jeff quadratic root 1

Average Error: 19.9 → 8.1

Time: 18.2s

Precision: binary64

Cost: 51300

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\\ t_1 := \frac{\left(-b\right) - t_0}{a \cdot 2}\\ t_2 := \frac{c \cdot 2}{t_0 - b}\\ t_3 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array}\\ t_4 := \mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\\ \mathbf{elif}\;t_3 \leq -1 \cdot 10^{-174}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{t_4}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t_4}\\ \end{array}\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{+266}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + \mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(c, a \cdot -4, a \cdot \left(4 \cdot c\right)\right)\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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 (/ (- (- b) t_0) (* a 2.0)))
        (t_2 (/ (* c 2.0) (- t_0 b)))
        (t_3 (if (>= b 0.0) t_1 t_2))
        (t_4 (fma 2.0 (/ c (/ b a)) (* b -2.0))))
   (if (<= t_3 (- INFINITY))
     (if (>= b 0.0) (/ (- (- b) b) (* a 2.0)) (/ (- b) a))
     (if (<= t_3 -1e-174)
       t_3
       (if (<= t_3 0.0)
         (if (>= b 0.0) (/ t_4 (* a 2.0)) (/ (* c 2.0) t_4))
         (if (<= t_3 2e+266)
           (if (>= b 0.0)
             t_1
             (/
              (* c 2.0)
              (-
               (sqrt
                (+
                 (* b b)
                 (fma c (* a -4.0) (fma c (* a -4.0) (* a (* 4.0 c))))))
               b)))
           (if (>= b 0.0) (- (/ c b) (/ b a)) t_2)))))))

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 t_2 = (c * 2.0) / (t_0 - b);
	double tmp;
	if (b >= 0.0) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	double t_3 = tmp;
	double t_4 = fma(2.0, (c / (b / a)), (b * -2.0));
	double tmp_2;
	if (t_3 <= -((double) INFINITY)) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - b) / (a * 2.0);
		} else {
			tmp_3 = -b / a;
		}
		tmp_2 = tmp_3;
	} else if (t_3 <= -1e-174) {
		tmp_2 = t_3;
	} else if (t_3 <= 0.0) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = t_4 / (a * 2.0);
		} else {
			tmp_4 = (c * 2.0) / t_4;
		}
		tmp_2 = tmp_4;
	} else if (t_3 <= 2e+266) {
		double tmp_5;
		if (b >= 0.0) {
			tmp_5 = t_1;
		} else {
			tmp_5 = (c * 2.0) / (sqrt(((b * b) + fma(c, (a * -4.0), fma(c, (a * -4.0), (a * (4.0 * c)))))) - b);
		}
		tmp_2 = tmp_5;
	} else if (b >= 0.0) {
		tmp_2 = (c / b) - (b / a);
	} else {
		tmp_2 = t_2;
	}
	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))
	t_2 = Float64(Float64(c * 2.0) / Float64(t_0 - b))
	tmp = 0.0
	if (b >= 0.0)
		tmp = t_1;
	else
		tmp = t_2;
	end
	t_3 = tmp
	t_4 = fma(2.0, Float64(c / Float64(b / a)), Float64(b * -2.0))
	tmp_2 = 0.0
	if (t_3 <= Float64(-Inf))
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(-b) - b) / Float64(a * 2.0));
		else
			tmp_3 = Float64(Float64(-b) / a);
		end
		tmp_2 = tmp_3;
	elseif (t_3 <= -1e-174)
		tmp_2 = t_3;
	elseif (t_3 <= 0.0)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(t_4 / Float64(a * 2.0));
		else
			tmp_4 = Float64(Float64(c * 2.0) / t_4);
		end
		tmp_2 = tmp_4;
	elseif (t_3 <= 2e+266)
		tmp_5 = 0.0
		if (b >= 0.0)
			tmp_5 = t_1;
		else
			tmp_5 = Float64(Float64(c * 2.0) / Float64(sqrt(Float64(Float64(b * b) + fma(c, Float64(a * -4.0), fma(c, Float64(a * -4.0), Float64(a * Float64(4.0 * c)))))) - b));
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp_2 = t_2;
	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 = N[(N[(c * 2.0), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = If[GreaterEqual[b, 0.0], t$95$1, t$95$2]}, Block[{t$95$4 = N[(2.0 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(b * -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], If[GreaterEqual[b, 0.0], N[(N[((-b) - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]], If[LessEqual[t$95$3, -1e-174], t$95$3, If[LessEqual[t$95$3, 0.0], If[GreaterEqual[b, 0.0], N[(t$95$4 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / t$95$4), $MachinePrecision]], If[LessEqual[t$95$3, 2e+266], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(c * 2.0), $MachinePrecision] / N[(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] - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]]]

