quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.1% → 85.0%

Time: 14.7s

Alternatives: 9

Speedup: 11.2×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

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

Python

def code(a, b_2, c):
	return (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a

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

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
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]

Initial Program: 52.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

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

Python

def code(a, b_2, c):
	return (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a

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

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
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]

Alternative 1: 85.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -6.8 \cdot 10^{-62}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \frac{-2}{\frac{c}{b\_2}}\right)}\\ \mathbf{elif}\;b\_2 \leq -5.1 \cdot 10^{-124}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\mathsf{fma}\left(-1, b\_2, \mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)\right)}}{a}\\ \mathbf{elif}\;b\_2 \leq -7 \cdot 10^{-135}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5.6 \cdot 10^{+81}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -6.8e-62)
   (/ 1.0 (fma 0.5 (/ a b_2) (/ -2.0 (/ c b_2))))
   (if (<= b_2 -5.1e-124)
     (/ (/ (* a c) (fma -1.0 b_2 (hypot b_2 (sqrt (* a (- c)))))) a)
     (if (<= b_2 -7e-135)
       (/ (* c -0.5) b_2)
       (if (<= b_2 5.6e+81)
         (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
         (* -2.0 (/ b_2 a)))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6.8e-62) {
		tmp = 1.0 / fma(0.5, (a / b_2), (-2.0 / (c / b_2)));
	} else if (b_2 <= -5.1e-124) {
		tmp = ((a * c) / fma(-1.0, b_2, hypot(b_2, sqrt((a * -c))))) / a;
	} else if (b_2 <= -7e-135) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 5.6e+81) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -6.8e-62)
		tmp = Float64(1.0 / fma(0.5, Float64(a / b_2), Float64(-2.0 / Float64(c / b_2))));
	elseif (b_2 <= -5.1e-124)
		tmp = Float64(Float64(Float64(a * c) / fma(-1.0, b_2, hypot(b_2, sqrt(Float64(a * Float64(-c)))))) / a);
	elseif (b_2 <= -7e-135)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 5.6e+81)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -6.8e-62], N[(1.0 / N[(0.5 * N[(a / b$95$2), $MachinePrecision] + N[(-2.0 / N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, -5.1e-124], N[(N[(N[(a * c), $MachinePrecision] / N[(-1.0 * b$95$2 + N[Sqrt[b$95$2 ^ 2 + N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, -7e-135], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 5.6e+81], 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], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b_2 < -6.79999999999999975e-62

   1. Initial program 14.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 12.3%

         \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\sqrt{\sqrt{b\_2 \cdot b\_2 - a \cdot c}} \cdot \sqrt{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}}}{a} \]

      2. pow2 12.3%

         \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{{\left(\sqrt{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}\right)}^{2}}}{a} \]

      3. pow1/2 12.3%

         \[\leadsto \frac{\left(-b\_2\right) - {\left(\sqrt{\color{blue}{{\left(b\_2 \cdot b\_2 - a \cdot c\right)}^{0.5}}}\right)}^{2}}{a} \]

      4. sqrt-pow1 12.2%

         \[\leadsto \frac{\left(-b\_2\right) - {\color{blue}{\left({\left(b\_2 \cdot b\_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}}{a} \]

      5. pow2 12.2%

         \[\leadsto \frac{\left(-b\_2\right) - {\left({\left(\color{blue}{{b\_2}^{2}} - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{a} \]

      6. metadata-eval 12.2%

         \[\leadsto \frac{\left(-b\_2\right) - {\left({\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

   4. Applied egg-rr 12.2%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]
   5. Step-by-step derivation

      1. clear-num 12.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b\_2\right) - {\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}}} \]

      2. inv-pow 12.2%

         \[\leadsto \color{blue}{{\left(\frac{a}{\left(-b\_2\right) - {\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}\right)}^{-1}} \]

      3. pow-pow 14.4%

         \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - \color{blue}{{\left({b\_2}^{2} - a \cdot c\right)}^{\left(0.25 \cdot 2\right)}}}\right)}^{-1} \]

      4. metadata-eval 14.4%

         \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - {\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.5}}}\right)}^{-1} \]

      5. pow1/2 14.4%

         \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - \color{blue}{\sqrt{{b\_2}^{2} - a \cdot c}}}\right)}^{-1} \]

      6. sub-neg 14.4%

         \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - \sqrt{\color{blue}{{b\_2}^{2} + \left(-a \cdot c\right)}}}\right)}^{-1} \]

      7. unpow2 14.4%

         \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2} + \left(-a \cdot c\right)}}\right)}^{-1} \]

      8. distribute-rgt-neg-out 14.4%

         \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 + \color{blue}{a \cdot \left(-c\right)}}}\right)}^{-1} \]

      9. add-sqr-sqrt 10.6%

         \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 + \color{blue}{\sqrt{a \cdot \left(-c\right)} \cdot \sqrt{a \cdot \left(-c\right)}}}}\right)}^{-1} \]

      10. hypot-undefine 16.1%

          \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - \color{blue}{\mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)}}\right)}^{-1} \]

   6. Applied egg-rr 16.1%

      \[\leadsto \color{blue}{{\left(\frac{a}{\left(-b\_2\right) - \mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 16.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b\_2\right) - \mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)}}} \]

      2. distribute-rgt-neg-out 16.1%

         \[\leadsto \frac{1}{\frac{a}{\left(-b\_2\right) - \mathsf{hypot}\left(b\_2, \sqrt{\color{blue}{-a \cdot c}}\right)}} \]

