quad2m (problem 3.2.1, negative)

Percentage Accurate: 51.9% → 86.5%

Time: 9.2s

Alternatives: 9

Speedup: 15.9×

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: 51.9% 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: 86.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - a \cdot c}\\ \mathbf{if}\;b_2 \leq -1.9 \cdot 10^{+69}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}\\ \mathbf{elif}\;b_2 \leq -1.1 \cdot 10^{-105}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\mathsf{fma}\left(-1, b_2, t_0\right)}}{a}\\ \mathbf{elif}\;b_2 \leq 7.5 \cdot 10^{+118}:\\ \;\;\;\;\frac{\left(-b_2\right) - t_0}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b_2 b_2) (* a c)))))
   (if (<= b_2 -1.9e+69)
     (/ 1.0 (+ (* 0.5 (/ a b_2)) (* -2.0 (/ b_2 c))))
     (if (<= b_2 -1.1e-105)
       (/ (/ (* a c) (fma -1.0 b_2 t_0)) a)
       (if (<= b_2 7.5e+118)
         (/ (- (- b_2) t_0) a)
         (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2))))))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(((b_2 * b_2) - (a * c)));
	double tmp;
	if (b_2 <= -1.9e+69) {
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	} else if (b_2 <= -1.1e-105) {
		tmp = ((a * c) / fma(-1.0, b_2, t_0)) / a;
	} else if (b_2 <= 7.5e+118) {
		tmp = (-b_2 - t_0) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	t_0 = sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))
	tmp = 0.0
	if (b_2 <= -1.9e+69)
		tmp = Float64(1.0 / Float64(Float64(0.5 * Float64(a / b_2)) + Float64(-2.0 * Float64(b_2 / c))));
	elseif (b_2 <= -1.1e-105)
		tmp = Float64(Float64(Float64(a * c) / fma(-1.0, b_2, t_0)) / a);
	elseif (b_2 <= 7.5e+118)
		tmp = Float64(Float64(Float64(-b_2) - t_0) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$2, -1.9e+69], N[(1.0 / N[(N[(0.5 * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, -1.1e-105], N[(N[(N[(a * c), $MachinePrecision] / N[(-1.0 * b$95$2 + t$95$0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 7.5e+118], N[(N[((-b$95$2) - t$95$0), $MachinePrecision] / a), $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 4 regimes

2. if b_2 < -1.90000000000000014e69

   1. Initial program 12.5%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. prod-diff 12.3%

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

      2. *-commutative 12.3%

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

      3. fma-neg 12.3%

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

      4. prod-diff 12.3%

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

      5. *-commutative 12.3%

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

      6. fma-neg 12.3%

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

      7. associate-+l+ 12.3%

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

      8. *-commutative 12.3%

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

      9. fma-udef 12.3%

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

      10. distribute-lft-neg-in 12.3%

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

      11. *-commutative 12.3%

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

      12. distribute-rgt-neg-in 12.3%

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

      13. fma-def 12.3%

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

      14. *-commutative 12.3%

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

      15. fma-udef 12.3%

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

      16. distribute-lft-neg-in 12.3%

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

      17. *-commutative 12.3%

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

      18. distribute-rgt-neg-in 12.3%

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

      19. fma-def 12.3%

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

   3. Applied egg-rr 12.3%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}}{a} \]
   4. Step-by-step derivation

      1. *-commutative 12.3%

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

      2. count-2 12.3%

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

      3. *-commutative 12.3%

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

   5. Simplified 12.3%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}}{a} \]
   6. Step-by-step derivation

      1. clear-num 12.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}}} \]

      2. inv-pow 12.3%

         \[\leadsto \color{blue}{{\left(\frac{a}{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}\right)}^{-1}} \]

      3. +-commutative 12.3%

         \[\leadsto {\left(\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right) + \left(b_2 \cdot b_2 - c \cdot a\right)}}}\right)}^{-1} \]

      4. *-commutative 12.3%

         \[\leadsto {\left(\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(a, -c, c \cdot a\right) \cdot 2} + \left(b_2 \cdot b_2 - c \cdot a\right)}}\right)}^{-1} \]

      5. fma-def 12.3%

         \[\leadsto {\left(\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, -c, c \cdot a\right), 2, b_2 \cdot b_2 - c \cdot a\right)}}}\right)}^{-1} \]

      6. *-commutative 12.3%

         \[\leadsto {\left(\frac{a}{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(a, -c, \color{blue}{a \cdot c}\right), 2, b_2 \cdot b_2 - c \cdot a\right)}}\right)}^{-1} \]

      7. *-commutative 12.3%

         \[\leadsto {\left(\frac{a}{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(a, -c, a \cdot c\right), 2, b_2 \cdot b_2 - \color{blue}{a \cdot c}\right)}}\right)}^{-1} \]

