jeff quadratic root 2

Average Error: 19.8 → 6.0

Time: 23.3s

Precision: binary64

Cost: 26960

Math

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

↓

\[\begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ t_1 := \frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}\\ \mathbf{if}\;b \leq -1.5 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-34}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) + 4 \cdot \mathsf{fma}\left(a \cdot -4, c, a \cdot \left(c \cdot 4\right)\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+149}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b, {\left(\sqrt[3]{a \cdot -4} \cdot \sqrt[3]{c}\right)}^{1.5}\right) - b}{2 \cdot a}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (if (>= b 0.0) (/ (* 2.0 c) (- (- b) b)) (- (/ c b) (/ b a))))
        (t_1 (/ (* 2.0 c) (- (- b) (sqrt (+ (* b b) (* c (* a -4.0))))))))
   (if (<= b -1.5e+154)
     t_0
     (if (<= b -5e-34)
       (if (>= b 0.0)
         t_1
         (/
          (-
           (sqrt
            (+
             (fma c (* a -4.0) (* b b))
             (* 4.0 (fma (* a -4.0) c (* a (* c 4.0))))))
           b)
          (* 2.0 a)))
       (if (<= b 5e+149)
         (if (>= b 0.0)
           t_1
           (/
            (- (hypot b (pow (* (cbrt (* a -4.0)) (cbrt c)) 1.5)) b)
            (* 2.0 a)))
         t_0)))))

C

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

↓

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (-b - b);
	} else {
		tmp = (c / b) - (b / a);
	}
	double t_0 = tmp;
	double t_1 = (2.0 * c) / (-b - sqrt(((b * b) + (c * (a * -4.0)))));
	double tmp_1;
	if (b <= -1.5e+154) {
		tmp_1 = t_0;
	} else if (b <= -5e-34) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = (sqrt((fma(c, (a * -4.0), (b * b)) + (4.0 * fma((a * -4.0), c, (a * (c * 4.0)))))) - b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 5e+149) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = (hypot(b, pow((cbrt((a * -4.0)) * cbrt(c)), 1.5)) - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Julia

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

↓

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) - b));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	t_0 = tmp
	t_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0))))))
	tmp_1 = 0.0
	if (b <= -1.5e+154)
		tmp_1 = t_0;
	elseif (b <= -5e-34)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(Float64(sqrt(Float64(fma(c, Float64(a * -4.0), Float64(b * b)) + Float64(4.0 * fma(Float64(a * -4.0), c, Float64(a * Float64(c * 4.0)))))) - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 5e+149)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = Float64(Float64(hypot(b, (Float64(cbrt(Float64(a * -4.0)) * cbrt(c)) ^ 1.5)) - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]

↓

code[a_, b_, c_] := Block[{t$95$0 = If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - b), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]}, Block[{t$95$1 = N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.5e+154], t$95$0, If[LessEqual[b, -5e-34], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(N[Sqrt[N[(N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[(a * -4.0), $MachinePrecision] * c + N[(a * N[(c * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 5e+149], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(N[Sqrt[b ^ 2 + N[Power[N[(N[Power[N[(a * -4.0), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[c, 1/3], $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision] ^ 2], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.50000000000000013e154 or 4.9999999999999999e149 < b

   1. Initial program 48.0

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

   2. Taylor expanded in b around inf 24.8

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

   3. Taylor expanded in b around -inf 2.0

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

   4. Simplified 2.0

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

      [Start] 2.0	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + -1 \cdot \frac{b}{a}\\ \end{array} \]
      mul-1-neg [=>] 2.0	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{c}{b} + \left(-\frac{b}{a}\right)}\\ \end{array} \]
      unsub-neg [=>] 2.0	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

