quad2m (problem 3.2.1, negative)

Percentage Accurate: 51.7% → 85.3%

Time: 13.4s

Alternatives: 9

Speedup: 11.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1}{\frac{a \cdot -0.5}{b\_2} + \frac{b\_2 \cdot 2}{c}}\\ \mathbf{elif}\;b\_2 \leq 1.02 \cdot 10^{+120}:\\ \;\;\;\;\frac{\left(0 - b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.2e-47)
   (/ -1.0 (+ (/ (* a -0.5) b_2) (/ (* b_2 2.0) c)))
   (if (<= b_2 1.02e+120)
     (/ (- (- 0.0 b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
     (/ (* b_2 -2.0) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.2e-47) {
		tmp = -1.0 / (((a * -0.5) / b_2) + ((b_2 * 2.0) / c));
	} else if (b_2 <= 1.02e+120) {
		tmp = ((0.0 - b_2) - sqrt(((b_2 * b_2) - (a * c)))) / a;
	} 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 <= (-3.2d-47)) then
        tmp = (-1.0d0) / (((a * (-0.5d0)) / b_2) + ((b_2 * 2.0d0) / c))
    else if (b_2 <= 1.02d+120) then
        tmp = ((0.0d0 - b_2) - sqrt(((b_2 * b_2) - (a * c)))) / a
    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 <= -3.2e-47) {
		tmp = -1.0 / (((a * -0.5) / b_2) + ((b_2 * 2.0) / c));
	} else if (b_2 <= 1.02e+120) {
		tmp = ((0.0 - b_2) - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.2e-47:
		tmp = -1.0 / (((a * -0.5) / b_2) + ((b_2 * 2.0) / c))
	elif b_2 <= 1.02e+120:
		tmp = ((0.0 - b_2) - math.sqrt(((b_2 * b_2) - (a * c)))) / a
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.2e-47)
		tmp = Float64(-1.0 / Float64(Float64(Float64(a * -0.5) / b_2) + Float64(Float64(b_2 * 2.0) / c)));
	elseif (b_2 <= 1.02e+120)
		tmp = Float64(Float64(Float64(0.0 - b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	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 <= -3.2e-47)
		tmp = -1.0 / (((a * -0.5) / b_2) + ((b_2 * 2.0) / c));
	elseif (b_2 <= 1.02e+120)
		tmp = ((0.0 - b_2) - sqrt(((b_2 * b_2) - (a * c)))) / a;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.2e-47], N[(-1.0 / N[(N[(N[(a * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision] + N[(N[(b$95$2 * 2.0), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.02e+120], N[(N[(N[(0.0 - b$95$2), $MachinePrecision] - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.1999999999999999e-47

   1. Initial program 11.4%

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

   4. Applied egg-rr 11.5%

      \[\leadsto \color{blue}{\frac{-1}{\frac{a}{b\_2 + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}}} \]

