quad2m (problem 3.2.1, negative)

Average Accuracy: 46.5% → 90.4%

Time: 22.8s

Precision: binary64

Cost: 7692

Math

\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

↓

\[\begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - c \cdot a}\\ \mathbf{if}\;b_2 \leq -2 \cdot 10^{+148}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(b_2, -2, \frac{c}{b_2} \cdot \left(a \cdot 0.5\right)\right)}\\ \mathbf{elif}\;b_2 \leq 7.2 \cdot 10^{-152}:\\ \;\;\;\;\frac{c}{t_0 - b_2}\\ \mathbf{elif}\;b_2 \leq 5 \cdot 10^{+117}:\\ \;\;\;\;\frac{-t_0}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

↓

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b_2 b_2) (* c a)))))
   (if (<= b_2 -2e+148)
     (/ c (fma b_2 -2.0 (* (/ c b_2) (* a 0.5))))
     (if (<= b_2 7.2e-152)
       (/ c (- t_0 b_2))
       (if (<= b_2 5e+117)
         (- (/ (- t_0) a) (/ b_2 a))
         (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5)))))))

C

double code(double a, double b_2, double c) {
	return (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
}

↓

double code(double a, double b_2, double c) {
	double t_0 = sqrt(((b_2 * b_2) - (c * a)));
	double tmp;
	if (b_2 <= -2e+148) {
		tmp = c / fma(b_2, -2.0, ((c / b_2) * (a * 0.5)));
	} else if (b_2 <= 7.2e-152) {
		tmp = c / (t_0 - b_2);
	} else if (b_2 <= 5e+117) {
		tmp = (-t_0 / a) - (b_2 / a);
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

↓

function code(a, b_2, c)
	t_0 = sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))
	tmp = 0.0
	if (b_2 <= -2e+148)
		tmp = Float64(c / fma(b_2, -2.0, Float64(Float64(c / b_2) * Float64(a * 0.5))));
	elseif (b_2 <= 7.2e-152)
		tmp = Float64(c / Float64(t_0 - b_2));
	elseif (b_2 <= 5e+117)
		tmp = Float64(Float64(Float64(-t_0) / a) - Float64(b_2 / a));
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c / b_2) * 0.5));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

↓

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$2, -2e+148], N[(c / N[(b$95$2 * -2.0 + N[(N[(c / b$95$2), $MachinePrecision] * N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 7.2e-152], N[(c / N[(t$95$0 - b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 5e+117], N[(N[((-t$95$0) / a), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -2.0000000000000001e148

   1. Initial program 0.9%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Applied egg-rr 0.9%

      \[\leadsto \color{blue}{\left(b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{-1}{a}} \]
      Proof

      [Start] 0.9	\[ \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      frac-2neg [=>] 0.9	\[ \color{blue}{\frac{-\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{-a}} \]
      div-inv [=>] 0.9	\[ \color{blue}{\left(-\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)\right) \cdot \frac{1}{-a}} \]
      neg-sub0 [=>] 0.9	\[ \color{blue}{\left(0 - \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)\right)} \cdot \frac{1}{-a} \]
      add-sqr-sqrt [=>] 0.6	\[ \left(0 - \left(\color{blue}{\sqrt{-b_2} \cdot \sqrt{-b_2}} - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)\right) \cdot \frac{1}{-a} \]
      sqrt-unprod [=>] 0.9	\[ \left(0 - \left(\color{blue}{\sqrt{\left(-b_2\right) \cdot \left(-b_2\right)}} - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)\right) \cdot \frac{1}{-a} \]
      sqr-neg [=>] 0.9	\[ \left(0 - \left(\sqrt{\color{blue}{b_2 \cdot b_2}} - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)\right) \cdot \frac{1}{-a} \]
      sqrt-prod [=>] 0.0	\[ \left(0 - \left(\color{blue}{\sqrt{b_2} \cdot \sqrt{b_2}} - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)\right) \cdot \frac{1}{-a} \]
      add-sqr-sqrt [<=] 0.1	\[ \left(0 - \left(\color{blue}{b_2} - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)\right) \cdot \frac{1}{-a} \]
      associate-+l- [<=] 0.1	\[ \color{blue}{\left(\left(0 - b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)} \cdot \frac{1}{-a} \]
      neg-sub0 [<=] 0.1	\[ \left(\color{blue}{\left(-b_2\right)} + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{-a} \]
      add-sqr-sqrt [=>] 0.1	\[ \left(\color{blue}{\sqrt{-b_2} \cdot \sqrt{-b_2}} + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{-a} \]
      sqrt-unprod [=>] 0.1	\[ \left(\color{blue}{\sqrt{\left(-b_2\right) \cdot \left(-b_2\right)}} + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{-a} \]
      sqr-neg [=>] 0.1	\[ \left(\sqrt{\color{blue}{b_2 \cdot b_2}} + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{-a} \]
      sqrt-prod [=>] 0.0	\[ \left(\color{blue}{\sqrt{b_2} \cdot \sqrt{b_2}} + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{-a} \]
      add-sqr-sqrt [<=] 0.9	\[ \left(\color{blue}{b_2} + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{-a} \]
      neg-mul-1 [=>] 0.9	\[ \left(b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{\color{blue}{-1 \cdot a}} \]
      associate-/r* [=>] 0.9	\[ \left(b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \color{blue}{\frac{\frac{1}{-1}}{a}} \]
      metadata-eval [=>] 0.9	\[ \left(b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{\color{blue}{-1}}{a} \]