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 19.3

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

   3. Taylor expanded in c around 0 19.3

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

   4. Simplified 19.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
      Proof
      (/.f64 (neg.f64 b) a): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 b)) a): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_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)))))) < -1e-174

   1. Initial program 2.9

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

   if -1e-174 < (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 31.6

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

   2. Applied egg-rr 31.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{0.25}\right)}^{2}}}\\ \end{array} \]

   3. Taylor expanded in b around -inf 12.2

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

   4. Simplified 10.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, -2 \cdot b\right)}}\\ \end{array} \]
      Proof
      (fma.f64 2 (/.f64 c (/.f64 b a)) (*.f64 -2 b)): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 c a) b)) (*.f64 -2 b)): 27 points increase in error, 16 points decrease in error
      (fma.f64 2 (/.f64 (*.f64 c a) b) (*.f64 (Rewrite<= metadata-eval (*.f64 2 -1)) b)): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 (*.f64 c a) b) (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 -1 b)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (/.f64 (*.f64 c a) b)) (*.f64 2 (*.f64 -1 b)))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in b around inf 12.7

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

   6. Simplified 12.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, -2 \cdot b\right)}\\ \end{array} \]
      Proof
      (fma.f64 2 (/.f64 c (/.f64 b a)) (*.f64 b -2)): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 c a) b)) (*.f64 b -2)): 27 points increase in error, 16 points decrease in error
      (fma.f64 2 (/.f64 (*.f64 c a) b) (Rewrite<= *-commutative_binary64 (*.f64 -2 b))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (/.f64 (*.f64 c a) b)) (*.f64 -2 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)))))) < 2.0000000000000001e266

   1. Initial program 2.6

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

   2. Applied egg-rr 2.6

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

   if 2.0000000000000001e266 < (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 58.9

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

   2. Taylor expanded in b around inf 28.0

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

   3. Simplified 19.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
      Proof
      (fma.f64 2 (/.f64 c (/.f64 b a)) (*.f64 b -2)): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 c a) b)) (*.f64 b -2)): 27 points increase in error, 16 points decrease in error
      (fma.f64 2 (/.f64 (*.f64 c a) b) (Rewrite<= *-commutative_binary64 (*.f64 -2 b))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (/.f64 (*.f64 c a) b)) (*.f64 -2 b))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in c around 0 19.5

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

   5. Simplified 19.5

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

4. Final simplification 8.1

   \[\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 -1 \cdot 10^{-174}:\\ \;\;\;\;\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{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, 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 2 \cdot 10^{+266}:\\ \;\;\;\;\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 + \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{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array} \]

Alternatives

Alternative 1
Error	8.1
Cost	38052

\[\begin{array}{l} t_0 := \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\\ t_1 := \frac{c \cdot 2}{t_0 - b}\\ t_2 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ t_3 := \mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-174}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{t_3}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t_3}\\ \end{array}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+266}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	6.3
Cost	7952

\[\begin{array}{l} t_0 := \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\\ t_1 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t_0 - b}\\ \end{array}\\ \mathbf{if}\;b \leq -4 \cdot 10^{+146}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-313}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{+142}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	14.1
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+150}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array} \]

Alternative 4
Error	17.3
Cost	7496

\[\begin{array}{l} t_0 := \mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)\\ \mathbf{if}\;b \leq -1.25 \cdot 10^{-80}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{t_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t_0}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{c \cdot \left(a \cdot -4\right)} - b}\\ \end{array} \]

Alternative 5
Error	17.4
Cost	7368

\[\begin{array}{l} t_0 := \frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{if}\;b \leq -1.25 \cdot 10^{-80}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{c \cdot \left(a \cdot -4\right)} - b}\\ \end{array} \]

Alternative 6
Error	45.5
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 7
Error	22.4
Cost	644

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

Reproduce

herbie shell --seed 2022328 
(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)))))))