      3. distribute-lft-neg-in 16.1%

         \[\leadsto \frac{1}{\frac{a}{\left(-b\_2\right) - \mathsf{hypot}\left(b\_2, \sqrt{\color{blue}{\left(-a\right) \cdot c}}\right)}} \]

   8. Simplified 16.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b\_2\right) - \mathsf{hypot}\left(b\_2, \sqrt{\left(-a\right) \cdot c}\right)}}} \]

   9. Taylor expanded in b_2 around -inf 0.0%

      \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \frac{a}{b\_2} + 2 \cdot \frac{b\_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}}} \]
   10. Step-by-step derivation

       1. fma-define 0.0%

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, 2 \cdot \frac{b\_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}\right)}} \]

       2. associate-*r/ 0.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \color{blue}{\frac{2 \cdot b\_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}}\right)} \]

       3. *-commutative 0.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \frac{2 \cdot b\_2}{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot c}}\right)} \]

       4. unpow2 0.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \frac{2 \cdot b\_2}{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot c}\right)} \]

       5. rem-square-sqrt 90.3%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \frac{2 \cdot b\_2}{\color{blue}{-1} \cdot c}\right)} \]

       6. times-frac 90.3%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \color{blue}{\frac{2}{-1} \cdot \frac{b\_2}{c}}\right)} \]

       7. metadata-eval 90.3%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \color{blue}{-2} \cdot \frac{b\_2}{c}\right)} \]

   11. Simplified 90.3%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, -2 \cdot \frac{b\_2}{c}\right)}} \]
   12. Step-by-step derivation

       1. clear-num 90.4%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, -2 \cdot \color{blue}{\frac{1}{\frac{c}{b\_2}}}\right)} \]

       2. un-div-inv 90.4%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \color{blue}{\frac{-2}{\frac{c}{b\_2}}}\right)} \]

   13. Applied egg-rr 90.4%

       \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \color{blue}{\frac{-2}{\frac{c}{b\_2}}}\right)} \]

   if -6.79999999999999975e-62 < b_2 < -5.1000000000000001e-124

   1. Initial program 67.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 67.5%

         \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\sqrt{\sqrt{b\_2 \cdot b\_2 - a \cdot c}} \cdot \sqrt{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}}}{a} \]

      2. pow2 67.5%

         \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{{\left(\sqrt{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}\right)}^{2}}}{a} \]

      3. pow1/2 67.5%

         \[\leadsto \frac{\left(-b\_2\right) - {\left(\sqrt{\color{blue}{{\left(b\_2 \cdot b\_2 - a \cdot c\right)}^{0.5}}}\right)}^{2}}{a} \]

      4. sqrt-pow1 67.5%

         \[\leadsto \frac{\left(-b\_2\right) - {\color{blue}{\left({\left(b\_2 \cdot b\_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}}{a} \]

      5. pow2 67.5%

         \[\leadsto \frac{\left(-b\_2\right) - {\left({\left(\color{blue}{{b\_2}^{2}} - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{a} \]

      6. metadata-eval 67.5%

         \[\leadsto \frac{\left(-b\_2\right) - {\left({\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

   4. Applied egg-rr 67.5%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]
   5. Step-by-step derivation

      1. flip-- 67.3%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\_2\right) \cdot \left(-b\_2\right) - {\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2} \cdot {\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}{\left(-b\_2\right) + {\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}}}{a} \]

      2. pow2 67.3%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b\_2\right)}^{2}} - {\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2} \cdot {\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}{\left(-b\_2\right) + {\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

      3. pow-pow 67.5%

         \[\leadsto \frac{\frac{{\left(-b\_2\right)}^{2} - \color{blue}{{\left({b\_2}^{2} - a \cdot c\right)}^{\left(0.25 \cdot 2\right)}} \cdot {\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}{\left(-b\_2\right) + {\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

      4. metadata-eval 67.5%

         \[\leadsto \frac{\frac{{\left(-b\_2\right)}^{2} - {\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.5}} \cdot {\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}{\left(-b\_2\right) + {\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

      5. pow-pow 66.9%

         \[\leadsto \frac{\frac{{\left(-b\_2\right)}^{2} - {\left({b\_2}^{2} - a \cdot c\right)}^{0.5} \cdot \color{blue}{{\left({b\_2}^{2} - a \cdot c\right)}^{\left(0.25 \cdot 2\right)}}}{\left(-b\_2\right) + {\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

      6. metadata-eval 66.9%

         \[\leadsto \frac{\frac{{\left(-b\_2\right)}^{2} - {\left({b\_2}^{2} - a \cdot c\right)}^{0.5} \cdot {\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.5}}}{\left(-b\_2\right) + {\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

      7. pow1/2 66.9%

         \[\leadsto \frac{\frac{{\left(-b\_2\right)}^{2} - \color{blue}{\sqrt{{b\_2}^{2} - a \cdot c}} \cdot {\left({b\_2}^{2} - a \cdot c\right)}^{0.5}}{\left(-b\_2\right) + {\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

      8. pow1/2 66.9%

         \[\leadsto \frac{\frac{{\left(-b\_2\right)}^{2} - \sqrt{{b\_2}^{2} - a \cdot c} \cdot \color{blue}{\sqrt{{b\_2}^{2} - a \cdot c}}}{\left(-b\_2\right) + {\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

      9. add-sqr-sqrt 67.3%

         \[\leadsto \frac{\frac{{\left(-b\_2\right)}^{2} - \color{blue}{\left({b\_2}^{2} - a \cdot c\right)}}{\left(-b\_2\right) + {\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

