jeff quadratic root 1

Average Error: 20.3 → 7.0

Time: 20.0s

Precision: binary64

Cost: 51300

Math

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

↓

\[\begin{array}{l} t_0 := a \cdot \left(4 \cdot c\right)\\ t_1 := \sqrt{b \cdot b + \mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(c, a \cdot -4, t_0\right)\right)}\\ t_2 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ t_3 := \frac{\left(-b\right) - t_2}{a \cdot 2}\\ t_4 := \frac{c \cdot 2}{t_2 - b}\\ t_5 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array}\\ t_6 := \frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{if}\;t_5 \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(b, -2, \frac{a + a}{\frac{b}{c}}\right)}\\ \end{array}\\ \mathbf{elif}\;t_5 \leq -1 \cdot 10^{-274}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array}\\ \mathbf{elif}\;t_5 \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_6\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(\left(a \cdot 2\right) \cdot \frac{c}{b} - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;t_5 \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t_1 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_6\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{b \cdot b + t_0} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \end{array} \]

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 (* a (* 4.0 c)))
        (t_1 (sqrt (+ (* b b) (fma c (* a -4.0) (fma c (* a -4.0) t_0)))))
        (t_2 (sqrt (- (* b b) (* (* 4.0 a) c))))
        (t_3 (/ (- (- b) t_2) (* a 2.0)))
        (t_4 (/ (* c 2.0) (- t_2 b)))
        (t_5 (if (>= b 0.0) t_3 t_4))
        (t_6 (/ (- (- b) b) (* a 2.0))))
   (if (<= t_5 (- INFINITY))
     (if (>= b 0.0)
       (- (/ c b) (/ b a))
       (/ (* c 2.0) (fma b -2.0 (/ (+ a a) (/ b c)))))
     (if (<= t_5 -1e-274)
       (if (>= b 0.0) (/ (- (- b) t_1) (* a 2.0)) t_4)
       (if (<= t_5 0.0)
         (if (>= b 0.0) t_6 (/ (* c 2.0) (- (- (* (* a 2.0) (/ c b)) b) b)))
         (if (<= t_5 2e+260)
           (if (>= b 0.0) t_3 (/ (* c 2.0) (- t_1 b)))
           (if (>= b 0.0)
             t_6
             (*
              (/ (+ c c) (+ (* b b) t_0))
              (- b (sqrt (* a (* c -4.0))))))))))))