   7. Applied egg-rr 12.3%

      \[\leadsto \color{blue}{{\left(\frac{a}{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(a, -c, a \cdot c\right), 2, b_2 \cdot b_2 - a \cdot c\right)}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 12.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(a, -c, a \cdot c\right), 2, b_2 \cdot b_2 - a \cdot c\right)}}}} \]

      2. fma-udef 12.3%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(a, -c, a \cdot c\right) \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}}} \]

      3. fma-def 12.3%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{\left(a \cdot \left(-c\right) + a \cdot c\right)} \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      4. *-commutative 12.3%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(\color{blue}{\left(-c\right) \cdot a} + a \cdot c\right) \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      5. neg-mul-1 12.3%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(\color{blue}{\left(-1 \cdot c\right)} \cdot a + a \cdot c\right) \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      6. associate-*r* 12.3%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(\color{blue}{-1 \cdot \left(c \cdot a\right)} + a \cdot c\right) \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      7. *-commutative 12.3%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(-1 \cdot \color{blue}{\left(a \cdot c\right)} + a \cdot c\right) \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      8. distribute-lft1-in 12.3%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{\left(\left(-1 + 1\right) \cdot \left(a \cdot c\right)\right)} \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      9. metadata-eval 12.3%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(\color{blue}{0} \cdot \left(a \cdot c\right)\right) \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      10. mul0-lft 12.5%

          \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{0} \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      11. metadata-eval 12.5%

          \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{0} + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      12. associate-+r- 12.5%

          \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{\left(0 + b_2 \cdot b_2\right) - a \cdot c}}}} \]

   9. Simplified 12.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(0 + b_2 \cdot b_2\right) - a \cdot c}}}} \]

   10. Taylor expanded in b_2 around -inf 94.9%

       \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}} \]

   if -1.90000000000000014e69 < b_2 < -1.10000000000000002e-105

   1. Initial program 44.5%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. prod-diff 44.4%

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

      2. *-commutative 44.4%

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

      3. fma-neg 44.3%

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

      4. prod-diff 44.4%

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

      5. *-commutative 44.4%

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

      6. fma-neg 44.3%

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

      7. associate-+l+ 44.3%

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

      8. *-commutative 44.3%

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

      9. fma-udef 44.3%

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

      10. distribute-lft-neg-in 44.3%

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

      11. *-commutative 44.3%

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

      12. distribute-rgt-neg-in 44.3%

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

      13. fma-def 44.3%

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

      14. *-commutative 44.3%

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

      15. fma-udef 44.3%

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

      16. distribute-lft-neg-in 44.3%

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

      17. *-commutative 44.3%

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

      18. distribute-rgt-neg-in 44.3%

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

      19. fma-def 44.3%

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

   3. Applied egg-rr 44.3%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}}{a} \]
   4. Step-by-step derivation

      1. *-commutative 44.3%

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

      2. count-2 44.3%

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

      3. *-commutative 44.3%

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

   5. Simplified 44.3%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}}{a} \]
   6. Step-by-step derivation

      1. flip-- 44.0%

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

      2. pow2 44.0%

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

      3. add-sqr-sqrt 44.5%

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

      4. +-commutative 44.5%

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

      5. *-commutative 44.5%

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

      6. fma-def 44.5%

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

      7. *-commutative 44.5%

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

      8. *-commutative 44.5%

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

   7. Applied egg-rr 44.5%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b_2\right)}^{2} - \mathsf{fma}\left(\mathsf{fma}\left(a, -c, a \cdot c\right), 2, b_2 \cdot b_2 - a \cdot c\right)}{\mathsf{fma}\left(-1, b_2, \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(a, -c, a \cdot c\right), 2, b_2 \cdot b_2 - a \cdot c\right)}\right)}}}{a} \]

   9. Simplified 81.8%

      \[\leadsto \frac{\color{blue}{\frac{a \cdot c}{\mathsf{fma}\left(-1, b_2, \sqrt{\left(0 + b_2 \cdot b_2\right) - a \cdot c}\right)}}}{a} \]