   5. Taylor expanded in b_2 around -inf

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(-1 \cdot \left(b\_2 \cdot \left(\frac{1}{2} \cdot \frac{a}{{b\_2}^{2}} - 2 \cdot \frac{1}{c}\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(b\_2 \cdot \left(\frac{1}{2} \cdot \frac{a}{{b\_2}^{2}} - 2 \cdot \frac{1}{c}\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot \frac{a}{{b\_2}^{2}} - 2 \cdot \frac{1}{c}\right) \cdot b\_2\right)\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(\frac{1}{2} \cdot \frac{a}{{b\_2}^{2}} - 2 \cdot \frac{1}{c}\right) \cdot \color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)}\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(\frac{1}{2} \cdot \frac{a}{{b\_2}^{2}} - 2 \cdot \frac{1}{c}\right) \cdot \left(-1 \cdot \color{blue}{b\_2}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \frac{a}{{b\_2}^{2}} - 2 \cdot \frac{1}{c}\right), \color{blue}{\left(-1 \cdot b\_2\right)}\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \frac{a}{{b\_2}^{2}} + \left(\mathsf{neg}\left(2 \cdot \frac{1}{c}\right)\right)\right), \left(\color{blue}{-1} \cdot b\_2\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{a}{{b\_2}^{2}}\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{c}\right)\right)\right), \left(\color{blue}{-1} \cdot b\_2\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot a}{{b\_2}^{2}}\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{c}\right)\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot a\right), \left({b\_2}^{2}\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{c}\right)\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(a \cdot \frac{1}{2}\right), \left({b\_2}^{2}\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{c}\right)\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \left({b\_2}^{2}\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{c}\right)\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \left(b\_2 \cdot b\_2\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{c}\right)\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{c}\right)\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right), \left(\mathsf{neg}\left(\frac{2 \cdot 1}{c}\right)\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right), \left(\mathsf{neg}\left(\frac{2}{c}\right)\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      16. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right), \left(\frac{\mathsf{neg}\left(2\right)}{c}\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right), \left(\frac{-2}{c}\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right), \mathsf{/.f64}\left(-2, c\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right), \mathsf{/.f64}\left(-2, c\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right)\right) \]

      20. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right), \mathsf{/.f64}\left(-2, c\right)\right), \left(0 - \color{blue}{b\_2}\right)\right)\right) \]

      21. --lowering--.f64 92.2%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right), \mathsf{/.f64}\left(-2, c\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{b\_2}\right)\right)\right) \]

   7. Simplified 92.2%

      \[\leadsto \frac{-1}{\color{blue}{\left(\frac{a \cdot 0.5}{b\_2 \cdot b\_2} + \frac{-2}{c}\right) \cdot \left(0 - b\_2\right)}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\frac{-1}{2} \cdot \frac{a}{b\_2} + 2 \cdot \frac{b\_2}{c}\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{a}{b\_2}\right), \color{blue}{\left(2 \cdot \frac{b\_2}{c}\right)}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2} \cdot a}{b\_2}\right), \left(\color{blue}{2} \cdot \frac{b\_2}{c}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot a\right), b\_2\right), \left(\color{blue}{2} \cdot \frac{b\_2}{c}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(a \cdot \frac{-1}{2}\right), b\_2\right), \left(2 \cdot \frac{b\_2}{c}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), b\_2\right), \left(2 \cdot \frac{b\_2}{c}\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), b\_2\right), \left(\frac{2 \cdot b\_2}{\color{blue}{c}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), b\_2\right), \mathsf{/.f64}\left(\left(2 \cdot b\_2\right), \color{blue}{c}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), b\_2\right), \mathsf{/.f64}\left(\left(b\_2 \cdot 2\right), c\right)\right)\right) \]

      9. *-lowering-*.f64 92.3%

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), b\_2\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b\_2, 2\right), c\right)\right)\right) \]

   10. Simplified 92.3%

       \[\leadsto \frac{-1}{\color{blue}{\frac{a \cdot -0.5}{b\_2} + \frac{b\_2 \cdot 2}{c}}} \]

   if -3.1999999999999999e-47 < b_2 < 1.01999999999999997e120

   1. Initial program 80.0%

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

   if 1.01999999999999997e120 < b_2

   1. Initial program 58.5%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-2 \cdot b\_2\right)}, a\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b\_2 \cdot -2\right), a\right) \]

      2. *-lowering-*.f64 98.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b\_2, -2\right), a\right) \]