      10. neg-mul-1 67.3%

          \[\leadsto \frac{\frac{{\left(-b\_2\right)}^{2} - \left({b\_2}^{2} - a \cdot c\right)}{\color{blue}{-1 \cdot b\_2} + {\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

      11. fma-define 67.3%

          \[\leadsto \frac{\frac{{\left(-b\_2\right)}^{2} - \left({b\_2}^{2} - a \cdot c\right)}{\color{blue}{\mathsf{fma}\left(-1, b\_2, {\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}\right)}}}{a} \]

      12. pow-pow 67.5%

          \[\leadsto \frac{\frac{{\left(-b\_2\right)}^{2} - \left({b\_2}^{2} - a \cdot c\right)}{\mathsf{fma}\left(-1, b\_2, \color{blue}{{\left({b\_2}^{2} - a \cdot c\right)}^{\left(0.25 \cdot 2\right)}}\right)}}{a} \]

   6. Applied egg-rr 67.5%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\_2\right)}^{2} - \left({b\_2}^{2} - a \cdot c\right)}{\mathsf{fma}\left(-1, b\_2, \mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)\right)}}}{a} \]
   7. Step-by-step derivation

      1. associate--r- 99.8%

         \[\leadsto \frac{\frac{\color{blue}{\left({\left(-b\_2\right)}^{2} - {b\_2}^{2}\right) + a \cdot c}}{\mathsf{fma}\left(-1, b\_2, \mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)\right)}}{a} \]

      2. unpow2 99.8%

         \[\leadsto \frac{\frac{\left(\color{blue}{\left(-b\_2\right) \cdot \left(-b\_2\right)} - {b\_2}^{2}\right) + a \cdot c}{\mathsf{fma}\left(-1, b\_2, \mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)\right)}}{a} \]

      3. sqr-neg 99.8%

         \[\leadsto \frac{\frac{\left(\color{blue}{b\_2 \cdot b\_2} - {b\_2}^{2}\right) + a \cdot c}{\mathsf{fma}\left(-1, b\_2, \mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)\right)}}{a} \]

      4. unpow2 99.8%

         \[\leadsto \frac{\frac{\left(\color{blue}{{b\_2}^{2}} - {b\_2}^{2}\right) + a \cdot c}{\mathsf{fma}\left(-1, b\_2, \mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)\right)}}{a} \]

      5. +-inverses 99.8%

         \[\leadsto \frac{\frac{\color{blue}{0} + a \cdot c}{\mathsf{fma}\left(-1, b\_2, \mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)\right)}}{a} \]

      6. distribute-rgt-neg-out 99.8%

         \[\leadsto \frac{\frac{0 + a \cdot c}{\mathsf{fma}\left(-1, b\_2, \mathsf{hypot}\left(b\_2, \sqrt{\color{blue}{-a \cdot c}}\right)\right)}}{a} \]

      7. *-commutative 99.8%

         \[\leadsto \frac{\frac{0 + a \cdot c}{\mathsf{fma}\left(-1, b\_2, \mathsf{hypot}\left(b\_2, \sqrt{-\color{blue}{c \cdot a}}\right)\right)}}{a} \]

      8. distribute-rgt-neg-in 99.8%

         \[\leadsto \frac{\frac{0 + a \cdot c}{\mathsf{fma}\left(-1, b\_2, \mathsf{hypot}\left(b\_2, \sqrt{\color{blue}{c \cdot \left(-a\right)}}\right)\right)}}{a} \]

   8. Simplified 99.8%

      \[\leadsto \frac{\color{blue}{\frac{0 + a \cdot c}{\mathsf{fma}\left(-1, b\_2, \mathsf{hypot}\left(b\_2, \sqrt{c \cdot \left(-a\right)}\right)\right)}}}{a} \]

   if -5.1000000000000001e-124 < b_2 < -6.9999999999999997e-135

   1. Initial program 3.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf 100.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b\_2}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b\_2}} \]

   if -6.9999999999999997e-135 < b_2 < 5.5999999999999999e81

   1. Initial program 80.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   if 5.5999999999999999e81 < b_2

   1. Initial program 59.0%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf 96.9%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -6.8 \cdot 10^{-62}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \frac{-2}{\frac{c}{b\_2}}\right)}\\ \mathbf{elif}\;b\_2 \leq -5.1 \cdot 10^{-124}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\mathsf{fma}\left(-1, b\_2, \mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)\right)}}{a}\\ \mathbf{elif}\;b\_2 \leq -7 \cdot 10^{-135}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5.6 \cdot 10^{+81}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -7 \cdot 10^{-135}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \frac{-2}{\frac{c}{b\_2}}\right)}\\ \mathbf{elif}\;b\_2 \leq 3.5 \cdot 10^{+89}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7e-135)
   (/ 1.0 (fma 0.5 (/ a b_2) (/ -2.0 (/ c b_2))))
   (if (<= b_2 3.5e+89)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
     (* -2.0 (/ b_2 a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7e-135) {
		tmp = 1.0 / fma(0.5, (a / b_2), (-2.0 / (c / b_2)));
	} else if (b_2 <= 3.5e+89) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -7e-135)
		tmp = Float64(1.0 / fma(0.5, Float64(a / b_2), Float64(-2.0 / Float64(c / b_2))));
	elseif (b_2 <= 3.5e+89)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7e-135], N[(1.0 / N[(0.5 * N[(a / b$95$2), $MachinePrecision] + N[(-2.0 / N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 3.5e+89], 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], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -6.9999999999999997e-135