   if -1.10000000000000002e-105 < b_2 < 7.50000000000000003e118

   1. Initial program 82.4%

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

   if 7.50000000000000003e118 < b_2

   1. Initial program 50.1%

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

   2. Taylor expanded in b_2 around inf 97.9%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.9 \cdot 10^{+69}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}\\ \mathbf{elif}\;b_2 \leq -1.1 \cdot 10^{-105}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\mathsf{fma}\left(-1, b_2, \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a}\\ \mathbf{elif}\;b_2 \leq 7.5 \cdot 10^{+118}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 2: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -1.5 \cdot 10^{-62}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}\\ \mathbf{elif}\;b_2 \leq 2.15 \cdot 10^{+120}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.5e-62)
   (/ 1.0 (+ (* 0.5 (/ a b_2)) (* -2.0 (/ b_2 c))))
   (if (<= b_2 2.15e+120)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
     (+ (* -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 <= -1.5e-62) {
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	} else if (b_2 <= 2.15e+120) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
	} 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 <= (-1.5d-62)) then
        tmp = 1.0d0 / ((0.5d0 * (a / b_2)) + ((-2.0d0) * (b_2 / c)))
    else if (b_2 <= 2.15d+120) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
    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 <= -1.5e-62) {
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	} else if (b_2 <= 2.15e+120) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
	} 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 <= -1.5e-62:
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)))
	elif b_2 <= 2.15e+120:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a
	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 <= -1.5e-62)
		tmp = Float64(1.0 / Float64(Float64(0.5 * Float64(a / b_2)) + Float64(-2.0 * Float64(b_2 / c))));
	elseif (b_2 <= 2.15e+120)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	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 <= -1.5e-62)
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	elseif (b_2 <= 2.15e+120)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
	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, -1.5e-62], N[(1.0 / N[(N[(0.5 * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 2.15e+120], 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[(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 3 regimes

2. if b_2 < -1.5000000000000001e-62

   1. Initial program 21.2%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. prod-diff 21.1%

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

      2. *-commutative 21.1%

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

      3. fma-neg 21.1%

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

      4. prod-diff 21.1%

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

      5. *-commutative 21.1%

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

      6. fma-neg 21.1%

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

      7. associate-+l+ 21.1%

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

      8. *-commutative 21.1%

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

      9. fma-udef 21.1%

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

      10. distribute-lft-neg-in 21.1%

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

      11. *-commutative 21.1%

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

      12. distribute-rgt-neg-in 21.1%

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

      13. fma-def 21.1%

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

      14. *-commutative 21.1%

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

      15. fma-udef 21.1%

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

      16. distribute-lft-neg-in 21.1%

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

      17. *-commutative 21.1%

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

      18. distribute-rgt-neg-in 21.1%

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

      19. fma-def 21.1%

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

   3. Applied egg-rr 21.1%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}}{a} \]
   4. Step-by-step derivation

      1. *-commutative 21.1%

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

      2. count-2 21.1%

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

      3. *-commutative 21.1%

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

   5. Simplified 21.1%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}}{a} \]
   6. Step-by-step derivation

      1. clear-num 21.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}}} \]

      2. inv-pow 21.1%

         \[\leadsto \color{blue}{{\left(\frac{a}{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}\right)}^{-1}} \]

      3. +-commutative 21.1%

         \[\leadsto {\left(\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right) + \left(b_2 \cdot b_2 - c \cdot a\right)}}}\right)}^{-1} \]

      4. *-commutative 21.1%

         \[\leadsto {\left(\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(a, -c, c \cdot a\right) \cdot 2} + \left(b_2 \cdot b_2 - c \cdot a\right)}}\right)}^{-1} \]

      5. fma-def 21.1%

         \[\leadsto {\left(\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, -c, c \cdot a\right), 2, b_2 \cdot b_2 - c \cdot a\right)}}}\right)}^{-1} \]

      6. *-commutative 21.1%

         \[\leadsto {\left(\frac{a}{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(a, -c, \color{blue}{a \cdot c}\right), 2, b_2 \cdot b_2 - c \cdot a\right)}}\right)}^{-1} \]

      7. *-commutative 21.1%

         \[\leadsto {\left(\frac{a}{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(a, -c, a \cdot c\right), 2, b_2 \cdot b_2 - \color{blue}{a \cdot c}\right)}}\right)}^{-1} \]

   7. Applied egg-rr 21.1%

      \[\leadsto \color{blue}{{\left(\frac{a}{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(a, -c, a \cdot c\right), 2, b_2 \cdot b_2 - a \cdot c\right)}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 21.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(a, -c, a \cdot c\right), 2, b_2 \cdot b_2 - a \cdot c\right)}}}} \]

      2. fma-udef 21.1%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(a, -c, a \cdot c\right) \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}}} \]

      3. fma-def 21.0%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{\left(a \cdot \left(-c\right) + a \cdot c\right)} \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      4. *-commutative 21.0%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(\color{blue}{\left(-c\right) \cdot a} + a \cdot c\right) \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      5. neg-mul-1 21.0%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(\color{blue}{\left(-1 \cdot c\right)} \cdot a + a \cdot c\right) \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      6. associate-*r* 21.0%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(\color{blue}{-1 \cdot \left(c \cdot a\right)} + a \cdot c\right) \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      7. *-commutative 21.0%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(-1 \cdot \color{blue}{\left(a \cdot c\right)} + a \cdot c\right) \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      8. distribute-lft1-in 21.0%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{\left(\left(-1 + 1\right) \cdot \left(a \cdot c\right)\right)} \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      9. metadata-eval 21.0%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(\color{blue}{0} \cdot \left(a \cdot c\right)\right) \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      10. mul0-lft 21.2%

          \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{0} \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      11. metadata-eval 21.2%

          \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{0} + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      12. associate-+r- 21.2%

          \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{\left(0 + b_2 \cdot b_2\right) - a \cdot c}}}} \]