   5. Simplified 98.5%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1}{\frac{a \cdot -0.5}{b\_2} + \frac{b\_2 \cdot 2}{c}}\\ \mathbf{elif}\;b\_2 \leq 1.02 \cdot 10^{+120}:\\ \;\;\;\;\frac{\left(0 - b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.1 \cdot 10^{-44}:\\ \;\;\;\;\frac{-1}{\frac{a \cdot -0.5}{b\_2} + \frac{b\_2 \cdot 2}{c}}\\ \mathbf{elif}\;b\_2 \leq 10^{-48}:\\ \;\;\;\;\frac{\left(0 - b\_2\right) - \sqrt{0 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a} + \frac{c \cdot 0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.1e-44)
   (/ -1.0 (+ (/ (* a -0.5) b_2) (/ (* b_2 2.0) c)))
   (if (<= b_2 1e-48)
     (/ (- (- 0.0 b_2) (sqrt (- 0.0 (* a c)))) a)
     (+ (/ (* b_2 -2.0) a) (/ (* c 0.5) b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.1e-44) {
		tmp = -1.0 / (((a * -0.5) / b_2) + ((b_2 * 2.0) / c));
	} else if (b_2 <= 1e-48) {
		tmp = ((0.0 - b_2) - sqrt((0.0 - (a * c)))) / a;
	} else {
		tmp = ((b_2 * -2.0) / a) + ((c * 0.5) / 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 <= (-1.1d-44)) then
        tmp = (-1.0d0) / (((a * (-0.5d0)) / b_2) + ((b_2 * 2.0d0) / c))
    else if (b_2 <= 1d-48) then
        tmp = ((0.0d0 - b_2) - sqrt((0.0d0 - (a * c)))) / a
    else
        tmp = ((b_2 * (-2.0d0)) / a) + ((c * 0.5d0) / 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 <= -1.1e-44) {
		tmp = -1.0 / (((a * -0.5) / b_2) + ((b_2 * 2.0) / c));
	} else if (b_2 <= 1e-48) {
		tmp = ((0.0 - b_2) - Math.sqrt((0.0 - (a * c)))) / a;
	} else {
		tmp = ((b_2 * -2.0) / a) + ((c * 0.5) / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.1e-44:
		tmp = -1.0 / (((a * -0.5) / b_2) + ((b_2 * 2.0) / c))
	elif b_2 <= 1e-48:
		tmp = ((0.0 - b_2) - math.sqrt((0.0 - (a * c)))) / a
	else:
		tmp = ((b_2 * -2.0) / a) + ((c * 0.5) / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.1e-44)
		tmp = Float64(-1.0 / Float64(Float64(Float64(a * -0.5) / b_2) + Float64(Float64(b_2 * 2.0) / c)));
	elseif (b_2 <= 1e-48)
		tmp = Float64(Float64(Float64(0.0 - b_2) - sqrt(Float64(0.0 - Float64(a * c)))) / a);
	else
		tmp = Float64(Float64(Float64(b_2 * -2.0) / a) + Float64(Float64(c * 0.5) / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.1e-44)
		tmp = -1.0 / (((a * -0.5) / b_2) + ((b_2 * 2.0) / c));
	elseif (b_2 <= 1e-48)
		tmp = ((0.0 - b_2) - sqrt((0.0 - (a * c)))) / a;
	else
		tmp = ((b_2 * -2.0) / a) + ((c * 0.5) / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.1e-44], N[(-1.0 / N[(N[(N[(a * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision] + N[(N[(b$95$2 * 2.0), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1e-48], N[(N[(N[(0.0 - b$95$2), $MachinePrecision] - N[Sqrt[N[(0.0 - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision] + N[(N[(c * 0.5), $MachinePrecision] / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.10000000000000006e-44

   1. Initial program 11.4%

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

   4. Applied egg-rr 11.5%

      \[\leadsto \color{blue}{\frac{-1}{\frac{a}{b\_2 + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}}} \]