   1. Initial program 17.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 15.9%

         \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\sqrt{\sqrt{b\_2 \cdot b\_2 - a \cdot c}} \cdot \sqrt{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}}}{a} \]

      2. pow2 15.9%

         \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{{\left(\sqrt{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}\right)}^{2}}}{a} \]

      3. pow1/2 15.9%

         \[\leadsto \frac{\left(-b\_2\right) - {\left(\sqrt{\color{blue}{{\left(b\_2 \cdot b\_2 - a \cdot c\right)}^{0.5}}}\right)}^{2}}{a} \]

      4. sqrt-pow1 15.9%

         \[\leadsto \frac{\left(-b\_2\right) - {\color{blue}{\left({\left(b\_2 \cdot b\_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}}{a} \]

      5. pow2 15.9%

         \[\leadsto \frac{\left(-b\_2\right) - {\left({\left(\color{blue}{{b\_2}^{2}} - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{a} \]

      6. metadata-eval 15.9%

         \[\leadsto \frac{\left(-b\_2\right) - {\left({\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

   4. Applied egg-rr 15.9%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]
   5. Step-by-step derivation

      1. clear-num 15.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b\_2\right) - {\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}}} \]

      2. inv-pow 15.9%

         \[\leadsto \color{blue}{{\left(\frac{a}{\left(-b\_2\right) - {\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}\right)}^{-1}} \]

      3. pow-pow 17.8%

         \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - \color{blue}{{\left({b\_2}^{2} - a \cdot c\right)}^{\left(0.25 \cdot 2\right)}}}\right)}^{-1} \]

      4. metadata-eval 17.8%

         \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - {\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.5}}}\right)}^{-1} \]

      5. pow1/2 17.8%

         \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - \color{blue}{\sqrt{{b\_2}^{2} - a \cdot c}}}\right)}^{-1} \]

      6. sub-neg 17.8%

         \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - \sqrt{\color{blue}{{b\_2}^{2} + \left(-a \cdot c\right)}}}\right)}^{-1} \]

      7. unpow2 17.8%

         \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2} + \left(-a \cdot c\right)}}\right)}^{-1} \]

      8. distribute-rgt-neg-out 17.8%

         \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 + \color{blue}{a \cdot \left(-c\right)}}}\right)}^{-1} \]

      9. add-sqr-sqrt 14.4%

         \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 + \color{blue}{\sqrt{a \cdot \left(-c\right)} \cdot \sqrt{a \cdot \left(-c\right)}}}}\right)}^{-1} \]

      10. hypot-undefine 19.3%

          \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - \color{blue}{\mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)}}\right)}^{-1} \]

   6. Applied egg-rr 19.3%

      \[\leadsto \color{blue}{{\left(\frac{a}{\left(-b\_2\right) - \mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 19.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b\_2\right) - \mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)}}} \]

      2. distribute-rgt-neg-out 19.3%

         \[\leadsto \frac{1}{\frac{a}{\left(-b\_2\right) - \mathsf{hypot}\left(b\_2, \sqrt{\color{blue}{-a \cdot c}}\right)}} \]

      3. distribute-lft-neg-in 19.3%

         \[\leadsto \frac{1}{\frac{a}{\left(-b\_2\right) - \mathsf{hypot}\left(b\_2, \sqrt{\color{blue}{\left(-a\right) \cdot c}}\right)}} \]

   8. Simplified 19.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b\_2\right) - \mathsf{hypot}\left(b\_2, \sqrt{\left(-a\right) \cdot c}\right)}}} \]

   9. Taylor expanded in b_2 around -inf 0.0%

      \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \frac{a}{b\_2} + 2 \cdot \frac{b\_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}}} \]
   10. Step-by-step derivation

       1. fma-define 0.0%

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, 2 \cdot \frac{b\_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}\right)}} \]

       2. associate-*r/ 0.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \color{blue}{\frac{2 \cdot b\_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}}\right)} \]

       3. *-commutative 0.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \frac{2 \cdot b\_2}{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot c}}\right)} \]

       4. unpow2 0.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \frac{2 \cdot b\_2}{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot c}\right)} \]

       5. rem-square-sqrt 87.1%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \frac{2 \cdot b\_2}{\color{blue}{-1} \cdot c}\right)} \]

       6. times-frac 87.1%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \color{blue}{\frac{2}{-1} \cdot \frac{b\_2}{c}}\right)} \]

       7. metadata-eval 87.1%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \color{blue}{-2} \cdot \frac{b\_2}{c}\right)} \]

   11. Simplified 87.1%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, -2 \cdot \frac{b\_2}{c}\right)}} \]
   12. Step-by-step derivation

       1. clear-num 87.2%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, -2 \cdot \color{blue}{\frac{1}{\frac{c}{b\_2}}}\right)} \]

       2. un-div-inv 87.2%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \color{blue}{\frac{-2}{\frac{c}{b\_2}}}\right)} \]