   9. Simplified 21.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(0 + b_2 \cdot b_2\right) - a \cdot c}}}} \]

   10. Taylor expanded in b_2 around -inf 85.3%

       \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}} \]

   if -1.5000000000000001e-62 < b_2 < 2.1500000000000001e120

   1. Initial program 80.2%

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

   if 2.1500000000000001e120 < b_2

   1. Initial program 50.1%

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

   2. Taylor expanded in b_2 around inf 97.9%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.5 \cdot 10^{-62}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}\\ \mathbf{elif}\;b_2 \leq 2.15 \cdot 10^{+120}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 3: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -7.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}\\ \mathbf{elif}\;b_2 \leq 3.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7.5e-67)
   (/ 1.0 (+ (* 0.5 (/ a b_2)) (* -2.0 (/ b_2 c))))
   (if (<= b_2 3.2e-42)
     (/ (- (- b_2) (sqrt (* a (- c)))) a)
     (+ (* -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 <= -7.5e-67) {
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	} else if (b_2 <= 3.2e-42) {
		tmp = (-b_2 - sqrt((a * -c))) / a;
	} 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 <= (-7.5d-67)) then
        tmp = 1.0d0 / ((0.5d0 * (a / b_2)) + ((-2.0d0) * (b_2 / c)))
    else if (b_2 <= 3.2d-42) then
        tmp = (-b_2 - sqrt((a * -c))) / a
    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 <= -7.5e-67) {
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	} else if (b_2 <= 3.2e-42) {
		tmp = (-b_2 - Math.sqrt((a * -c))) / a;
	} 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 <= -7.5e-67:
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)))
	elif b_2 <= 3.2e-42:
		tmp = (-b_2 - math.sqrt((a * -c))) / a
	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 <= -7.5e-67)
		tmp = Float64(1.0 / Float64(Float64(0.5 * Float64(a / b_2)) + Float64(-2.0 * Float64(b_2 / c))));
	elseif (b_2 <= 3.2e-42)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(a * Float64(-c)))) / a);
	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 <= -7.5e-67)
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	elseif (b_2 <= 3.2e-42)
		tmp = (-b_2 - sqrt((a * -c))) / a;
	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, -7.5e-67], N[(1.0 / N[(N[(0.5 * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 3.2e-42], N[(N[((-b$95$2) - N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $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 3 regimes

2. if b_2 < -7.5000000000000005e-67

   1. Initial program 22.4%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. prod-diff 22.2%

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

      2. *-commutative 22.2%

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

      3. fma-neg 22.2%

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

      4. prod-diff 22.2%

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

      5. *-commutative 22.2%

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

      6. fma-neg 22.2%

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

      7. associate-+l+ 22.2%

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

      8. *-commutative 22.2%

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

      9. fma-udef 22.2%

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

      10. distribute-lft-neg-in 22.2%

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

      11. *-commutative 22.2%

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

      12. distribute-rgt-neg-in 22.2%

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

      13. fma-def 22.2%

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

      14. *-commutative 22.2%

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

      15. fma-udef 22.2%

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

      16. distribute-lft-neg-in 22.2%

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

      17. *-commutative 22.2%

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

      18. distribute-rgt-neg-in 22.2%

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

      19. fma-def 22.2%

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

   3. Applied egg-rr 22.2%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}}{a} \]
   4. Step-by-step derivation

      1. *-commutative 22.2%

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

      2. count-2 22.2%

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

      3. *-commutative 22.2%

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

   5. Simplified 22.2%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}}{a} \]
   6. Step-by-step derivation

      1. clear-num 22.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}}} \]

      2. inv-pow 22.2%

         \[\leadsto \color{blue}{{\left(\frac{a}{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}\right)}^{-1}} \]

      3. +-commutative 22.2%

         \[\leadsto {\left(\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right) + \left(b_2 \cdot b_2 - c \cdot a\right)}}}\right)}^{-1} \]

      4. *-commutative 22.2%

         \[\leadsto {\left(\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(a, -c, c \cdot a\right) \cdot 2} + \left(b_2 \cdot b_2 - c \cdot a\right)}}\right)}^{-1} \]

      5. fma-def 22.2%

         \[\leadsto {\left(\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, -c, c \cdot a\right), 2, b_2 \cdot b_2 - c \cdot a\right)}}}\right)}^{-1} \]

      6. *-commutative 22.2%

         \[\leadsto {\left(\frac{a}{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(a, -c, \color{blue}{a \cdot c}\right), 2, b_2 \cdot b_2 - c \cdot a\right)}}\right)}^{-1} \]

      7. *-commutative 22.2%

         \[\leadsto {\left(\frac{a}{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(a, -c, a \cdot c\right), 2, b_2 \cdot b_2 - \color{blue}{a \cdot c}\right)}}\right)}^{-1} \]