   5. Taylor expanded in b_2 around -inf

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(-1 \cdot \left(b\_2 \cdot \left(\frac{1}{2} \cdot \frac{a}{{b\_2}^{2}} - 2 \cdot \frac{1}{c}\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(b\_2 \cdot \left(\frac{1}{2} \cdot \frac{a}{{b\_2}^{2}} - 2 \cdot \frac{1}{c}\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot \frac{a}{{b\_2}^{2}} - 2 \cdot \frac{1}{c}\right) \cdot b\_2\right)\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(\frac{1}{2} \cdot \frac{a}{{b\_2}^{2}} - 2 \cdot \frac{1}{c}\right) \cdot \color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)}\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(\frac{1}{2} \cdot \frac{a}{{b\_2}^{2}} - 2 \cdot \frac{1}{c}\right) \cdot \left(-1 \cdot \color{blue}{b\_2}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \frac{a}{{b\_2}^{2}} - 2 \cdot \frac{1}{c}\right), \color{blue}{\left(-1 \cdot b\_2\right)}\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \frac{a}{{b\_2}^{2}} + \left(\mathsf{neg}\left(2 \cdot \frac{1}{c}\right)\right)\right), \left(\color{blue}{-1} \cdot b\_2\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{a}{{b\_2}^{2}}\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{c}\right)\right)\right), \left(\color{blue}{-1} \cdot b\_2\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot a}{{b\_2}^{2}}\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{c}\right)\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot a\right), \left({b\_2}^{2}\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{c}\right)\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(a \cdot \frac{1}{2}\right), \left({b\_2}^{2}\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{c}\right)\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \left({b\_2}^{2}\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{c}\right)\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \left(b\_2 \cdot b\_2\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{c}\right)\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{c}\right)\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right), \left(\mathsf{neg}\left(\frac{2 \cdot 1}{c}\right)\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right), \left(\mathsf{neg}\left(\frac{2}{c}\right)\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      16. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right), \left(\frac{\mathsf{neg}\left(2\right)}{c}\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right), \left(\frac{-2}{c}\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right), \mathsf{/.f64}\left(-2, c\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right), \mathsf{/.f64}\left(-2, c\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right)\right) \]

      20. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right), \mathsf{/.f64}\left(-2, c\right)\right), \left(0 - \color{blue}{b\_2}\right)\right)\right) \]

      21. --lowering--.f64 92.2%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right), \mathsf{/.f64}\left(-2, c\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{b\_2}\right)\right)\right) \]

   7. Simplified 92.2%

      \[\leadsto \frac{-1}{\color{blue}{\left(\frac{a \cdot 0.5}{b\_2 \cdot b\_2} + \frac{-2}{c}\right) \cdot \left(0 - b\_2\right)}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\frac{-1}{2} \cdot \frac{a}{b\_2} + 2 \cdot \frac{b\_2}{c}\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{a}{b\_2}\right), \color{blue}{\left(2 \cdot \frac{b\_2}{c}\right)}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2} \cdot a}{b\_2}\right), \left(\color{blue}{2} \cdot \frac{b\_2}{c}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot a\right), b\_2\right), \left(\color{blue}{2} \cdot \frac{b\_2}{c}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(a \cdot \frac{-1}{2}\right), b\_2\right), \left(2 \cdot \frac{b\_2}{c}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), b\_2\right), \left(2 \cdot \frac{b\_2}{c}\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), b\_2\right), \left(\frac{2 \cdot b\_2}{\color{blue}{c}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), b\_2\right), \mathsf{/.f64}\left(\left(2 \cdot b\_2\right), \color{blue}{c}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), b\_2\right), \mathsf{/.f64}\left(\left(b\_2 \cdot 2\right), c\right)\right)\right) \]

      9. *-lowering-*.f64 92.3%

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), b\_2\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b\_2, 2\right), c\right)\right)\right) \]

   10. Simplified 92.3%

       \[\leadsto \frac{-1}{\color{blue}{\frac{a \cdot -0.5}{b\_2} + \frac{b\_2 \cdot 2}{c}}} \]

   if -1.10000000000000006e-44 < b_2 < 9.9999999999999997e-49

   1. Initial program 72.5%

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

   3. Taylor expanded in b_2 around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot c\right)\right)}\right)\right), a\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(a \cdot c\right)\right)\right)\right), a\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\left(0 - a \cdot c\right)\right)\right), a\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot c\right)\right)\right)\right), a\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \left(c \cdot a\right)\right)\right)\right), a\right) \]