   13. Applied egg-rr 87.2%

       \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \color{blue}{\frac{-2}{\frac{c}{b\_2}}}\right)} \]

   if -6.9999999999999997e-135 < b_2 < 3.5000000000000001e89

   1. Initial program 80.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   if 3.5000000000000001e89 < b_2

   1. Initial program 59.0%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf 96.9%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -7 \cdot 10^{-135}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \frac{-2}{\frac{c}{b\_2}}\right)}\\ \mathbf{elif}\;b\_2 \leq 3.5 \cdot 10^{+89}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -7 \cdot 10^{-135}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.65 \cdot 10^{-20}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7e-135)
   (/ (* c -0.5) b_2)
   (if (<= b_2 2.65e-20)
     (/ (- (- b_2) (sqrt (* a (- c)))) a)
     (* -2.0 (/ b_2 a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7e-135) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 2.65e-20) {
		tmp = (-b_2 - sqrt((a * -c))) / a;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-7d-135)) then
        tmp = (c * (-0.5d0)) / b_2
    else if (b_2 <= 2.65d-20) then
        tmp = (-b_2 - sqrt((a * -c))) / a
    else
        tmp = (-2.0d0) * (b_2 / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7e-135) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 2.65e-20) {
		tmp = (-b_2 - Math.sqrt((a * -c))) / a;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -7e-135:
		tmp = (c * -0.5) / b_2
	elif b_2 <= 2.65e-20:
		tmp = (-b_2 - math.sqrt((a * -c))) / a
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -7e-135)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 2.65e-20)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(a * Float64(-c)))) / a);
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -7e-135)
		tmp = (c * -0.5) / b_2;
	elseif (b_2 <= 2.65e-20)
		tmp = (-b_2 - sqrt((a * -c))) / a;
	else
		tmp = -2.0 * (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7e-135], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 2.65e-20], N[(N[((-b$95$2) - N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -6.9999999999999997e-135

   1. Initial program 17.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf 86.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. *-commutative 86.9%

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]

      2. associate-*l/ 86.9%

         \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b\_2}} \]

   5. Simplified 86.9%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b\_2}} \]

   if -6.9999999999999997e-135 < b_2 < 2.6500000000000001e-20

   1. Initial program 78.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around 0 69.5%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]
   4. Step-by-step derivation

      1. mul-1-neg 69.5%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{-a \cdot c}}}{a} \]

      2. distribute-rgt-neg-out 69.5%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{a \cdot \left(-c\right)}}}{a} \]

   5. Simplified 69.5%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{a \cdot \left(-c\right)}}}{a} \]

   if 2.6500000000000001e-20 < b_2

   1. Initial program 64.0%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf 94.6%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -7 \cdot 10^{-135}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.65 \cdot 10^{-20}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -7 \cdot 10^{-135}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, -2 \cdot \frac{b\_2}{c}\right)}\\ \mathbf{elif}\;b\_2 \leq 1.05 \cdot 10^{-17}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7e-135)
   (/ 1.0 (fma 0.5 (/ a b_2) (* -2.0 (/ b_2 c))))
   (if (<= b_2 1.05e-17)
     (/ (- (- b_2) (sqrt (* a (- c)))) a)
     (* -2.0 (/ b_2 a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7e-135) {
		tmp = 1.0 / fma(0.5, (a / b_2), (-2.0 * (b_2 / c)));
	} else if (b_2 <= 1.05e-17) {
		tmp = (-b_2 - sqrt((a * -c))) / a;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -7e-135)
		tmp = Float64(1.0 / fma(0.5, Float64(a / b_2), Float64(-2.0 * Float64(b_2 / c))));
	elseif (b_2 <= 1.05e-17)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(a * Float64(-c)))) / a);
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7e-135], N[(1.0 / N[(0.5 * N[(a / b$95$2), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.05e-17], N[(N[((-b$95$2) - N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -6.9999999999999997e-135

   1. Initial program 17.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 15.9%

         \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\sqrt{\sqrt{b\_2 \cdot b\_2 - a \cdot c}} \cdot \sqrt{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}}}{a} \]

      2. pow2 15.9%

         \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{{\left(\sqrt{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}\right)}^{2}}}{a} \]

      3. pow1/2 15.9%

         \[\leadsto \frac{\left(-b\_2\right) - {\left(\sqrt{\color{blue}{{\left(b\_2 \cdot b\_2 - a \cdot c\right)}^{0.5}}}\right)}^{2}}{a} \]

      4. sqrt-pow1 15.9%

         \[\leadsto \frac{\left(-b\_2\right) - {\color{blue}{\left({\left(b\_2 \cdot b\_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}}{a} \]

      5. pow2 15.9%

         \[\leadsto \frac{\left(-b\_2\right) - {\left({\left(\color{blue}{{b\_2}^{2}} - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{a} \]

      6. metadata-eval 15.9%

         \[\leadsto \frac{\left(-b\_2\right) - {\left({\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

   4. Applied egg-rr 15.9%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]
   5. Step-by-step derivation

      1. clear-num 15.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b\_2\right) - {\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}}} \]

      2. inv-pow 15.9%

         \[\leadsto \color{blue}{{\left(\frac{a}{\left(-b\_2\right) - {\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}\right)}^{-1}} \]

      3. pow-pow 17.8%

         \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - \color{blue}{{\left({b\_2}^{2} - a \cdot c\right)}^{\left(0.25 \cdot 2\right)}}}\right)}^{-1} \]

      4. metadata-eval 17.8%

         \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - {\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.5}}}\right)}^{-1} \]

      5. pow1/2 17.8%

         \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - \color{blue}{\sqrt{{b\_2}^{2} - a \cdot c}}}\right)}^{-1} \]

      6. sub-neg 17.8%

         \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - \sqrt{\color{blue}{{b\_2}^{2} + \left(-a \cdot c\right)}}}\right)}^{-1} \]

      7. unpow2 17.8%

         \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2} + \left(-a \cdot c\right)}}\right)}^{-1} \]

      8. distribute-rgt-neg-out 17.8%

         \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 + \color{blue}{a \cdot \left(-c\right)}}}\right)}^{-1} \]