   7. Applied egg-rr 22.2%

      \[\leadsto \color{blue}{{\left(\frac{a}{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(a, -c, a \cdot c\right), 2, b_2 \cdot b_2 - a \cdot c\right)}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 22.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(a, -c, a \cdot c\right), 2, b_2 \cdot b_2 - a \cdot c\right)}}}} \]

      2. fma-udef 22.2%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(a, -c, a \cdot c\right) \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}}} \]

      3. fma-def 22.2%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{\left(a \cdot \left(-c\right) + a \cdot c\right)} \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      4. *-commutative 22.2%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(\color{blue}{\left(-c\right) \cdot a} + a \cdot c\right) \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      5. neg-mul-1 22.2%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(\color{blue}{\left(-1 \cdot c\right)} \cdot a + a \cdot c\right) \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      6. associate-*r* 22.2%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(\color{blue}{-1 \cdot \left(c \cdot a\right)} + a \cdot c\right) \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      7. *-commutative 22.2%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(-1 \cdot \color{blue}{\left(a \cdot c\right)} + a \cdot c\right) \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      8. distribute-lft1-in 22.2%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{\left(\left(-1 + 1\right) \cdot \left(a \cdot c\right)\right)} \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      9. metadata-eval 22.2%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(\color{blue}{0} \cdot \left(a \cdot c\right)\right) \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      10. mul0-lft 22.3%

          \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{0} \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      11. metadata-eval 22.3%

          \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{0} + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      12. associate-+r- 22.3%

          \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{\left(0 + b_2 \cdot b_2\right) - a \cdot c}}}} \]

   9. Simplified 22.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(0 + b_2 \cdot b_2\right) - a \cdot c}}}} \]

   10. Taylor expanded in b_2 around -inf 84.9%

       \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}} \]

   if -7.5000000000000005e-67 < b_2 < 3.20000000000000025e-42

   1. Initial program 74.3%

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

   2. Taylor expanded in b_2 around 0 64.1%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{-1 \cdot \left(c \cdot a\right)}}}{a} \]
   3. Step-by-step derivation

      1. mul-1-neg 64.1%

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

      2. distribute-rgt-neg-out 64.1%

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

   4. Simplified 64.1%

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

   if 3.20000000000000025e-42 < b_2

   1. Initial program 70.2%

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

   2. Taylor expanded in b_2 around inf 91.6%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -7.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}\\ \mathbf{elif}\;b_2 \leq 3.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 4: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -4.6 \cdot 10^{-69}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}\\ \mathbf{elif}\;b_2 \leq 5.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{-\sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.6e-69)
   (/ 1.0 (+ (* 0.5 (/ a b_2)) (* -2.0 (/ b_2 c))))
   (if (<= b_2 5.5e-42)
     (/ (- (sqrt (* a (- c)))) a)
     (+ (* -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 <= -4.6e-69) {
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	} else if (b_2 <= 5.5e-42) {
		tmp = -sqrt((a * -c)) / a;
	} 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 <= (-4.6d-69)) then
        tmp = 1.0d0 / ((0.5d0 * (a / b_2)) + ((-2.0d0) * (b_2 / c)))
    else if (b_2 <= 5.5d-42) then
        tmp = -sqrt((a * -c)) / a
    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 <= -4.6e-69) {
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	} else if (b_2 <= 5.5e-42) {
		tmp = -Math.sqrt((a * -c)) / a;
	} 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 <= -4.6e-69:
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)))
	elif b_2 <= 5.5e-42:
		tmp = -math.sqrt((a * -c)) / a
	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 <= -4.6e-69)
		tmp = Float64(1.0 / Float64(Float64(0.5 * Float64(a / b_2)) + Float64(-2.0 * Float64(b_2 / c))));
	elseif (b_2 <= 5.5e-42)
		tmp = Float64(Float64(-sqrt(Float64(a * Float64(-c)))) / a);
	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 <= -4.6e-69)
		tmp = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	elseif (b_2 <= 5.5e-42)
		tmp = -sqrt((a * -c)) / a;
	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, -4.6e-69], N[(1.0 / N[(N[(0.5 * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 5.5e-42], N[((-N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision]) / a), $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 3 regimes

2. if b_2 < -4.6000000000000001e-69

   1. Initial program 22.4%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. prod-diff 22.2%

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

      2. *-commutative 22.2%

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

      3. fma-neg 22.2%

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

      4. prod-diff 22.2%

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

      5. *-commutative 22.2%

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

      6. fma-neg 22.2%

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

      7. associate-+l+ 22.2%

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

      8. *-commutative 22.2%

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

      9. fma-udef 22.2%

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

      10. distribute-lft-neg-in 22.2%

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

      11. *-commutative 22.2%

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

      12. distribute-rgt-neg-in 22.2%

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

      13. fma-def 22.2%

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

      14. *-commutative 22.2%

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

      15. fma-udef 22.2%

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

      16. distribute-lft-neg-in 22.2%

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

      17. *-commutative 22.2%

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

      18. distribute-rgt-neg-in 22.2%

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

      19. fma-def 22.2%

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

   3. Applied egg-rr 22.2%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}}{a} \]
   4. Step-by-step derivation

      1. *-commutative 22.2%

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

      2. count-2 22.2%

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

      3. *-commutative 22.2%

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

   5. Simplified 22.2%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}}{a} \]
   6. Step-by-step derivation

      1. clear-num 22.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}}} \]