   3. Simplified 0.9%

      \[\leadsto \color{blue}{\left(b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}\right) \cdot \frac{-1}{a}} \]
      Proof

      [Start] 0.9	\[ \left(b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{-1}{a} \]
      *-commutative [=>] 0.9	\[ \left(b_2 + \sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}}\right) \cdot \frac{-1}{a} \]

   4. Applied egg-rr 0.9%

      \[\leadsto \color{blue}{\frac{b_2 \cdot b_2 - \mathsf{fma}\left(b_2, b_2, c \cdot a\right)}{\left(-a\right) \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)}} \]
      Proof

      [Start] 0.9	\[ \left(b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}\right) \cdot \frac{-1}{a} \]
      frac-2neg [=>] 0.9	\[ \left(b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}\right) \cdot \color{blue}{\frac{--1}{-a}} \]
      metadata-eval [=>] 0.9	\[ \left(b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}\right) \cdot \frac{\color{blue}{1}}{-a} \]
      un-div-inv [=>] 0.9	\[ \color{blue}{\frac{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}{-a}} \]
      +-commutative [=>] 0.9	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}}{-a} \]
      flip-+ [=>] 0.9	\[ \frac{\color{blue}{\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} \cdot \sqrt{b_2 \cdot b_2 - c \cdot a} - b_2 \cdot b_2}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}}{-a} \]
      associate-/l/ [=>] 0.9	\[ \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} \cdot \sqrt{b_2 \cdot b_2 - c \cdot a} - b_2 \cdot b_2}{\left(-a\right) \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)}} \]
      add-sqr-sqrt [<=] 0.9	\[ \frac{\color{blue}{\left(b_2 \cdot b_2 - c \cdot a\right)} - b_2 \cdot b_2}{\left(-a\right) \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)} \]
      associate--l- [=>] 0.9	\[ \frac{\color{blue}{b_2 \cdot b_2 - \left(c \cdot a + b_2 \cdot b_2\right)}}{\left(-a\right) \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)} \]
      +-commutative [<=] 0.9	\[ \frac{b_2 \cdot b_2 - \color{blue}{\left(b_2 \cdot b_2 + c \cdot a\right)}}{\left(-a\right) \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)} \]
      fma-def [=>] 0.9	\[ \frac{b_2 \cdot b_2 - \color{blue}{\mathsf{fma}\left(b_2, b_2, c \cdot a\right)}}{\left(-a\right) \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)} \]

   5. Simplified 39.7%

      \[\leadsto \color{blue}{\frac{\frac{0 - c \cdot a}{-a}}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}} \]
      Proof

      [Start] 0.9	\[ \frac{b_2 \cdot b_2 - \mathsf{fma}\left(b_2, b_2, c \cdot a\right)}{\left(-a\right) \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)} \]
      associate-/r* [=>] 0.9	\[ \color{blue}{\frac{\frac{b_2 \cdot b_2 - \mathsf{fma}\left(b_2, b_2, c \cdot a\right)}{-a}}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}} \]
      fma-udef [=>] 0.9	\[ \frac{\frac{b_2 \cdot b_2 - \color{blue}{\left(b_2 \cdot b_2 + c \cdot a\right)}}{-a}}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2} \]
      associate--r+ [=>] 2.5	\[ \frac{\frac{\color{blue}{\left(b_2 \cdot b_2 - b_2 \cdot b_2\right) - c \cdot a}}{-a}}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2} \]
      +-inverses [=>] 39.7	\[ \frac{\frac{\color{blue}{0} - c \cdot a}{-a}}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2} \]