      9. add-sqr-sqrt 14.4%

         \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 + \color{blue}{\sqrt{a \cdot \left(-c\right)} \cdot \sqrt{a \cdot \left(-c\right)}}}}\right)}^{-1} \]

      10. hypot-undefine 19.3%

          \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - \color{blue}{\mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)}}\right)}^{-1} \]

   6. Applied egg-rr 19.3%

      \[\leadsto \color{blue}{{\left(\frac{a}{\left(-b\_2\right) - \mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 19.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b\_2\right) - \mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)}}} \]

      2. distribute-rgt-neg-out 19.3%

         \[\leadsto \frac{1}{\frac{a}{\left(-b\_2\right) - \mathsf{hypot}\left(b\_2, \sqrt{\color{blue}{-a \cdot c}}\right)}} \]

      3. distribute-lft-neg-in 19.3%

         \[\leadsto \frac{1}{\frac{a}{\left(-b\_2\right) - \mathsf{hypot}\left(b\_2, \sqrt{\color{blue}{\left(-a\right) \cdot c}}\right)}} \]

   8. Simplified 19.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b\_2\right) - \mathsf{hypot}\left(b\_2, \sqrt{\left(-a\right) \cdot c}\right)}}} \]

   9. Taylor expanded in b_2 around -inf 0.0%

      \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \frac{a}{b\_2} + 2 \cdot \frac{b\_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}}} \]
   10. Step-by-step derivation

       1. fma-define 0.0%

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, 2 \cdot \frac{b\_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}\right)}} \]

       2. associate-*r/ 0.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \color{blue}{\frac{2 \cdot b\_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}}\right)} \]

       3. *-commutative 0.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \frac{2 \cdot b\_2}{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot c}}\right)} \]

       4. unpow2 0.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \frac{2 \cdot b\_2}{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot c}\right)} \]

       5. rem-square-sqrt 87.1%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \frac{2 \cdot b\_2}{\color{blue}{-1} \cdot c}\right)} \]

       6. times-frac 87.1%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \color{blue}{\frac{2}{-1} \cdot \frac{b\_2}{c}}\right)} \]

       7. metadata-eval 87.1%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \color{blue}{-2} \cdot \frac{b\_2}{c}\right)} \]

   11. Simplified 87.1%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, -2 \cdot \frac{b\_2}{c}\right)}} \]

   if -6.9999999999999997e-135 < b_2 < 1.04999999999999996e-17

   1. Initial program 78.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around 0 69.5%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]
   4. Step-by-step derivation

      1. mul-1-neg 69.5%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{-a \cdot c}}}{a} \]

      2. distribute-rgt-neg-out 69.5%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{a \cdot \left(-c\right)}}}{a} \]

   5. Simplified 69.5%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{a \cdot \left(-c\right)}}}{a} \]

   if 1.04999999999999996e-17 < b_2

   1. Initial program 64.0%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf 94.6%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -7 \cdot 10^{-135}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, -2 \cdot \frac{b\_2}{c}\right)}\\ \mathbf{elif}\;b\_2 \leq 1.05 \cdot 10^{-17}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -7 \cdot 10^{-135}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \frac{-2}{\frac{c}{b\_2}}\right)}\\ \mathbf{elif}\;b\_2 \leq 8.4 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7e-135)
   (/ 1.0 (fma 0.5 (/ a b_2) (/ -2.0 (/ c b_2))))
   (if (<= b_2 8.4e-18)
     (/ (- (- b_2) (sqrt (* a (- c)))) a)
     (* -2.0 (/ b_2 a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7e-135) {
		tmp = 1.0 / fma(0.5, (a / b_2), (-2.0 / (c / b_2)));
	} else if (b_2 <= 8.4e-18) {
		tmp = (-b_2 - sqrt((a * -c))) / a;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -7e-135)
		tmp = Float64(1.0 / fma(0.5, Float64(a / b_2), Float64(-2.0 / Float64(c / b_2))));
	elseif (b_2 <= 8.4e-18)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(a * Float64(-c)))) / a);
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7e-135], N[(1.0 / N[(0.5 * N[(a / b$95$2), $MachinePrecision] + N[(-2.0 / N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 8.4e-18], N[(N[((-b$95$2) - N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -6.9999999999999997e-135

   1. Initial program 17.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 15.9%

         \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\sqrt{\sqrt{b\_2 \cdot b\_2 - a \cdot c}} \cdot \sqrt{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}}}{a} \]

      2. pow2 15.9%

         \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{{\left(\sqrt{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}\right)}^{2}}}{a} \]

      3. pow1/2 15.9%

         \[\leadsto \frac{\left(-b\_2\right) - {\left(\sqrt{\color{blue}{{\left(b\_2 \cdot b\_2 - a \cdot c\right)}^{0.5}}}\right)}^{2}}{a} \]

      4. sqrt-pow1 15.9%

         \[\leadsto \frac{\left(-b\_2\right) - {\color{blue}{\left({\left(b\_2 \cdot b\_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}}{a} \]

      5. pow2 15.9%

         \[\leadsto \frac{\left(-b\_2\right) - {\left({\left(\color{blue}{{b\_2}^{2}} - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{a} \]

      6. metadata-eval 15.9%

         \[\leadsto \frac{\left(-b\_2\right) - {\left({\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

   4. Applied egg-rr 15.9%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]
   5. Step-by-step derivation

      1. clear-num 15.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b\_2\right) - {\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}}} \]

      2. inv-pow 15.9%

         \[\leadsto \color{blue}{{\left(\frac{a}{\left(-b\_2\right) - {\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}}\right)}^{-1}} \]