   if -1.50000000000000013e154 < b < -5.0000000000000003e-34

   1. Initial program 5.4

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

   2. Applied egg-rr 5.4

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

   3. Simplified 5.4

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

      [Start] 5.4	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right) + \left(\left(\left(4 \cdot \left(a \cdot c\right) + c \cdot \left(a \cdot -4\right)\right) + \left(4 \cdot \left(a \cdot c\right) + c \cdot \left(a \cdot -4\right)\right)\right) + \left(\left(4 \cdot \left(a \cdot c\right) + c \cdot \left(a \cdot -4\right)\right) + \left(4 \cdot \left(a \cdot c\right) + c \cdot \left(a \cdot -4\right)\right)\right)\right)}}{2 \cdot a}\\ \end{array} \]
      fma-def [<=] 5.4	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left(b \cdot b + c \cdot \left(a \cdot -4\right)\right) + \left(\left(\left(4 \cdot \left(a \cdot c\right) + c \cdot \left(a \cdot -4\right)\right) + \left(4 \cdot \left(a \cdot c\right) + c \cdot \left(a \cdot -4\right)\right)\right) + \left(\left(4 \cdot \left(a \cdot c\right) + c \cdot \left(a \cdot -4\right)\right) + \left(4 \cdot \left(a \cdot c\right) + c \cdot \left(a \cdot -4\right)\right)\right)\right)}}{2 \cdot a}\\ \end{array} \]
      +-commutative [=>] 5.4	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left(c \cdot \left(a \cdot -4\right) + b \cdot b\right) + \left(\left(\left(4 \cdot \left(a \cdot c\right) + c \cdot \left(a \cdot -4\right)\right) + \left(4 \cdot \left(a \cdot c\right) + c \cdot \left(a \cdot -4\right)\right)\right) + \left(\left(4 \cdot \left(a \cdot c\right) + c \cdot \left(a \cdot -4\right)\right) + \left(4 \cdot \left(a \cdot c\right) + c \cdot \left(a \cdot -4\right)\right)\right)\right)}}{2 \cdot a}\\ \end{array} \]
      fma-def [=>] 5.4	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) + \left(\left(\left(4 \cdot \left(a \cdot c\right) + c \cdot \left(a \cdot -4\right)\right) + \left(4 \cdot \left(a \cdot c\right) + c \cdot \left(a \cdot -4\right)\right)\right) + \left(\left(4 \cdot \left(a \cdot c\right) + c \cdot \left(a \cdot -4\right)\right) + \left(4 \cdot \left(a \cdot c\right) + c \cdot \left(a \cdot -4\right)\right)\right)\right)}}{2 \cdot a}\\ \end{array} \]
      count-2 [=>] 5.4	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) + 2 \cdot \left(\left(4 \cdot \left(a \cdot c\right) + c \cdot \left(a \cdot -4\right)\right) + \left(4 \cdot \left(a \cdot c\right) + c \cdot \left(a \cdot -4\right)\right)\right)}}{2 \cdot a}\\ \end{array} \]
      count-2 [=>] 5.4	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) + 2 \cdot \left(2 \cdot \left(4 \cdot \left(a \cdot c\right) + c \cdot \left(a \cdot -4\right)\right)\right)}}{2 \cdot a}\\ \end{array} \]
      associate-*r* [=>] 5.4	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) + \left(2 \cdot 2\right) \cdot \left(4 \cdot \left(a \cdot c\right) + c \cdot \left(a \cdot -4\right)\right)}}{2 \cdot a}\\ \end{array} \]
      metadata-eval [=>] 5.4	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) + 4 \cdot \left(4 \cdot \left(a \cdot c\right) + c \cdot \left(a \cdot -4\right)\right)}}{2 \cdot a}\\ \end{array} \]
      +-commutative [=>] 5.4	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) + 4 \cdot \left(c \cdot \left(a \cdot -4\right) + 4 \cdot \left(a \cdot c\right)\right)}}{2 \cdot a}\\ \end{array} \]
      *-commutative [=>] 5.4	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) + 4 \cdot \left(\left(a \cdot -4\right) \cdot c + 4 \cdot \left(a \cdot c\right)\right)}}{2 \cdot a}\\ \end{array} \]
      fma-udef [<=] 5.4	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) + 4 \cdot \mathsf{fma}\left(a \cdot -4, c, 4 \cdot \left(a \cdot c\right)\right)}}{2 \cdot a}\\ \end{array} \]
      *-commutative [=>] 5.4	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) + 4 \cdot \mathsf{fma}\left(a \cdot -4, c, 4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \end{array} \]
      associate-*r* [=>] 5.4	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) + 4 \cdot \mathsf{fma}\left(a \cdot -4, c, \left(4 \cdot c\right) \cdot a\right)}}{2 \cdot a}\\ \end{array} \]

   if -5.0000000000000003e-34 < b < 4.9999999999999999e149

   1. Initial program 9.1

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

   2. Applied egg-rr 10.3

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

   3. Applied egg-rr 9.5

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

   4. Applied egg-rr 9.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b, \sqrt{{\left(\sqrt[3]{c \cdot \left(a \cdot -4\right)}\right)}^{2}} \cdot \sqrt{\sqrt[3]{c \cdot \left(a \cdot -4\right)}}\right) - b}{2 \cdot a}\\ \end{array} \]

