jeff quadratic root 1

Average Error: 19.8 → 6.2

Time: 15.6s

Precision: binary64

Cost: 51364

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\\ t_1 := \frac{c \cdot 2}{t_0 - b}\\ t_2 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ t_3 := \frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot c}{a \cdot c}\\ \end{array}\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-289}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{-1}{{\left(\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)\right)}^{-0.5}} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(\frac{a + a}{\frac{b}{c}} - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;t_2 \leq 10^{+280}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + \mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(c, a \cdot -4, a \cdot \left(4 \cdot c\right)\right)\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2}\\ \end{array} \]

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 (/ (* c 2.0) (- t_0 b)))
        (t_2 (if (>= b 0.0) (/ (- (- b) t_0) (* a 2.0)) t_1))
        (t_3 (/ (- (- b) b) (* a 2.0))))
   (if (<= t_2 (- INFINITY))
     (if (>= b 0.0) t_3 (/ (* b c) (* a c)))
     (if (<= t_2 -1e-289)
       (if (>= b 0.0)
         (/ (- (/ -1.0 (pow (fma -4.0 (* a c) (* b b)) -0.5)) b) (* a 2.0))
         t_1)
       (if (<= t_2 0.0)
         (if (>= b 0.0) t_3 (/ (* c 2.0) (- (- (/ (+ a a) (/ b c)) b) b)))
         (if (<= t_2 1e+280)
           (if (>= b 0.0)
             (/
              (-
               (- b)
               (sqrt
                (+
                 (* b b)
                 (fma c (* a -4.0) (fma c (* a -4.0) (* a (* 4.0 c)))))))
              (* a 2.0))
             t_1)
           (if (>= b 0.0) t_3 (/ (* c 2.0) (* b -2.0)))))))))

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 = (c * 2.0) / (t_0 - b);
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - t_0) / (a * 2.0);
	} else {
		tmp = t_1;
	}
	double t_2 = tmp;
	double t_3 = (-b - b) / (a * 2.0);
	double tmp_2;
	if (t_2 <= -((double) INFINITY)) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_3;
		} else {
			tmp_3 = (b * c) / (a * c);
		}
		tmp_2 = tmp_3;
	} else if (t_2 <= -1e-289) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = ((-1.0 / pow(fma(-4.0, (a * c), (b * b)), -0.5)) - b) / (a * 2.0);
		} else {
			tmp_4 = t_1;
		}
		tmp_2 = tmp_4;
	} else if (t_2 <= 0.0) {
		double tmp_5;
		if (b >= 0.0) {
			tmp_5 = t_3;
		} else {
			tmp_5 = (c * 2.0) / ((((a + a) / (b / c)) - b) - b);
		}
		tmp_2 = tmp_5;
	} else if (t_2 <= 1e+280) {
		double tmp_6;
		if (b >= 0.0) {
			tmp_6 = (-b - sqrt(((b * b) + fma(c, (a * -4.0), fma(c, (a * -4.0), (a * (4.0 * c))))))) / (a * 2.0);
		} else {
			tmp_6 = t_1;
		}
		tmp_2 = tmp_6;
	} else if (b >= 0.0) {
		tmp_2 = t_3;
	} else {
		tmp_2 = (c * 2.0) / (b * -2.0);
	}
	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(c * 2.0) / Float64(t_0 - b))
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(Float64(-b) - t_0) / Float64(a * 2.0));
	else
		tmp = t_1;
	end
	t_2 = tmp
	t_3 = Float64(Float64(Float64(-b) - b) / Float64(a * 2.0))
	tmp_2 = 0.0
	if (t_2 <= Float64(-Inf))
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_3;
		else
			tmp_3 = Float64(Float64(b * c) / Float64(a * c));
		end
		tmp_2 = tmp_3;
	elseif (t_2 <= -1e-289)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(Float64(-1.0 / (fma(-4.0, Float64(a * c), Float64(b * b)) ^ -0.5)) - b) / Float64(a * 2.0));
		else
			tmp_4 = t_1;
		end
		tmp_2 = tmp_4;
	elseif (t_2 <= 0.0)
		tmp_5 = 0.0
		if (b >= 0.0)
			tmp_5 = t_3;
		else
			tmp_5 = Float64(Float64(c * 2.0) / Float64(Float64(Float64(Float64(a + a) / Float64(b / c)) - b) - b));
		end
		tmp_2 = tmp_5;
	elseif (t_2 <= 1e+280)
		tmp_6 = 0.0
		if (b >= 0.0)
			tmp_6 = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) + fma(c, Float64(a * -4.0), fma(c, Float64(a * -4.0), Float64(a * Float64(4.0 * c))))))) / Float64(a * 2.0));
		else
			tmp_6 = t_1;
		end
		tmp_2 = tmp_6;
	elseif (b >= 0.0)
		tmp_2 = t_3;
	else
		tmp_2 = Float64(Float64(c * 2.0) / Float64(b * -2.0));
	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[(c * 2.0), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], t$95$1]}, Block[{t$95$3 = N[(N[((-b) - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], If[GreaterEqual[b, 0.0], t$95$3, N[(N[(b * c), $MachinePrecision] / N[(a * c), $MachinePrecision]), $MachinePrecision]], If[LessEqual[t$95$2, -1e-289], If[GreaterEqual[b, 0.0], N[(N[(N[(-1.0 / N[Power[N[(-4.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], t$95$1], If[LessEqual[t$95$2, 0.0], If[GreaterEqual[b, 0.0], t$95$3, N[(N[(c * 2.0), $MachinePrecision] / N[(N[(N[(N[(a + a), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[t$95$2, 1e+280], If[GreaterEqual[b, 0.0], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision] + N[(a * N[(4.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], t$95$1], If[GreaterEqual[b, 0.0], t$95$3, N[(N[(c * 2.0), $MachinePrecision] / N[(b * -2.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 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 15.6