   6. Taylor expanded in b_2 around -inf 88.4%

      \[\leadsto \frac{\frac{0 - c \cdot a}{-a}}{\color{blue}{-2 \cdot b_2 + 0.5 \cdot \frac{c \cdot a}{b_2}}} \]

   7. Applied egg-rr 77.8%

      \[\leadsto \color{blue}{\left(\frac{0}{a \cdot \mathsf{fma}\left(0.5, \frac{c}{\frac{b_2}{a}}, -2 \cdot b_2\right)} - \frac{0}{a \cdot \mathsf{fma}\left(0.5, \frac{c}{\frac{b_2}{a}}, -2 \cdot b_2\right)}\right) + \frac{c \cdot a}{a \cdot \mathsf{fma}\left(0.5, \frac{c}{\frac{b_2}{a}}, -2 \cdot b_2\right)}} \]
      Proof

      [Start] 88.4	\[ \frac{\frac{0 - c \cdot a}{-a}}{-2 \cdot b_2 + 0.5 \cdot \frac{c \cdot a}{b_2}} \]
      associate-/l/ [=>] 77.8	\[ \color{blue}{\frac{0 - c \cdot a}{\left(-2 \cdot b_2 + 0.5 \cdot \frac{c \cdot a}{b_2}\right) \cdot \left(-a\right)}} \]
      div-sub [=>] 77.8	\[ \color{blue}{\frac{0}{\left(-2 \cdot b_2 + 0.5 \cdot \frac{c \cdot a}{b_2}\right) \cdot \left(-a\right)} - \frac{c \cdot a}{\left(-2 \cdot b_2 + 0.5 \cdot \frac{c \cdot a}{b_2}\right) \cdot \left(-a\right)}} \]
      add-sqr-sqrt [=>] 43.7	\[ \frac{0}{\left(-2 \cdot b_2 + 0.5 \cdot \frac{c \cdot a}{b_2}\right) \cdot \left(-a\right)} - \frac{\color{blue}{\sqrt{c \cdot a} \cdot \sqrt{c \cdot a}}}{\left(-2 \cdot b_2 + 0.5 \cdot \frac{c \cdot a}{b_2}\right) \cdot \left(-a\right)} \]
      sqrt-unprod [=>] 48.7	\[ \frac{0}{\left(-2 \cdot b_2 + 0.5 \cdot \frac{c \cdot a}{b_2}\right) \cdot \left(-a\right)} - \frac{\color{blue}{\sqrt{\left(c \cdot a\right) \cdot \left(c \cdot a\right)}}}{\left(-2 \cdot b_2 + 0.5 \cdot \frac{c \cdot a}{b_2}\right) \cdot \left(-a\right)} \]
      sqr-neg [<=] 48.7	\[ \frac{0}{\left(-2 \cdot b_2 + 0.5 \cdot \frac{c \cdot a}{b_2}\right) \cdot \left(-a\right)} - \frac{\sqrt{\color{blue}{\left(-c \cdot a\right) \cdot \left(-c \cdot a\right)}}}{\left(-2 \cdot b_2 + 0.5 \cdot \frac{c \cdot a}{b_2}\right) \cdot \left(-a\right)} \]
      sub0-neg [<=] 48.7	\[ \frac{0}{\left(-2 \cdot b_2 + 0.5 \cdot \frac{c \cdot a}{b_2}\right) \cdot \left(-a\right)} - \frac{\sqrt{\color{blue}{\left(0 - c \cdot a\right)} \cdot \left(-c \cdot a\right)}}{\left(-2 \cdot b_2 + 0.5 \cdot \frac{c \cdot a}{b_2}\right) \cdot \left(-a\right)} \]
      sub0-neg [<=] 48.7	\[ \frac{0}{\left(-2 \cdot b_2 + 0.5 \cdot \frac{c \cdot a}{b_2}\right) \cdot \left(-a\right)} - \frac{\sqrt{\left(0 - c \cdot a\right) \cdot \color{blue}{\left(0 - c \cdot a\right)}}}{\left(-2 \cdot b_2 + 0.5 \cdot \frac{c \cdot a}{b_2}\right) \cdot \left(-a\right)} \]
      sqrt-unprod [<=] 22.5	\[ \frac{0}{\left(-2 \cdot b_2 + 0.5 \cdot \frac{c \cdot a}{b_2}\right) \cdot \left(-a\right)} - \frac{\color{blue}{\sqrt{0 - c \cdot a} \cdot \sqrt{0 - c \cdot a}}}{\left(-2 \cdot b_2 + 0.5 \cdot \frac{c \cdot a}{b_2}\right) \cdot \left(-a\right)} \]
      add-sqr-sqrt [<=] 37.4	\[ \frac{0}{\left(-2 \cdot b_2 + 0.5 \cdot \frac{c \cdot a}{b_2}\right) \cdot \left(-a\right)} - \frac{\color{blue}{0 - c \cdot a}}{\left(-2 \cdot b_2 + 0.5 \cdot \frac{c \cdot a}{b_2}\right) \cdot \left(-a\right)} \]
      div-sub [=>] 37.4	\[ \frac{0}{\left(-2 \cdot b_2 + 0.5 \cdot \frac{c \cdot a}{b_2}\right) \cdot \left(-a\right)} - \color{blue}{\left(\frac{0}{\left(-2 \cdot b_2 + 0.5 \cdot \frac{c \cdot a}{b_2}\right) \cdot \left(-a\right)} - \frac{c \cdot a}{\left(-2 \cdot b_2 + 0.5 \cdot \frac{c \cdot a}{b_2}\right) \cdot \left(-a\right)}\right)} \]