      2. inv-pow 22.2%

         \[\leadsto \color{blue}{{\left(\frac{a}{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}\right)}^{-1}} \]

      3. +-commutative 22.2%

         \[\leadsto {\left(\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right) + \left(b_2 \cdot b_2 - c \cdot a\right)}}}\right)}^{-1} \]

      4. *-commutative 22.2%

         \[\leadsto {\left(\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(a, -c, c \cdot a\right) \cdot 2} + \left(b_2 \cdot b_2 - c \cdot a\right)}}\right)}^{-1} \]

      5. fma-def 22.2%

         \[\leadsto {\left(\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, -c, c \cdot a\right), 2, b_2 \cdot b_2 - c \cdot a\right)}}}\right)}^{-1} \]

      6. *-commutative 22.2%

         \[\leadsto {\left(\frac{a}{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(a, -c, \color{blue}{a \cdot c}\right), 2, b_2 \cdot b_2 - c \cdot a\right)}}\right)}^{-1} \]

      7. *-commutative 22.2%

         \[\leadsto {\left(\frac{a}{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(a, -c, a \cdot c\right), 2, b_2 \cdot b_2 - \color{blue}{a \cdot c}\right)}}\right)}^{-1} \]

   7. Applied egg-rr 22.2%

      \[\leadsto \color{blue}{{\left(\frac{a}{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(a, -c, a \cdot c\right), 2, b_2 \cdot b_2 - a \cdot c\right)}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 22.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(a, -c, a \cdot c\right), 2, b_2 \cdot b_2 - a \cdot c\right)}}}} \]

      2. fma-udef 22.2%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(a, -c, a \cdot c\right) \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}}} \]

      3. fma-def 22.2%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{\left(a \cdot \left(-c\right) + a \cdot c\right)} \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      4. *-commutative 22.2%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(\color{blue}{\left(-c\right) \cdot a} + a \cdot c\right) \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      5. neg-mul-1 22.2%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(\color{blue}{\left(-1 \cdot c\right)} \cdot a + a \cdot c\right) \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      6. associate-*r* 22.2%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(\color{blue}{-1 \cdot \left(c \cdot a\right)} + a \cdot c\right) \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      7. *-commutative 22.2%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(-1 \cdot \color{blue}{\left(a \cdot c\right)} + a \cdot c\right) \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      8. distribute-lft1-in 22.2%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{\left(\left(-1 + 1\right) \cdot \left(a \cdot c\right)\right)} \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      9. metadata-eval 22.2%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(\color{blue}{0} \cdot \left(a \cdot c\right)\right) \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      10. mul0-lft 22.3%

          \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{0} \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      11. metadata-eval 22.3%

          \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{0} + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      12. associate-+r- 22.3%

          \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{\left(0 + b_2 \cdot b_2\right) - a \cdot c}}}} \]

   9. Simplified 22.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(0 + b_2 \cdot b_2\right) - a \cdot c}}}} \]

   10. Taylor expanded in b_2 around -inf 84.9%

       \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}} \]

   if -4.6000000000000001e-69 < b_2 < 5.5e-42

   1. Initial program 74.3%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. prod-diff 73.7%

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

      2. *-commutative 73.7%

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

      3. fma-neg 73.7%

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

      4. prod-diff 73.7%

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

      5. *-commutative 73.7%

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

      6. fma-neg 73.7%

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

      7. associate-+l+ 73.7%

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

      8. *-commutative 73.7%

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

      9. fma-udef 73.7%

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

      10. distribute-lft-neg-in 73.7%

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

      11. *-commutative 73.7%

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

      12. distribute-rgt-neg-in 73.7%

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

      13. fma-def 73.7%

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

      14. *-commutative 73.7%

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

      15. fma-udef 73.7%

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

      16. distribute-lft-neg-in 73.7%

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

      17. *-commutative 73.7%

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

      18. distribute-rgt-neg-in 73.7%

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

      19. fma-def 73.7%

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

   3. Applied egg-rr 73.7%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}}{a} \]
   4. Step-by-step derivation

      1. *-commutative 73.7%

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

      2. count-2 73.7%

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

      3. *-commutative 73.7%

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

   5. Simplified 73.7%

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

   6. Taylor expanded in b_2 around 0 62.0%

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

      1. mul-1-neg 62.0%

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

      2. distribute-lft1-in 62.0%

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

      3. metadata-eval 62.0%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(\color{blue}{0} \cdot \left(c \cdot a\right)\right) - c \cdot a}}{a} \]