      5. *-lowering-*.f64 68.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(c, a\right)\right)\right)\right), a\right) \]

   5. Simplified 68.9%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{0 - c \cdot a}}}{a} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(c \cdot a\right)\right)\right)\right), a\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(a \cdot c\right)\right)\right)\right), a\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\mathsf{neg.f64}\left(\left(a \cdot c\right)\right)\right)\right), a\right) \]

      4. *-lowering-*.f64 68.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(a, c\right)\right)\right)\right), a\right) \]

   7. Applied egg-rr 68.9%

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

   if 9.9999999999999997e-49 < b_2

   1. Initial program 72.4%

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

   4. Applied egg-rr 72.3%

      \[\leadsto \color{blue}{\frac{-1}{\frac{a}{b\_2 + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}}} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. distribute-lft-in N/A

         \[\leadsto \frac{-1}{a} \cdot b\_2 + \color{blue}{\frac{-1}{a} \cdot \sqrt{b\_2 \cdot b\_2 - a \cdot c}} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\frac{-1}{a}\right), \color{blue}{b\_2}, \left(\frac{-1}{a} \cdot \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(-1, a\right), b\_2, \left(\frac{-1}{a} \cdot \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(-1, a\right), b\_2, \mathsf{*.f64}\left(\left(\frac{-1}{a}\right), \left(\sqrt{b\_2 \cdot b\_2 - a \cdot c}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(-1, a\right), b\_2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(-1, a\right), \left(\sqrt{b\_2 \cdot b\_2 - a \cdot c}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(-1, a\right), b\_2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(-1, a\right), \mathsf{sqrt.f64}\left(\left(b\_2 \cdot b\_2 - a \cdot c\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(-1, a\right), b\_2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(-1, a\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(b\_2 \cdot b\_2\right), \left(a \cdot c\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(-1, a\right), b\_2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(-1, a\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \left(a \cdot c\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 72.3%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(-1, a\right), b\_2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(-1, a\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \mathsf{*.f64}\left(a, c\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 72.3%

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

   7. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + \frac{1}{2} \cdot \frac{c}{b\_2}} \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(-2 \cdot \frac{b\_2}{a}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{c}{b\_2}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{-2 \cdot b\_2}{a}\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{c}{b\_2}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot b\_2\right), a\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{c}{b\_2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, b\_2\right), a\right), \left(\frac{1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, b\_2\right), a\right), \left(\frac{\frac{1}{2} \cdot c}{\color{blue}{b\_2}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, b\_2\right), a\right), \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot c\right), \color{blue}{b\_2}\right)\right) \]

      7. *-lowering-*.f64 91.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, b\_2\right), a\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, c\right), b\_2\right)\right) \]

   9. Simplified 91.4%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.1 \cdot 10^{-44}:\\ \;\;\;\;\frac{-1}{\frac{a \cdot -0.5}{b\_2} + \frac{b\_2 \cdot 2}{c}}\\ \mathbf{elif}\;b\_2 \leq 10^{-48}:\\ \;\;\;\;\frac{\left(0 - b\_2\right) - \sqrt{0 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a} + \frac{c \cdot 0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.7% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-1}{\frac{a \cdot -0.5}{b\_2} + \frac{b\_2 \cdot 2}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a} + \frac{c \cdot 0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310)
   (/ -1.0 (+ (/ (* a -0.5) b_2) (/ (* b_2 2.0) c)))
   (+ (/ (* b_2 -2.0) a) (/ (* c 0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = -1.0 / (((a * -0.5) / b_2) + ((b_2 * 2.0) / c));
	} else {
		tmp = ((b_2 * -2.0) / a) + ((c * 0.5) / 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 <= (-5d-310)) then
        tmp = (-1.0d0) / (((a * (-0.5d0)) / b_2) + ((b_2 * 2.0d0) / c))
    else
        tmp = ((b_2 * (-2.0d0)) / a) + ((c * 0.5d0) / 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 <= -5e-310) {
		tmp = -1.0 / (((a * -0.5) / b_2) + ((b_2 * 2.0) / c));
	} else {
		tmp = ((b_2 * -2.0) / a) + ((c * 0.5) / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = -1.0 / (((a * -0.5) / b_2) + ((b_2 * 2.0) / c))
	else:
		tmp = ((b_2 * -2.0) / a) + ((c * 0.5) / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(-1.0 / Float64(Float64(Float64(a * -0.5) / b_2) + Float64(Float64(b_2 * 2.0) / c)));
	else
		tmp = Float64(Float64(Float64(b_2 * -2.0) / a) + Float64(Float64(c * 0.5) / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = -1.0 / (((a * -0.5) / b_2) + ((b_2 * 2.0) / c));
	else
		tmp = ((b_2 * -2.0) / a) + ((c * 0.5) / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[(-1.0 / N[(N[(N[(a * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision] + N[(N[(b$95$2 * 2.0), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision] + N[(N[(c * 0.5), $MachinePrecision] / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 32.4%