   8. Simplified 98.1%

      \[\leadsto \color{blue}{\frac{c}{\mathsf{fma}\left(b_2, -2, \frac{c}{b_2} \cdot \left(a \cdot 0.5\right)\right)}} \]
      Proof

      [Start] 77.8	\[ \left(\frac{0}{a \cdot \mathsf{fma}\left(0.5, \frac{c}{\frac{b_2}{a}}, -2 \cdot b_2\right)} - \frac{0}{a \cdot \mathsf{fma}\left(0.5, \frac{c}{\frac{b_2}{a}}, -2 \cdot b_2\right)}\right) + \frac{c \cdot a}{a \cdot \mathsf{fma}\left(0.5, \frac{c}{\frac{b_2}{a}}, -2 \cdot b_2\right)} \]
      +-inverses [=>] 77.8	\[ \color{blue}{0} + \frac{c \cdot a}{a \cdot \mathsf{fma}\left(0.5, \frac{c}{\frac{b_2}{a}}, -2 \cdot b_2\right)} \]
      +-lft-identity [=>] 77.8	\[ \color{blue}{\frac{c \cdot a}{a \cdot \mathsf{fma}\left(0.5, \frac{c}{\frac{b_2}{a}}, -2 \cdot b_2\right)}} \]
      times-frac [=>] 71.3	\[ \color{blue}{\frac{c}{a} \cdot \frac{a}{\mathsf{fma}\left(0.5, \frac{c}{\frac{b_2}{a}}, -2 \cdot b_2\right)}} \]
      associate-/r/ [<=] 74.1	\[ \color{blue}{\frac{c}{\frac{a}{\frac{a}{\mathsf{fma}\left(0.5, \frac{c}{\frac{b_2}{a}}, -2 \cdot b_2\right)}}}} \]
      associate-/r/ [=>] 98.1	\[ \frac{c}{\color{blue}{\frac{a}{a} \cdot \mathsf{fma}\left(0.5, \frac{c}{\frac{b_2}{a}}, -2 \cdot b_2\right)}} \]
      *-inverses [=>] 98.1	\[ \frac{c}{\color{blue}{1} \cdot \mathsf{fma}\left(0.5, \frac{c}{\frac{b_2}{a}}, -2 \cdot b_2\right)} \]
      *-lft-identity [=>] 98.1	\[ \frac{c}{\color{blue}{\mathsf{fma}\left(0.5, \frac{c}{\frac{b_2}{a}}, -2 \cdot b_2\right)}} \]
      fma-udef [=>] 98.1	\[ \frac{c}{\color{blue}{0.5 \cdot \frac{c}{\frac{b_2}{a}} + -2 \cdot b_2}} \]
      +-commutative [<=] 98.1	\[ \frac{c}{\color{blue}{-2 \cdot b_2 + 0.5 \cdot \frac{c}{\frac{b_2}{a}}}} \]
      *-commutative [=>] 98.1	\[ \frac{c}{\color{blue}{b_2 \cdot -2} + 0.5 \cdot \frac{c}{\frac{b_2}{a}}} \]
      fma-udef [<=] 98.1	\[ \frac{c}{\color{blue}{\mathsf{fma}\left(b_2, -2, 0.5 \cdot \frac{c}{\frac{b_2}{a}}\right)}} \]
      *-commutative [=>] 98.1	\[ \frac{c}{\mathsf{fma}\left(b_2, -2, \color{blue}{\frac{c}{\frac{b_2}{a}} \cdot 0.5}\right)} \]
      associate-/r/ [=>] 98.1	\[ \frac{c}{\mathsf{fma}\left(b_2, -2, \color{blue}{\left(\frac{c}{b_2} \cdot a\right)} \cdot 0.5\right)} \]
      associate-*l* [=>] 98.1	\[ \frac{c}{\mathsf{fma}\left(b_2, -2, \color{blue}{\frac{c}{b_2} \cdot \left(a \cdot 0.5\right)}\right)} \]