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

   3. Applied egg-rr 15.6

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

   4. Taylor expanded in b around -inf 15.6

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

   5. Simplified 15.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{a}}{c} \cdot c\\ \end{array} \]
      Proof
      (/.f64 (/.f64 b a) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 b a))) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (Rewrite<= metadata-eval (/.f64 -4 -4)) (/.f64 b a)) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -4 b) (*.f64 -4 a))) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (*.f64 -4 b) (*.f64 (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 -4) (sqrt.f64 -4))) a)) c): 201 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (*.f64 -4 b) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 -4) 2)) a)) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (*.f64 -4 b) (Rewrite<= *-commutative_binary64 (*.f64 a (pow.f64 (sqrt.f64 -4) 2)))) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 b -4)) (*.f64 a (pow.f64 (sqrt.f64 -4) 2))) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 b (*.f64 a (pow.f64 (sqrt.f64 -4) 2))) -4)) c): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (/.f64 b (*.f64 a (pow.f64 (sqrt.f64 -4) 2))) c) -4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> associate-/l/_binary64 (/.f64 b (*.f64 c (*.f64 a (pow.f64 (sqrt.f64 -4) 2))))) -4): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 -4 (/.f64 b (*.f64 c (*.f64 a (pow.f64 (sqrt.f64 -4) 2)))))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 15.6

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

   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)))))) < -1e-289

   1. Initial program 2.3

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

   2. Applied egg-rr 16.0

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

   3. Applied egg-rr 2.5

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

   4. Applied egg-rr 2.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \frac{1}{\color{blue}{{\left(\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)\right)}^{-0.5}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   if -1e-289 < (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 36.9