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

   4. Applied egg-rr 32.3%

      \[\leadsto \color{blue}{\frac{-1}{\frac{a}{b\_2 + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}}} \]

   5. Taylor expanded in b_2 around -inf

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(-1 \cdot \left(b\_2 \cdot \left(\frac{1}{2} \cdot \frac{a}{{b\_2}^{2}} - 2 \cdot \frac{1}{c}\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(b\_2 \cdot \left(\frac{1}{2} \cdot \frac{a}{{b\_2}^{2}} - 2 \cdot \frac{1}{c}\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot \frac{a}{{b\_2}^{2}} - 2 \cdot \frac{1}{c}\right) \cdot b\_2\right)\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(\frac{1}{2} \cdot \frac{a}{{b\_2}^{2}} - 2 \cdot \frac{1}{c}\right) \cdot \color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)}\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(\frac{1}{2} \cdot \frac{a}{{b\_2}^{2}} - 2 \cdot \frac{1}{c}\right) \cdot \left(-1 \cdot \color{blue}{b\_2}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \frac{a}{{b\_2}^{2}} - 2 \cdot \frac{1}{c}\right), \color{blue}{\left(-1 \cdot b\_2\right)}\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \frac{a}{{b\_2}^{2}} + \left(\mathsf{neg}\left(2 \cdot \frac{1}{c}\right)\right)\right), \left(\color{blue}{-1} \cdot b\_2\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{a}{{b\_2}^{2}}\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{c}\right)\right)\right), \left(\color{blue}{-1} \cdot b\_2\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot a}{{b\_2}^{2}}\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{c}\right)\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot a\right), \left({b\_2}^{2}\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{c}\right)\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(a \cdot \frac{1}{2}\right), \left({b\_2}^{2}\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{c}\right)\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \left({b\_2}^{2}\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{c}\right)\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \left(b\_2 \cdot b\_2\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{c}\right)\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{c}\right)\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right), \left(\mathsf{neg}\left(\frac{2 \cdot 1}{c}\right)\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right), \left(\mathsf{neg}\left(\frac{2}{c}\right)\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      16. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right), \left(\frac{\mathsf{neg}\left(2\right)}{c}\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right), \left(\frac{-2}{c}\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right), \mathsf{/.f64}\left(-2, c\right)\right), \left(-1 \cdot b\_2\right)\right)\right) \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right), \mathsf{/.f64}\left(-2, c\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right)\right) \]

      20. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right), \mathsf{/.f64}\left(-2, c\right)\right), \left(0 - \color{blue}{b\_2}\right)\right)\right) \]

      21. --lowering--.f64 65.9%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{1}{2}\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right), \mathsf{/.f64}\left(-2, c\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{b\_2}\right)\right)\right) \]