   if -2.0000000000000001e148 < b_2 < 7.2e-152

   1. Initial program 52.3%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Applied egg-rr 52.2%

      \[\leadsto \color{blue}{\left(b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{-1}{a}} \]
      Proof

      [Start] 52.3	\[ \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      frac-2neg [=>] 52.3	\[ \color{blue}{\frac{-\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{-a}} \]
      div-inv [=>] 52.2	\[ \color{blue}{\left(-\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)\right) \cdot \frac{1}{-a}} \]
      neg-sub0 [=>] 52.2	\[ \color{blue}{\left(0 - \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)\right)} \cdot \frac{1}{-a} \]
      add-sqr-sqrt [=>] 34.1	\[ \left(0 - \left(\color{blue}{\sqrt{-b_2} \cdot \sqrt{-b_2}} - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)\right) \cdot \frac{1}{-a} \]
      sqrt-unprod [=>] 51.7	\[ \left(0 - \left(\color{blue}{\sqrt{\left(-b_2\right) \cdot \left(-b_2\right)}} - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)\right) \cdot \frac{1}{-a} \]
      sqr-neg [=>] 51.7	\[ \left(0 - \left(\sqrt{\color{blue}{b_2 \cdot b_2}} - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)\right) \cdot \frac{1}{-a} \]
      sqrt-prod [=>] 15.0	\[ \left(0 - \left(\color{blue}{\sqrt{b_2} \cdot \sqrt{b_2}} - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)\right) \cdot \frac{1}{-a} \]
      add-sqr-sqrt [<=] 44.9	\[ \left(0 - \left(\color{blue}{b_2} - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)\right) \cdot \frac{1}{-a} \]
      associate-+l- [<=] 44.9	\[ \color{blue}{\left(\left(0 - b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)} \cdot \frac{1}{-a} \]
      neg-sub0 [<=] 44.9	\[ \left(\color{blue}{\left(-b_2\right)} + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{-a} \]
      add-sqr-sqrt [=>] 29.9	\[ \left(\color{blue}{\sqrt{-b_2} \cdot \sqrt{-b_2}} + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{-a} \]
      sqrt-unprod [=>] 45.0	\[ \left(\color{blue}{\sqrt{\left(-b_2\right) \cdot \left(-b_2\right)}} + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{-a} \]
      sqr-neg [=>] 45.0	\[ \left(\sqrt{\color{blue}{b_2 \cdot b_2}} + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{-a} \]
      sqrt-prod [=>] 15.5	\[ \left(\color{blue}{\sqrt{b_2} \cdot \sqrt{b_2}} + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{-a} \]
      add-sqr-sqrt [<=] 52.2	\[ \left(\color{blue}{b_2} + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{-a} \]
      neg-mul-1 [=>] 52.2	\[ \left(b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{\color{blue}{-1 \cdot a}} \]
      associate-/r* [=>] 52.2	\[ \left(b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \color{blue}{\frac{\frac{1}{-1}}{a}} \]
      metadata-eval [=>] 52.2	\[ \left(b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{\color{blue}{-1}}{a} \]

   3. Simplified 52.2%

      \[\leadsto \color{blue}{\left(b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}\right) \cdot \frac{-1}{a}} \]
      Proof

      [Start] 52.2	\[ \left(b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{-1}{a} \]
      *-commutative [=>] 52.2	\[ \left(b_2 + \sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}}\right) \cdot \frac{-1}{a} \]