      3. pow-pow 17.8%

         \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - \color{blue}{{\left({b\_2}^{2} - a \cdot c\right)}^{\left(0.25 \cdot 2\right)}}}\right)}^{-1} \]

      4. metadata-eval 17.8%

         \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - {\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.5}}}\right)}^{-1} \]

      5. pow1/2 17.8%

         \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - \color{blue}{\sqrt{{b\_2}^{2} - a \cdot c}}}\right)}^{-1} \]

      6. sub-neg 17.8%

         \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - \sqrt{\color{blue}{{b\_2}^{2} + \left(-a \cdot c\right)}}}\right)}^{-1} \]

      7. unpow2 17.8%

         \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2} + \left(-a \cdot c\right)}}\right)}^{-1} \]

      8. distribute-rgt-neg-out 17.8%

         \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 + \color{blue}{a \cdot \left(-c\right)}}}\right)}^{-1} \]

      9. add-sqr-sqrt 14.4%

         \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 + \color{blue}{\sqrt{a \cdot \left(-c\right)} \cdot \sqrt{a \cdot \left(-c\right)}}}}\right)}^{-1} \]

      10. hypot-undefine 19.3%

          \[\leadsto {\left(\frac{a}{\left(-b\_2\right) - \color{blue}{\mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)}}\right)}^{-1} \]

   6. Applied egg-rr 19.3%

      \[\leadsto \color{blue}{{\left(\frac{a}{\left(-b\_2\right) - \mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 19.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b\_2\right) - \mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)}}} \]

      2. distribute-rgt-neg-out 19.3%

         \[\leadsto \frac{1}{\frac{a}{\left(-b\_2\right) - \mathsf{hypot}\left(b\_2, \sqrt{\color{blue}{-a \cdot c}}\right)}} \]

      3. distribute-lft-neg-in 19.3%

         \[\leadsto \frac{1}{\frac{a}{\left(-b\_2\right) - \mathsf{hypot}\left(b\_2, \sqrt{\color{blue}{\left(-a\right) \cdot c}}\right)}} \]

   8. Simplified 19.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b\_2\right) - \mathsf{hypot}\left(b\_2, \sqrt{\left(-a\right) \cdot c}\right)}}} \]

   9. Taylor expanded in b_2 around -inf 0.0%

      \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \frac{a}{b\_2} + 2 \cdot \frac{b\_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}}} \]
   10. Step-by-step derivation

       1. fma-define 0.0%

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, 2 \cdot \frac{b\_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}\right)}} \]

       2. associate-*r/ 0.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \color{blue}{\frac{2 \cdot b\_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}}\right)} \]

       3. *-commutative 0.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \frac{2 \cdot b\_2}{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot c}}\right)} \]

       4. unpow2 0.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \frac{2 \cdot b\_2}{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot c}\right)} \]

       5. rem-square-sqrt 87.1%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \frac{2 \cdot b\_2}{\color{blue}{-1} \cdot c}\right)} \]

       6. times-frac 87.1%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \color{blue}{\frac{2}{-1} \cdot \frac{b\_2}{c}}\right)} \]

       7. metadata-eval 87.1%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \color{blue}{-2} \cdot \frac{b\_2}{c}\right)} \]

   11. Simplified 87.1%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, -2 \cdot \frac{b\_2}{c}\right)}} \]
   12. Step-by-step derivation

       1. clear-num 87.2%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, -2 \cdot \color{blue}{\frac{1}{\frac{c}{b\_2}}}\right)} \]

       2. un-div-inv 87.2%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \color{blue}{\frac{-2}{\frac{c}{b\_2}}}\right)} \]

   13. Applied egg-rr 87.2%

       \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \color{blue}{\frac{-2}{\frac{c}{b\_2}}}\right)} \]

   if -6.9999999999999997e-135 < b_2 < 8.39999999999999998e-18

   1. Initial program 78.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around 0 69.5%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]
   4. Step-by-step derivation

      1. mul-1-neg 69.5%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{-a \cdot c}}}{a} \]

      2. distribute-rgt-neg-out 69.5%

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{a \cdot \left(-c\right)}}}{a} \]

   5. Simplified 69.5%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{a \cdot \left(-c\right)}}}{a} \]

   if 8.39999999999999998e-18 < b_2

   1. Initial program 64.0%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf 94.6%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -7 \cdot 10^{-135}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b\_2}, \frac{-2}{\frac{c}{b\_2}}\right)}\\ \mathbf{elif}\;b\_2 \leq 8.4 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.4% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e-310)
   (/ (* c -0.5) b_2)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-4d-310)) then
        tmp = (c * (-0.5d0)) / b_2
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4e-310:
		tmp = (c * -0.5) / b_2
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4e-310)
		tmp = Float64(Float64(c * -0.5) / b_2);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4e-310)
		tmp = (c * -0.5) / b_2;
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4e-310], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.999999999999988e-310

   1. Initial program 30.0%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf 71.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. *-commutative 71.2%

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]

      2. associate-*l/ 71.2%

         \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b\_2}} \]

   5. Simplified 71.2%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b\_2}} \]

   if -3.999999999999988e-310 < b_2

   1. Initial program 70.7%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf 63.3%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.8% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.2 \cdot 10^{+17}:\\ \;\;\;\;0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.2e+17) (* 0.5 (/ c b_2)) (* -2.0 (/ b_2 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.2e+17) {
		tmp = 0.5 * (c / b_2);
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.2d+17)) then
        tmp = 0.5d0 * (c / b_2)
    else
        tmp = (-2.0d0) * (b_2 / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.2e+17) {
		tmp = 0.5 * (c / b_2);
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.2e+17:
		tmp = 0.5 * (c / b_2)
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.2e+17)
		tmp = Float64(0.5 * Float64(c / b_2));
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.2e+17)
		tmp = 0.5 * (c / b_2);
	else
		tmp = -2.0 * (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.2e+17], N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.2e17