      4. *-commutative 62.0%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(0 \cdot \color{blue}{\left(a \cdot c\right)}\right) - c \cdot a}}{a} \]

      5. mul0-lft 62.5%

         \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{0} - c \cdot a}}{a} \]

      6. metadata-eval 62.5%

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

      7. *-commutative 62.5%

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

      8. neg-sub0 62.5%

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

      9. distribute-rgt-neg-out 62.5%

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

   8. Simplified 62.5%

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

   if 5.5e-42 < b_2

   1. Initial program 70.2%

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

   2. Taylor expanded in b_2 around inf 91.6%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4.6 \cdot 10^{-69}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}\\ \mathbf{elif}\;b_2 \leq 5.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{-\sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 5: 67.6% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e-310)
   (/ 1.0 (+ (* 0.5 (/ a b_2)) (* -2.0 (/ b_2 c))))
   (+ (* -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 = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	} 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 = 1.0d0 / ((0.5d0 * (a / b_2)) + ((-2.0d0) * (b_2 / c)))
    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 = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	} 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 = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)))
	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(1.0 / Float64(Float64(0.5 * Float64(a / b_2)) + Float64(-2.0 * Float64(b_2 / c))));
	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 = 1.0 / ((0.5 * (a / b_2)) + (-2.0 * (b_2 / c)));
	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[(1.0 / N[(N[(0.5 * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 33.9%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. prod-diff 33.7%

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

      2. *-commutative 33.7%

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

      3. fma-neg 33.7%

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

      4. prod-diff 33.7%

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

      5. *-commutative 33.7%

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

      6. fma-neg 33.7%

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

      7. associate-+l+ 33.7%

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

      8. *-commutative 33.7%

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

      9. fma-udef 33.7%

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

      10. distribute-lft-neg-in 33.7%

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

      11. *-commutative 33.7%

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

      12. distribute-rgt-neg-in 33.7%

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

      13. fma-def 33.7%

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

      14. *-commutative 33.7%

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

      15. fma-udef 33.7%

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

      16. distribute-lft-neg-in 33.7%

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

      17. *-commutative 33.7%

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

      18. distribute-rgt-neg-in 33.7%

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

      19. fma-def 33.7%

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

   3. Applied egg-rr 33.7%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}}{a} \]
   4. Step-by-step derivation

      1. *-commutative 33.7%

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

      2. count-2 33.7%

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

      3. *-commutative 33.7%

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

   5. Simplified 33.7%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}}{a} \]
   6. Step-by-step derivation

      1. clear-num 33.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}}} \]

      2. inv-pow 33.7%

         \[\leadsto \color{blue}{{\left(\frac{a}{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}\right)}^{-1}} \]

      3. +-commutative 33.7%

         \[\leadsto {\left(\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right) + \left(b_2 \cdot b_2 - c \cdot a\right)}}}\right)}^{-1} \]

      4. *-commutative 33.7%

         \[\leadsto {\left(\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(a, -c, c \cdot a\right) \cdot 2} + \left(b_2 \cdot b_2 - c \cdot a\right)}}\right)}^{-1} \]

      5. fma-def 33.7%

         \[\leadsto {\left(\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, -c, c \cdot a\right), 2, b_2 \cdot b_2 - c \cdot a\right)}}}\right)}^{-1} \]

      6. *-commutative 33.7%

         \[\leadsto {\left(\frac{a}{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(a, -c, \color{blue}{a \cdot c}\right), 2, b_2 \cdot b_2 - c \cdot a\right)}}\right)}^{-1} \]

      7. *-commutative 33.7%

         \[\leadsto {\left(\frac{a}{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(a, -c, a \cdot c\right), 2, b_2 \cdot b_2 - \color{blue}{a \cdot c}\right)}}\right)}^{-1} \]

   7. Applied egg-rr 33.7%

      \[\leadsto \color{blue}{{\left(\frac{a}{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(a, -c, a \cdot c\right), 2, b_2 \cdot b_2 - a \cdot c\right)}}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 33.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(a, -c, a \cdot c\right), 2, b_2 \cdot b_2 - a \cdot c\right)}}}} \]

      2. fma-udef 33.7%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(a, -c, a \cdot c\right) \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}}} \]

      3. fma-def 33.6%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{\left(a \cdot \left(-c\right) + a \cdot c\right)} \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      4. *-commutative 33.6%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(\color{blue}{\left(-c\right) \cdot a} + a \cdot c\right) \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      5. neg-mul-1 33.6%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(\color{blue}{\left(-1 \cdot c\right)} \cdot a + a \cdot c\right) \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      6. associate-*r* 33.6%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(\color{blue}{-1 \cdot \left(c \cdot a\right)} + a \cdot c\right) \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      7. *-commutative 33.6%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(-1 \cdot \color{blue}{\left(a \cdot c\right)} + a \cdot c\right) \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      8. distribute-lft1-in 33.6%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{\left(\left(-1 + 1\right) \cdot \left(a \cdot c\right)\right)} \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      9. metadata-eval 33.6%