   4. Applied egg-rr 43.7%

      \[\leadsto \color{blue}{\frac{b_2 \cdot b_2 - \mathsf{fma}\left(b_2, b_2, c \cdot a\right)}{\left(-a\right) \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)}} \]
      Proof

      [Start] 52.2	\[ \left(b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}\right) \cdot \frac{-1}{a} \]
      frac-2neg [=>] 52.2	\[ \left(b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}\right) \cdot \color{blue}{\frac{--1}{-a}} \]
      metadata-eval [=>] 52.2	\[ \left(b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}\right) \cdot \frac{\color{blue}{1}}{-a} \]
      un-div-inv [=>] 52.3	\[ \color{blue}{\frac{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}{-a}} \]
      +-commutative [=>] 52.3	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}}{-a} \]
      flip-+ [=>] 51.8	\[ \frac{\color{blue}{\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} \cdot \sqrt{b_2 \cdot b_2 - c \cdot a} - b_2 \cdot b_2}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}}{-a} \]
      associate-/l/ [=>] 43.6	\[ \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} \cdot \sqrt{b_2 \cdot b_2 - c \cdot a} - b_2 \cdot b_2}{\left(-a\right) \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)}} \]
      add-sqr-sqrt [<=] 43.7	\[ \frac{\color{blue}{\left(b_2 \cdot b_2 - c \cdot a\right)} - b_2 \cdot b_2}{\left(-a\right) \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)} \]
      associate--l- [=>] 43.7	\[ \frac{\color{blue}{b_2 \cdot b_2 - \left(c \cdot a + b_2 \cdot b_2\right)}}{\left(-a\right) \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)} \]
      +-commutative [<=] 43.7	\[ \frac{b_2 \cdot b_2 - \color{blue}{\left(b_2 \cdot b_2 + c \cdot a\right)}}{\left(-a\right) \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)} \]
      fma-def [=>] 43.7	\[ \frac{b_2 \cdot b_2 - \color{blue}{\mathsf{fma}\left(b_2, b_2, c \cdot a\right)}}{\left(-a\right) \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)} \]

   5. Simplified 77.6%

      \[\leadsto \color{blue}{\frac{\frac{0 - c \cdot a}{-a}}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}} \]
      Proof

      [Start] 43.7	\[ \frac{b_2 \cdot b_2 - \mathsf{fma}\left(b_2, b_2, c \cdot a\right)}{\left(-a\right) \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)} \]
      associate-/r* [=>] 51.8	\[ \color{blue}{\frac{\frac{b_2 \cdot b_2 - \mathsf{fma}\left(b_2, b_2, c \cdot a\right)}{-a}}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}} \]
      fma-udef [=>] 51.8	\[ \frac{\frac{b_2 \cdot b_2 - \color{blue}{\left(b_2 \cdot b_2 + c \cdot a\right)}}{-a}}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2} \]
      associate--r+ [=>] 77.6	\[ \frac{\frac{\color{blue}{\left(b_2 \cdot b_2 - b_2 \cdot b_2\right) - c \cdot a}}{-a}}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2} \]
      +-inverses [=>] 77.6	\[ \frac{\frac{\color{blue}{0} - c \cdot a}{-a}}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2} \]

   6. Taylor expanded in c around 0 85.6%

      \[\leadsto \frac{\color{blue}{c}}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2} \]

   if 7.2e-152 < b_2 < 4.99999999999999983e117

   1. Initial program 91.0%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Applied egg-rr 91.0%

      \[\leadsto \color{blue}{\frac{0}{a} - \left(\frac{b_2}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right)} \]
      Proof