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

   2. Taylor expanded in b around inf 37.0

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

   3. Taylor expanded in b around -inf 12.9

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

   4. Simplified 10.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(\frac{c}{b} \cdot \left(a \cdot 2\right) - b\right)}}\\ \end{array} \]
      Proof
      (-.f64 (*.f64 (/.f64 c b) (*.f64 a 2)) b): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 c b) a) 2)) b): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 c a) b)) 2) b): 14 points increase in error, 23 points decrease in error
      (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 (/.f64 (*.f64 c a) b))) b): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 2 (/.f64 (*.f64 c a) b)) (neg.f64 b))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 2 (/.f64 (*.f64 c a) b)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 b))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 10.5

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

   if 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)))))) < 1e280

   1. Initial program 2.2

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

   2. Applied egg-rr 2.2

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

   if 1e280 < (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 60.6

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

   2. Taylor expanded in b around inf 20.3

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

   3. Taylor expanded in b around -inf 20.5

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

   4. Simplified 14.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(\frac{c}{b} \cdot \left(a \cdot 2\right) - b\right)}}\\ \end{array} \]
      Proof
      (-.f64 (*.f64 (/.f64 c b) (*.f64 a 2)) b): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 c b) a) 2)) b): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 c a) b)) 2) b): 14 points increase in error, 23 points decrease in error
      (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 (/.f64 (*.f64 c a) b))) b): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 2 (/.f64 (*.f64 c a) b)) (neg.f64 b))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 2 (/.f64 (*.f64 c a) b)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 b))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in b around inf 14.9

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

   6. Simplified 14.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{b \cdot -2}}\\ \end{array} \]
      Proof
      (*.f64 b -2): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 -2 b)): 0 points increase in error, 0 points decrease in error
3. Recombined 5 regimes into one program.

4. Final simplification 6.2

   \[\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 \cdot c}{a \cdot c}\\ \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 -1 \cdot 10^{-289}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{-1}{{\left(\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)\right)}^{-0.5}} - b}{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) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(\frac{a + a}{\frac{b}{c}} - b\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 10^{+280}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + \mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(c, a \cdot -4, a \cdot \left(4 \cdot c\right)\right)\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{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.2
Cost	38052

\[\begin{array}{l} t_0 := \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\\ t_1 := \frac{\left(-b\right) - t_0}{a \cdot 2}\\ t_2 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + -4 \cdot \left(a \cdot c\right)} - b}\\ \end{array}\\ t_3 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t_0 - b}\\ \end{array}\\ t_4 := \frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot c}{a \cdot c}\\ \end{array}\\ \mathbf{elif}\;t_3 \leq -1 \cdot 10^{-289}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(\frac{a + a}{\frac{b}{c}} - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;t_3 \leq 10^{+280}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2}\\ \end{array} \]

Alternative 2
Error	6.2
Cost	38052

\[\begin{array}{l} t_0 := \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\\ t_1 := \frac{c \cdot 2}{t_0 - b}\\ t_2 := \frac{\left(-b\right) - t_0}{a \cdot 2}\\ t_3 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ t_4 := \frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot c}{a \cdot c}\\ \end{array}\\ \mathbf{elif}\;t_3 \leq -1 \cdot 10^{-289}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{-1}{{\left(\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)\right)}^{-0.5}} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(\frac{a + a}{\frac{b}{c}} - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;t_3 \leq 10^{+280}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + -4 \cdot \left(a \cdot c\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2}\\ \end{array} \]

Alternative 3
Error	14.2
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -3.4659304561553807 \cdot 10^{+159}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, -2, 2 \cdot \frac{a}{\frac{b}{c}}\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array} \]

Alternative 4
Error	17.5
Cost	7368

\[\begin{array}{l} t_0 := \frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{if}\;b \leq -1.5 \cdot 10^{-14}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(\frac{a + a}{\frac{b}{c}} - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{c \cdot \left(a \cdot -4\right)} - b}\\ \end{array} \]

Alternative 5
Error	22.0
Cost	1092

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

Alternative 6
Error	44.9
Cost	644

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

Alternative 7
Error	44.4
Cost	644

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

Alternative 8
Error	22.2
Cost	644

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

Reproduce

herbie shell --seed 2022294 
(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)))))))