\begin{array}{l} t_0 := \left(-b\right) - b\\ t_1 := \frac{c \cdot 2}{t_0}\\ t_2 := \frac{t_0}{a \cdot 2}\\ t_3 := \sqrt{c \cdot \left(a \cdot -4\right)}\\ \mathbf{if}\;b \leq -3.2 \cdot 10^{-114}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{-2 \cdot \left(c \cdot \frac{a}{b} - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq -8.9 \cdot 10^{-305}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{b + t_3}\\ \end{array}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-40}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b - t_3}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	13.9
Cost	7632

\[\begin{array}{l} t_0 := \left(-b\right) - b\\ t_1 := \frac{t_0}{a \cdot 2}\\ t_2 := \frac{c \cdot 2}{t_0}\\ \mathbf{if}\;b \leq -2.05 \cdot 10^{-109}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{-2 \cdot \left(c \cdot \frac{a}{b} - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq -8.9 \cdot 10^{-305}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{b - \sqrt{a \cdot \left(c \cdot -4\right)}}{c}}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-41}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	13.9
Cost	7632

\[\begin{array}{l} t_0 := \left(-b\right) - b\\ t_1 := \frac{c \cdot 2}{t_0}\\ t_2 := \frac{t_0}{a \cdot 2}\\ t_3 := \sqrt{c \cdot \left(a \cdot -4\right)}\\ \mathbf{if}\;b \leq -4.7 \cdot 10^{-110}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -2}{-2 \cdot \left(c \cdot \frac{a}{b} - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq -8.9 \cdot 10^{-305}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t_3 - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-41}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b - t_3}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	18.7
Cost	7368

\[\begin{array}{l} t_0 := \frac{c \cdot 2}{\left(-b\right) - b}\\ \mathbf{if}\;b \leq 1.65 \cdot 10^{-41}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0.5}{\frac{a}{b - \sqrt{a \cdot \left(c \cdot -4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	18.7
Cost	7368

\[\begin{array}{l} t_0 := \frac{c \cdot 2}{\left(-b\right) - b}\\ \mathbf{if}\;b \leq 2.5 \cdot 10^{-41}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	22.5
Cost	964

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

Alternative 11
Error	44.8
Cost	644

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

Alternative 12
Error	22.7
Cost	644

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

Alternative 13
Error	22.7
Cost	644

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

Reproduce

herbie shell --seed 2023066 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  :precision binary64
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))