   7. Simplified 65.9%

      \[\leadsto \frac{-1}{\color{blue}{\left(\frac{a \cdot 0.5}{b\_2 \cdot b\_2} + \frac{-2}{c}\right) \cdot \left(0 - b\_2\right)}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\frac{-1}{2} \cdot \frac{a}{b\_2} + 2 \cdot \frac{b\_2}{c}\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{a}{b\_2}\right), \color{blue}{\left(2 \cdot \frac{b\_2}{c}\right)}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2} \cdot a}{b\_2}\right), \left(\color{blue}{2} \cdot \frac{b\_2}{c}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot a\right), b\_2\right), \left(\color{blue}{2} \cdot \frac{b\_2}{c}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(a \cdot \frac{-1}{2}\right), b\_2\right), \left(2 \cdot \frac{b\_2}{c}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), b\_2\right), \left(2 \cdot \frac{b\_2}{c}\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), b\_2\right), \left(\frac{2 \cdot b\_2}{\color{blue}{c}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), b\_2\right), \mathsf{/.f64}\left(\left(2 \cdot b\_2\right), \color{blue}{c}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), b\_2\right), \mathsf{/.f64}\left(\left(b\_2 \cdot 2\right), c\right)\right)\right) \]

      9. *-lowering-*.f64 68.5%

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), b\_2\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b\_2, 2\right), c\right)\right)\right) \]

   10. Simplified 68.5%

       \[\leadsto \frac{-1}{\color{blue}{\frac{a \cdot -0.5}{b\_2} + \frac{b\_2 \cdot 2}{c}}} \]

   if -4.999999999999985e-310 < b_2

   1. Initial program 72.9%

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

   4. Applied egg-rr 72.7%

      \[\leadsto \color{blue}{\frac{-1}{\frac{a}{b\_2 + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}}} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. distribute-lft-in N/A

         \[\leadsto \frac{-1}{a} \cdot b\_2 + \color{blue}{\frac{-1}{a} \cdot \sqrt{b\_2 \cdot b\_2 - a \cdot c}} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\frac{-1}{a}\right), \color{blue}{b\_2}, \left(\frac{-1}{a} \cdot \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(-1, a\right), b\_2, \left(\frac{-1}{a} \cdot \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(-1, a\right), b\_2, \mathsf{*.f64}\left(\left(\frac{-1}{a}\right), \left(\sqrt{b\_2 \cdot b\_2 - a \cdot c}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(-1, a\right), b\_2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(-1, a\right), \left(\sqrt{b\_2 \cdot b\_2 - a \cdot c}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(-1, a\right), b\_2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(-1, a\right), \mathsf{sqrt.f64}\left(\left(b\_2 \cdot b\_2 - a \cdot c\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(-1, a\right), b\_2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(-1, a\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(b\_2 \cdot b\_2\right), \left(a \cdot c\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(-1, a\right), b\_2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(-1, a\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \left(a \cdot c\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 72.7%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(-1, a\right), b\_2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(-1, a\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \mathsf{*.f64}\left(a, c\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 72.7%

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

   7. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + \frac{1}{2} \cdot \frac{c}{b\_2}} \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(-2 \cdot \frac{b\_2}{a}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{c}{b\_2}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{-2 \cdot b\_2}{a}\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{c}{b\_2}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot b\_2\right), a\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{c}{b\_2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, b\_2\right), a\right), \left(\frac{1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, b\_2\right), a\right), \left(\frac{\frac{1}{2} \cdot c}{\color{blue}{b\_2}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, b\_2\right), a\right), \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot c\right), \color{blue}{b\_2}\right)\right) \]

      7. *-lowering-*.f64 69.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, b\_2\right), a\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, c\right), b\_2\right)\right) \]