         \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(\color{blue}{0} \cdot \left(a \cdot c\right)\right) \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      10. mul0-lft 33.9%

          \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{0} \cdot 2 + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      11. metadata-eval 33.9%

          \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{0} + \left(b_2 \cdot b_2 - a \cdot c\right)}}} \]

      12. associate-+r- 33.9%

          \[\leadsto \frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\color{blue}{\left(0 + b_2 \cdot b_2\right) - a \cdot c}}}} \]

   9. Simplified 33.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{\left(0 + b_2 \cdot b_2\right) - a \cdot c}}}} \]

   10. Taylor expanded in b_2 around -inf 68.0%

       \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}} \]

   if -3.999999999999988e-310 < b_2

   1. Initial program 73.0%

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

   2. Taylor expanded in b_2 around inf 69.1%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{a}{b_2} + -2 \cdot \frac{b_2}{c}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 6: 67.9% accurate, 8.6× 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 33.9%

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

   2. Taylor expanded in b_2 around -inf 66.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   3. Step-by-step derivation

      1. associate-*r/ 66.9%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   4. Simplified 66.9%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   if -3.999999999999988e-310 < b_2

   1. Initial program 73.0%

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

   2. Taylor expanded in b_2 around inf 69.1%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 7: 47.7% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \end{array} \]

FPCore

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

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 = -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 <= (-4d-310)) then
        tmp = (c * (-0.5d0)) / b_2
    else
        tmp = -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 <= -4e-310) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4e-310:
		tmp = (c * -0.5) / b_2
	else:
		tmp = -b_2 / a
	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(-b_2) / a);
	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 = -b_2 / a;
	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[((-b$95$2) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.999999999999988e-310

   1. Initial program 33.9%

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

   2. Taylor expanded in b_2 around -inf 66.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   3. Step-by-step derivation

      1. associate-*r/ 66.9%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   4. Simplified 66.9%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   if -3.999999999999988e-310 < b_2

   1. Initial program 73.0%

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

   2. Taylor expanded in b_2 around 0 38.2%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{-1 \cdot \left(c \cdot a\right)}}}{a} \]
   3. Step-by-step derivation

      1. mul-1-neg 38.2%

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

      2. distribute-rgt-neg-out 38.2%

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

   4. Simplified 38.2%

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

   5. Taylor expanded in b_2 around inf 26.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{b_2}{a}} \]
   6. Step-by-step derivation

      1. associate-*r/ 26.0%

         \[\leadsto \color{blue}{\frac{-1 \cdot b_2}{a}} \]

      2. mul-1-neg 26.0%

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

   7. Simplified 26.0%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 8: 67.7% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -3 \cdot 10^{-309}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3e-309) (/ (* c -0.5) b_2) (/ (* b_2 -2.0) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3e-309) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = (b_2 * -2.0) / 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 <= (-3d-309)) then
        tmp = (c * (-0.5d0)) / b_2
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3e-309) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3e-309:
		tmp = (c * -0.5) / b_2
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3e-309)
		tmp = Float64(Float64(c * -0.5) / b_2);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3e-309)
		tmp = (c * -0.5) / b_2;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3e-309], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.000000000000001e-309

   1. Initial program 33.9%

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

   2. Taylor expanded in b_2 around -inf 66.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   3. Step-by-step derivation

      1. associate-*r/ 66.9%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   4. Simplified 66.9%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   if -3.000000000000001e-309 < b_2

   1. Initial program 73.0%

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

   2. Taylor expanded in b_2 around inf 68.6%

      \[\leadsto \frac{\color{blue}{-2 \cdot b_2}}{a} \]
   3. Step-by-step derivation

      1. *-commutative 68.6%

         \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]

   4. Simplified 68.6%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -3 \cdot 10^{-309}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 9: 15.3% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ \frac{-b_2}{a} \end{array} \]

FPCore

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

C

double code(double a, double b_2, double c) {
	return -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 = -b_2 / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.0%

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

2. Taylor expanded in b_2 around 0 32.8%

   \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{-1 \cdot \left(c \cdot a\right)}}}{a} \]
3. Step-by-step derivation

   1. mul-1-neg 32.8%

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

   2. distribute-rgt-neg-out 32.8%

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

4. Simplified 32.8%

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

5. Taylor expanded in b_2 around inf 15.3%

   \[\leadsto \color{blue}{-1 \cdot \frac{b_2}{a}} \]
6. Step-by-step derivation

   1. associate-*r/ 15.3%

      \[\leadsto \color{blue}{\frac{-1 \cdot b_2}{a}} \]

   2. mul-1-neg 15.3%

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

7. Simplified 15.3%

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

8. Final simplification 15.3%

   \[\leadsto \frac{-b_2}{a} \]

Reproduce

herbie shell --seed 2023192 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))