   1. Initial program 10.0%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf 2.3%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2 + 0.5 \cdot \frac{a \cdot c}{b\_2}}}{a} \]

   4. Taylor expanded in b_2 around 0 22.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b\_2}} \]

   if -2.2e17 < b_2

   1. Initial program 65.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf 42.8%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.2 \cdot 10^{+17}:\\ \;\;\;\;0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.3% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -9.4 \cdot 10^{-291}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -9.4e-291) (/ (* c -0.5) b_2) (* -2.0 (/ b_2 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9.4e-291) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-9.4d-291)) then
        tmp = (c * (-0.5d0)) / b_2
    else
        tmp = (-2.0d0) * (b_2 / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9.4e-291) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -9.4e-291:
		tmp = (c * -0.5) / b_2
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -9.4e-291)
		tmp = Float64(Float64(c * -0.5) / b_2);
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -9.4e-291)
		tmp = (c * -0.5) / b_2;
	else
		tmp = -2.0 * (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -9.4e-291], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -9.3999999999999997e-291

   1. Initial program 28.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf 72.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. *-commutative 72.7%

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]

      2. associate-*l/ 72.7%

         \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b\_2}} \]

   5. Simplified 72.7%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b\_2}} \]

   if -9.3999999999999997e-291 < b_2

   1. Initial program 71.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf 61.6%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -9.4 \cdot 10^{-291}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.7% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \frac{b\_2}{a} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* -2.0 (/ b_2 a)))

C

double code(double a, double b_2, double c) {
	return -2.0 * (b_2 / a);
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-2.0d0) * (b_2 / a)
end function

Java

public static double code(double a, double b_2, double c) {
	return -2.0 * (b_2 / a);
}

Python

def code(a, b_2, c):
	return -2.0 * (b_2 / a)

Julia

function code(a, b_2, c)
	return Float64(-2.0 * Float64(b_2 / a))
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = -2.0 * (b_2 / a);
end

Wolfram

code[a_, b$95$2_, c_] := N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.4%

   \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
2. Add Preprocessing

3. Taylor expanded in b_2 around inf 30.2%

   \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]

4. Final simplification 30.2%

   \[\leadsto -2 \cdot \frac{b\_2}{a} \]
5. Add Preprocessing

Developer target: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_1 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{\left|b\_2\right| - t\_0} \cdot \sqrt{\left|b\_2\right| + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b\_2, t\_0\right)\\ \end{array}\\ \mathbf{if}\;b\_2 < 0:\\ \;\;\;\;\frac{c}{t\_1 - b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + t\_1}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_1
         (if (== (copysign a c) a)
           (* (sqrt (- (fabs b_2) t_0)) (sqrt (+ (fabs b_2) t_0)))
           (hypot b_2 t_0))))
   (if (< b_2 0.0) (/ c (- t_1 b_2)) (/ (+ b_2 t_1) (- a)))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((fabs(b_2) - t_0)) * sqrt((fabs(b_2) + t_0));
	} else {
		tmp = hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = c / (t_1 - b_2);
	} else {
		tmp_1 = (b_2 + t_1) / -a;
	}
	return tmp_1;
}

Java

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((Math.abs(b_2) - t_0)) * Math.sqrt((Math.abs(b_2) + t_0));
	} else {
		tmp = Math.hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = c / (t_1 - b_2);
	} else {
		tmp_1 = (b_2 + t_1) / -a;
	}
	return tmp_1;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((math.fabs(b_2) - t_0)) * math.sqrt((math.fabs(b_2) + t_0))
	else:
		tmp = math.hypot(b_2, t_0)
	t_1 = tmp
	tmp_1 = 0
	if b_2 < 0.0:
		tmp_1 = c / (t_1 - b_2)
	else:
		tmp_1 = (b_2 + t_1) / -a
	return tmp_1

Julia

function code(a, b_2, c)
	t_0 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(abs(b_2) - t_0)) * sqrt(Float64(abs(b_2) + t_0)));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp
	tmp_1 = 0.0
	if (b_2 < 0.0)
		tmp_1 = Float64(c / Float64(t_1 - b_2));
	else
		tmp_1 = Float64(Float64(b_2 + t_1) / Float64(-a));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b_2, c)
	t_0 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((abs(b_2) - t_0)) * sqrt((abs(b_2) + t_0));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp;
	tmp_2 = 0.0;
	if (b_2 < 0.0)
		tmp_2 = c / (t_1 - b_2);
	else
		tmp_2 = (b_2 + t_1) / -a;
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(N[Abs[b$95$2], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[Abs[b$95$2], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[b$95$2 ^ 2 + t$95$0 ^ 2], $MachinePrecision]]}, If[Less[b$95$2, 0.0], N[(c / N[(t$95$1 - b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$2 + t$95$1), $MachinePrecision] / (-a)), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2024046 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  :herbie-expected 10

  :alt
  (if (< b_2 0.0) (/ c (- (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot b_2 (* (sqrt (fabs a)) (sqrt (fabs c))))) b_2)) (/ (+ b_2 (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot b_2 (* (sqrt (fabs a)) (sqrt (fabs c)))))) (- a)))

  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))