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 = a * (4.0 * c);
	double t_1 = sqrt(((b * b) + fma(c, (a * -4.0), fma(c, (a * -4.0), t_0))));
	double t_2 = sqrt(((b * b) - ((4.0 * a) * c)));
	double t_3 = (-b - t_2) / (a * 2.0);
	double t_4 = (c * 2.0) / (t_2 - b);
	double tmp;
	if (b >= 0.0) {
		tmp = t_3;
	} else {
		tmp = t_4;
	}
	double t_5 = tmp;
	double t_6 = (-b - b) / (a * 2.0);
	double tmp_2;
	if (t_5 <= -((double) INFINITY)) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c / b) - (b / a);
		} else {
			tmp_3 = (c * 2.0) / fma(b, -2.0, ((a + a) / (b / c)));
		}
		tmp_2 = tmp_3;
	} else if (t_5 <= -1e-274) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (-b - t_1) / (a * 2.0);
		} else {
			tmp_4 = t_4;
		}
		tmp_2 = tmp_4;
	} else if (t_5 <= 0.0) {
		double tmp_5;
		if (b >= 0.0) {
			tmp_5 = t_6;
		} else {
			tmp_5 = (c * 2.0) / ((((a * 2.0) * (c / b)) - b) - b);
		}
		tmp_2 = tmp_5;
	} else if (t_5 <= 2e+260) {
		double tmp_6;
		if (b >= 0.0) {
			tmp_6 = t_3;
		} else {
			tmp_6 = (c * 2.0) / (t_1 - b);
		}
		tmp_2 = tmp_6;
	} else if (b >= 0.0) {
		tmp_2 = t_6;
	} else {
		tmp_2 = ((c + c) / ((b * b) + t_0)) * (b - sqrt((a * (c * -4.0))));
	}
	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 = Float64(a * Float64(4.0 * c))
	t_1 = sqrt(Float64(Float64(b * b) + fma(c, Float64(a * -4.0), fma(c, Float64(a * -4.0), t_0))))
	t_2 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	t_3 = Float64(Float64(Float64(-b) - t_2) / Float64(a * 2.0))
	t_4 = Float64(Float64(c * 2.0) / Float64(t_2 - b))
	tmp = 0.0
	if (b >= 0.0)
		tmp = t_3;
	else
		tmp = t_4;
	end
	t_5 = tmp
	t_6 = Float64(Float64(Float64(-b) - b) / Float64(a * 2.0))
	tmp_2 = 0.0
	if (t_5 <= Float64(-Inf))
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(c / b) - Float64(b / a));
		else
			tmp_3 = Float64(Float64(c * 2.0) / fma(b, -2.0, Float64(Float64(a + a) / Float64(b / c))));
		end
		tmp_2 = tmp_3;
	elseif (t_5 <= -1e-274)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(Float64(-b) - t_1) / Float64(a * 2.0));
		else
			tmp_4 = t_4;
		end
		tmp_2 = tmp_4;
	elseif (t_5 <= 0.0)
		tmp_5 = 0.0
		if (b >= 0.0)
			tmp_5 = t_6;
		else
			tmp_5 = Float64(Float64(c * 2.0) / Float64(Float64(Float64(Float64(a * 2.0) * Float64(c / b)) - b) - b));
		end
		tmp_2 = tmp_5;
	elseif (t_5 <= 2e+260)
		tmp_6 = 0.0
		if (b >= 0.0)
			tmp_6 = t_3;
		else
			tmp_6 = Float64(Float64(c * 2.0) / Float64(t_1 - b));
		end
		tmp_2 = tmp_6;
	elseif (b >= 0.0)
		tmp_2 = t_6;
	else
		tmp_2 = Float64(Float64(Float64(c + c) / Float64(Float64(b * b) + t_0)) * Float64(b - sqrt(Float64(a * Float64(c * -4.0)))));
	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[(a * N[(4.0 * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[((-b) - t$95$2), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(c * 2.0), $MachinePrecision] / N[(t$95$2 - b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = If[GreaterEqual[b, 0.0], t$95$3, t$95$4]}, Block[{t$95$6 = N[(N[((-b) - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, (-Infinity)], If[GreaterEqual[b, 0.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(b * -2.0 + N[(N[(a + a), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[t$95$5, -1e-274], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$1), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], t$95$4], If[LessEqual[t$95$5, 0.0], If[GreaterEqual[b, 0.0], t$95$6, N[(N[(c * 2.0), $MachinePrecision] / N[(N[(N[(N[(a * 2.0), $MachinePrecision] * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[t$95$5, 2e+260], If[GreaterEqual[b, 0.0], t$95$3, N[(N[(c * 2.0), $MachinePrecision] / N[(t$95$1 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$6, N[(N[(N[(c + c), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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 64.0

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

   3. Simplified 64.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\mathsf{fma}\left(b, -2, \frac{c}{b} \cdot \left(a \cdot 2\right)\right)}}\\ \end{array} \]
      Proof
      (fma.f64 b -2 (*.f64 (/.f64 c b) (*.f64 a 2))): 0 points increase in error, 0 points decrease in error
      (fma.f64 b -2 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 c b) a) 2))): 0 points increase in error, 0 points decrease in error
      (fma.f64 b -2 (*.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 c a) b)) 2)): 29 points increase in error, 18 points decrease in error
      (fma.f64 b -2 (Rewrite<= *-commutative_binary64 (*.f64 2 (/.f64 (*.f64 c a) b)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 b -2) (*.f64 2 (/.f64 (*.f64 c a) b)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 -2 b)) (*.f64 2 (/.f64 (*.f64 c a) b))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 2 (/.f64 (*.f64 c a) b)) (*.f64 -2 b))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 64.0

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

   5. Taylor expanded in b around inf 19.4

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

   6. Taylor expanded in b around 0 16.1

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

   7. Simplified 16.1

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

   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)))))) < -9.99999999999999966e-275

   1. Initial program 2.9

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

   2. Applied egg-rr 2.9

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

   if -9.99999999999999966e-275 < (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 38.4

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

   2. Taylor expanded in b around inf 38.5

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

   3. Taylor expanded in b around -inf 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 10.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(\frac{c}{b} \cdot \left(a \cdot 2\right) - b\right)}}\\ \end{array} \]
      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): 29 points increase in error, 18 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

   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)))))) < 2.00000000000000013e260

   1. Initial program 2.7

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

   2. Applied egg-rr 2.7

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

   if 2.00000000000000013e260 < (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 56.4

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

   2. Taylor expanded in b around inf 18.0

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

   3. Taylor expanded in b around 0 18.0

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

   4. Simplified 18.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \end{array} \]
      Proof
      (*.f64 c (*.f64 a -4)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 c a) -4)): 2 points increase in error, 2 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 -4 (*.f64 c a))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 18.0

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

   6. Applied egg-rr 18.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{b \cdot b - a \cdot \left(-4 \cdot c\right)} \cdot \left(b - \sqrt{a \cdot \left(-4 \cdot c\right)}\right)\\ \end{array} \]
3. Recombined 5 regimes into one program.