   5. Simplified 9.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b, {\left(\sqrt[3]{c \cdot \left(a \cdot -4\right)}\right)}^{1.5}\right) - b}{2 \cdot a}\\ \end{array} \]
      Proof

      [Start] 9.7	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b, \sqrt{{\left(\sqrt[3]{c \cdot \left(a \cdot -4\right)}\right)}^{2}} \cdot \sqrt{\sqrt[3]{c \cdot \left(a \cdot -4\right)}}\right) - b}{2 \cdot a}\\ \end{array} \]
      *-commutative [=>] 9.7	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b, \sqrt{\sqrt[3]{c \cdot \left(a \cdot -4\right)}} \cdot \sqrt{{\left(\sqrt[3]{c \cdot \left(a \cdot -4\right)}\right)}^{2}}\right) - b}{2 \cdot a}\\ \end{array} \]
      unpow1/2 [<=] 9.7	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b, {\left(\sqrt[3]{c \cdot \left(a \cdot -4\right)}\right)}^{0.5} \cdot \sqrt{{\left(\sqrt[3]{c \cdot \left(a \cdot -4\right)}\right)}^{2}}\right) - b}{2 \cdot a}\\ \end{array} \]
      sqr-pow [=>] 9.7	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b, {\left(\sqrt[3]{c \cdot \left(a \cdot -4\right)}\right)}^{0.5} \cdot \sqrt{{\left(\sqrt[3]{c \cdot \left(a \cdot -4\right)}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\sqrt[3]{c \cdot \left(a \cdot -4\right)}\right)}^{\left(\frac{2}{2}\right)}}\right) - b}{2 \cdot a}\\ \end{array} \]
      rem-sqrt-square [=>] 9.7	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b, {\left(\sqrt[3]{c \cdot \left(a \cdot -4\right)}\right)}^{0.5} \cdot \left|{\left(\sqrt[3]{c \cdot \left(a \cdot -4\right)}\right)}^{\left(\frac{2}{2}\right)}\right|\right) - b}{2 \cdot a}\\ \end{array} \]
      metadata-eval [=>] 9.7	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b, {\left(\sqrt[3]{c \cdot \left(a \cdot -4\right)}\right)}^{0.5} \cdot \left|{\left(\sqrt[3]{c \cdot \left(a \cdot -4\right)}\right)}^{1}\right|\right) - b}{2 \cdot a}\\ \end{array} \]
      metadata-eval [<=] 9.7	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b, {\left(\sqrt[3]{c \cdot \left(a \cdot -4\right)}\right)}^{0.5} \cdot \left|{\left(\sqrt[3]{c \cdot \left(a \cdot -4\right)}\right)}^{\left(2 \cdot 0.5\right)}\right|\right) - b}{2 \cdot a}\\ \end{array} \]
      pow-sqr [<=] 9.7	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b, {\left(\sqrt[3]{c \cdot \left(a \cdot -4\right)}\right)}^{0.5} \cdot \left|{\left(\sqrt[3]{c \cdot \left(a \cdot -4\right)}\right)}^{0.5} \cdot {\left(\sqrt[3]{c \cdot \left(a \cdot -4\right)}\right)}^{0.5}\right|\right) - b}{2 \cdot a}\\ \end{array} \]
      fabs-sqr [=>] 9.7	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b, {\left(\sqrt[3]{c \cdot \left(a \cdot -4\right)}\right)}^{0.5} \cdot \left({\left(\sqrt[3]{c \cdot \left(a \cdot -4\right)}\right)}^{0.5} \cdot {\left(\sqrt[3]{c \cdot \left(a \cdot -4\right)}\right)}^{0.5}\right)\right) - b}{2 \cdot a}\\ \end{array} \]
      pow-sqr [=>] 9.7	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b, {\left(\sqrt[3]{c \cdot \left(a \cdot -4\right)}\right)}^{0.5} \cdot {\left(\sqrt[3]{c \cdot \left(a \cdot -4\right)}\right)}^{\left(2 \cdot 0.5\right)}\right) - b}{2 \cdot a}\\ \end{array} \]
      metadata-eval [=>] 9.7	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b, {\left(\sqrt[3]{c \cdot \left(a \cdot -4\right)}\right)}^{0.5} \cdot {\left(\sqrt[3]{c \cdot \left(a \cdot -4\right)}\right)}^{1}\right) - b}{2 \cdot a}\\ \end{array} \]
      unpow1 [=>] 9.7	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b, {\left(\sqrt[3]{c \cdot \left(a \cdot -4\right)}\right)}^{0.5} \cdot \sqrt[3]{c \cdot \left(a \cdot -4\right)}\right) - b}{2 \cdot a}\\ \end{array} \]
      pow-plus [=>] 9.7	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b, {\left(\sqrt[3]{c \cdot \left(a \cdot -4\right)}\right)}^{\left(0.5 + 1\right)}\right) - b}{2 \cdot a}\\ \end{array} \]
      metadata-eval [=>] 9.7	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b, {\left(\sqrt[3]{c \cdot \left(a \cdot -4\right)}\right)}^{1.5}\right) - b}{2 \cdot a}\\ \end{array} \]