      [Start] 91.0	\[ \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      div-sub [=>] 91.0	\[ \color{blue}{\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}} \]
      neg-sub0 [=>] 91.0	\[ \frac{\color{blue}{0 - b_2}}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      div-sub [=>] 91.0	\[ \color{blue}{\left(\frac{0}{a} - \frac{b_2}{a}\right)} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      add-sqr-sqrt [=>] 90.8	\[ \left(\frac{0}{a} - \frac{\color{blue}{\sqrt{b_2} \cdot \sqrt{b_2}}}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      sqrt-prod [<=] 91.0	\[ \left(\frac{0}{a} - \frac{\color{blue}{\sqrt{b_2 \cdot b_2}}}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      sqr-neg [<=] 91.0	\[ \left(\frac{0}{a} - \frac{\sqrt{\color{blue}{\left(-b_2\right) \cdot \left(-b_2\right)}}}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      sqrt-unprod [<=] 0.0	\[ \left(\frac{0}{a} - \frac{\color{blue}{\sqrt{-b_2} \cdot \sqrt{-b_2}}}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      add-sqr-sqrt [<=] 31.4	\[ \left(\frac{0}{a} - \frac{\color{blue}{-b_2}}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      associate--l- [=>] 31.4	\[ \color{blue}{\frac{0}{a} - \left(\frac{-b_2}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right)} \]
      add-sqr-sqrt [=>] 0.0	\[ \frac{0}{a} - \left(\frac{\color{blue}{\sqrt{-b_2} \cdot \sqrt{-b_2}}}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]
      sqrt-unprod [=>] 91.0	\[ \frac{0}{a} - \left(\frac{\color{blue}{\sqrt{\left(-b_2\right) \cdot \left(-b_2\right)}}}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]
      sqr-neg [=>] 91.0	\[ \frac{0}{a} - \left(\frac{\sqrt{\color{blue}{b_2 \cdot b_2}}}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]
      sqrt-prod [=>] 90.8	\[ \frac{0}{a} - \left(\frac{\color{blue}{\sqrt{b_2} \cdot \sqrt{b_2}}}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]
      add-sqr-sqrt [<=] 91.0	\[ \frac{0}{a} - \left(\frac{\color{blue}{b_2}}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]

   3. Simplified 91.0%

      \[\leadsto \color{blue}{\frac{-\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{b_2}{a}} \]
      Proof

      [Start] 91.0	\[ \frac{0}{a} - \left(\frac{b_2}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]
      div0 [=>] 91.0	\[ \color{blue}{0} - \left(\frac{b_2}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]
      +-commutative [=>] 91.0	\[ 0 - \color{blue}{\left(\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} + \frac{b_2}{a}\right)} \]
      associate--r+ [=>] 91.0	\[ \color{blue}{\left(0 - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) - \frac{b_2}{a}} \]
      neg-sub0 [<=] 91.0	\[ \color{blue}{\left(-\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right)} - \frac{b_2}{a} \]
      distribute-neg-frac [=>] 91.0	\[ \color{blue}{\frac{-\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}} - \frac{b_2}{a} \]
      *-commutative [=>] 91.0	\[ \frac{-\sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}}}{a} - \frac{b_2}{a} \]

   if 4.99999999999999983e117 < b_2

   1. Initial program 20.3%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 94.5%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{+148}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(b_2, -2, \frac{c}{b_2} \cdot \left(a \cdot 0.5\right)\right)}\\ \mathbf{elif}\;b_2 \leq 7.2 \cdot 10^{-152}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\ \mathbf{elif}\;b_2 \leq 5 \cdot 10^{+117}:\\ \;\;\;\;\frac{-\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	90.4%
Cost	7564

\[\begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - c \cdot a}\\ \mathbf{if}\;b_2 \leq -5 \cdot 10^{+154}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(b_2, -2, \frac{c}{b_2} \cdot \left(a \cdot 0.5\right)\right)}\\ \mathbf{elif}\;b_2 \leq 7.5 \cdot 10^{-152}:\\ \;\;\;\;\frac{c}{t_0 - b_2}\\ \mathbf{elif}\;b_2 \leq 7.8 \cdot 10^{+118}:\\ \;\;\;\;\frac{\left(-b_2\right) - t_0}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 2
Accuracy	85.2%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{+149}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \mathbf{elif}\;b_2 \leq 5.8 \cdot 10^{-77}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 3
Accuracy	85.2%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(b_2, -2, \frac{c}{b_2} \cdot \left(a \cdot 0.5\right)\right)}\\ \mathbf{elif}\;b_2 \leq 3.8 \cdot 10^{-77}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 4
Accuracy	79.7%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -4.4 \cdot 10^{-51}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \mathbf{elif}\;b_2 \leq 1.7 \cdot 10^{-76}:\\ \;\;\;\;\frac{c}{\sqrt{c \cdot \left(-a\right)} - b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 5
Accuracy	43.8%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3.8 \cdot 10^{-219}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 6
Accuracy	65.8%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3.8 \cdot 10^{-219}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 7
Accuracy	16.9%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 8
Accuracy	12.1%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023126 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))