4. Final simplification 7.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array} \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(b, -2, \frac{a + a}{\frac{b}{c}}\right)}\\ \end{array}\\ \mathbf{elif}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array} \leq -1 \cdot 10^{-274}:\\ \;\;\;\;\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 - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\\ \mathbf{elif}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - 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(\left(a \cdot 2\right) \cdot \frac{c}{b} - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array} \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\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)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{b \cdot b + a \cdot \left(4 \cdot c\right)} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	6.9
Cost	51300

\[\begin{array}{l} t_0 := \frac{\left(-b\right) - b}{a \cdot 2}\\ t_1 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ t_2 := \frac{\left(-b\right) - t_1}{a \cdot 2}\\ t_3 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t_1 - b}\\ \end{array}\\ t_4 := a \cdot \left(4 \cdot c\right)\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(b, -2, \frac{a + a}{\frac{b}{c}}\right)}\\ \end{array}\\ \mathbf{elif}\;t_3 \leq -1 \cdot 10^{-274}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(\left(a \cdot 2\right) \cdot \frac{c}{b} - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{+260}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + \mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(c, a \cdot -4, t_4\right)\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{b \cdot b + t_4} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \end{array} \]

Alternative 2
Error	6.9
Cost	38372

\[\begin{array}{l} t_0 := \frac{\left(-b\right) - b}{a \cdot 2}\\ t_1 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ t_2 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t_1 - b}\\ \end{array}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(b, -2, \frac{a + a}{\frac{b}{c}}\right)}\\ \end{array}\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-274}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(\left(a \cdot 2\right) \cdot \frac{c}{b} - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+260}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{b \cdot b + a \cdot \left(4 \cdot c\right)} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \end{array} \]

Alternative 3
Error	6.9
Cost	38052

\[\begin{array}{l} t_0 := \frac{\left(-b\right) - b}{a \cdot 2}\\ t_1 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ t_2 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t_1 - b}\\ \end{array}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(b, -2, \frac{a + a}{\frac{b}{c}}\right)}\\ \end{array}\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-274}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(\left(a \cdot 2\right) \cdot \frac{c}{b} - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+260}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{b + \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \end{array} \]

Alternative 4
Error	10.0
Cost	7952

\[\begin{array}{l} t_0 := \frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{if}\;b \leq -1.62 \cdot 10^{-99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(\left(a \cdot 2\right) \cdot \frac{c}{b} - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-299}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{c \cdot \left(a \cdot -4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{+145}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(2 \cdot \left(c \cdot \left(a \cdot \frac{1}{b}\right)\right) - b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(b, -2, \frac{a + a}{\frac{b}{c}}\right)}\\ \end{array} \]

Alternative 5
Error	7.0
Cost	7952

\[\begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;b \leq -1.730925771395005 \cdot 10^{+76}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{b \cdot -2}\\ \end{array}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-299}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \mathsf{fma}\left(-2, a \cdot \frac{c}{b}, b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{+145}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(2 \cdot \left(c \cdot \left(a \cdot \frac{1}{b}\right)\right) - b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(b, -2, \frac{a + a}{\frac{b}{c}}\right)}\\ \end{array} \]

Alternative 6
Error	18.0
Cost	7368

\[\begin{array}{l} t_0 := \frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{if}\;b \leq -1.62 \cdot 10^{-99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(\left(a \cdot 2\right) \cdot \frac{c}{b} - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{b + \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \end{array} \]

Alternative 7
Error	18.0
Cost	7368

\[\begin{array}{l} t_0 := \frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{if}\;b \leq -1.62 \cdot 10^{-99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(\left(a \cdot 2\right) \cdot \frac{c}{b} - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{b + \sqrt{c \cdot \left(a \cdot -4\right)}}{c}}\\ \end{array} \]

Alternative 8
Error	17.8
Cost	7368

\[\begin{array}{l} t_0 := \frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{if}\;b \leq -1.62 \cdot 10^{-99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(\left(a \cdot 2\right) \cdot \frac{c}{b} - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{c \cdot \left(a \cdot -4\right)} - b}\\ \end{array} \]

Alternative 9
Error	22.4
Cost	1092

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

Alternative 10
Error	44.9
Cost	644

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

Alternative 11
Error	22.6
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 2022308 
(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)))))))