   6. Applied egg-rr 8.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b, {\left(\sqrt[3]{a \cdot -4} \cdot \sqrt[3]{c}\right)}^{1.5}\right) - b}{2 \cdot a}\\ \end{array} \]
3. Recombined 3 regimes into one program.

4. Final simplification 6.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-34}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) + 4 \cdot \mathsf{fma}\left(a \cdot -4, c, a \cdot \left(c \cdot 4\right)\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+149}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b, {\left(\sqrt[3]{a \cdot -4} \cdot \sqrt[3]{c}\right)}^{1.5}\right) - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.6
Cost	38052

\[\begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\\ t_1 := \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\\ t_2 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-275}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{-2 \cdot \left(c \cdot \frac{a}{b} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{c \cdot a}{b} - b\right) - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+277}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	6.7
Cost	38052

\[\begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\\ t_1 := \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\\ t_2 := \frac{t_1 - b}{2 \cdot a}\\ t_3 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array}\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_3 \leq -5 \cdot 10^{-219}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{b} \cdot a, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array}\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+277}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	10.4
Cost	7888

\[\begin{array}{l} t_0 := \frac{2 \cdot c}{\left(-b\right) - b}\\ t_1 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ t_2 := c \cdot \left(a \cdot -4\right)\\ \mathbf{if}\;b \leq -5.4 \cdot 10^{-30}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -2.1 \cdot 10^{-299}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{t_2} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{+150}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{b \cdot b + t_2}}\\ \mathbf{else}:\\ \;\;\;\;\left(b + b\right) \cdot \frac{-0.5}{a}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	6.8
Cost	7888

\[\begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ t_1 := \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\\ \mathbf{if}\;b \leq -1.12 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -2.1 \cdot 10^{-299}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 3.3 \cdot 10^{+149}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + t_1}\\ \mathbf{else}:\\ \;\;\;\;\left(b + b\right) \cdot \frac{-0.5}{a}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	13.6
Cost	7632

\[\begin{array}{l} t_0 := \frac{2 \cdot c}{\left(-b\right) - b}\\ t_1 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ t_2 := \sqrt{c \cdot \left(a \cdot -4\right)}\\ \mathbf{if}\;b \leq -3.1 \cdot 10^{-30}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -2.1 \cdot 10^{-299}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 3.35 \cdot 10^{-40}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + t_2}\\ \mathbf{else}:\\ \;\;\;\;\left(b + b\right) \cdot \frac{-0.5}{a}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	18.2
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq 5.6 \cdot 10^{-40}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b + b\right) \cdot \frac{-0.5}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 7
Error	23.2
Cost	644

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

Alternative 8
Error	23.2
Cost	644

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

Reproduce

herbie shell --seed 2023237 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))