   9. Simplified 69.0%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-1}{\frac{a \cdot -0.5}{b\_2} + \frac{b\_2 \cdot 2}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a} + \frac{c \cdot 0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.0% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310)
   (/ (* -0.5 c) b_2)
   (+ (/ (* b_2 -2.0) a) (/ (* c 0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = ((b_2 * -2.0) / a) + ((c * 0.5) / 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 <= (-5d-310)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = ((b_2 * (-2.0d0)) / a) + ((c * 0.5d0) / 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 <= -5e-310) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = ((b_2 * -2.0) / a) + ((c * 0.5) / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = ((b_2 * -2.0) / a) + ((c * 0.5) / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(Float64(Float64(b_2 * -2.0) / a) + Float64(Float64(c * 0.5) / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = ((b_2 * -2.0) / a) + ((c * 0.5) / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision] + N[(N[(c * 0.5), $MachinePrecision] / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 32.4%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b\_2}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\_2\right) \]

      4. *-lowering-*.f64 67.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\_2\right) \]

   5. Simplified 67.9%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 72.9%

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

   4. Applied egg-rr 72.7%

      \[\leadsto \color{blue}{\frac{-1}{\frac{a}{b\_2 + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}}} \]
   5. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. distribute-lft-in N/A

         \[\leadsto \frac{-1}{a} \cdot b\_2 + \color{blue}{\frac{-1}{a} \cdot \sqrt{b\_2 \cdot b\_2 - a \cdot c}} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\frac{-1}{a}\right), \color{blue}{b\_2}, \left(\frac{-1}{a} \cdot \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(-1, a\right), b\_2, \left(\frac{-1}{a} \cdot \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(-1, a\right), b\_2, \mathsf{*.f64}\left(\left(\frac{-1}{a}\right), \left(\sqrt{b\_2 \cdot b\_2 - a \cdot c}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(-1, a\right), b\_2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(-1, a\right), \left(\sqrt{b\_2 \cdot b\_2 - a \cdot c}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(-1, a\right), b\_2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(-1, a\right), \mathsf{sqrt.f64}\left(\left(b\_2 \cdot b\_2 - a \cdot c\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(-1, a\right), b\_2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(-1, a\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(b\_2 \cdot b\_2\right), \left(a \cdot c\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(-1, a\right), b\_2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(-1, a\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \left(a \cdot c\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 72.7%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(-1, a\right), b\_2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(-1, a\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \mathsf{*.f64}\left(a, c\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 72.7%

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

   7. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + \frac{1}{2} \cdot \frac{c}{b\_2}} \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(-2 \cdot \frac{b\_2}{a}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{c}{b\_2}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{-2 \cdot b\_2}{a}\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{c}{b\_2}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot b\_2\right), a\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{c}{b\_2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, b\_2\right), a\right), \left(\frac{1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, b\_2\right), a\right), \left(\frac{\frac{1}{2} \cdot c}{\color{blue}{b\_2}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, b\_2\right), a\right), \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot c\right), \color{blue}{b\_2}\right)\right) \]

      7. *-lowering-*.f64 69.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, b\_2\right), a\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, c\right), b\_2\right)\right) \]

   9. Simplified 69.0%

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

4. Final simplification 68.5%

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

Alternative 5: 67.9% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310)
   (/ (* -0.5 c) b_2)
   (+ (* b_2 (/ -2.0 a)) (* c (/ 0.5 b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (b_2 * (-2.0 / a)) + (c * (0.5 / 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 <= (-5d-310)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = (b_2 * ((-2.0d0) / a)) + (c * (0.5d0 / 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 <= -5e-310) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (b_2 * (-2.0 / a)) + (c * (0.5 / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = (b_2 * (-2.0 / a)) + (c * (0.5 / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(Float64(b_2 * Float64(-2.0 / a)) + Float64(c * Float64(0.5 / b_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = (b_2 * (-2.0 / a)) + (c * (0.5 / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision] + N[(c * N[(0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 32.4%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b\_2}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\_2\right) \]

      4. *-lowering-*.f64 67.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\_2\right) \]

   5. Simplified 67.9%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 72.9%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + \frac{1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(-2 \cdot \frac{b\_2}{a}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{c}{b\_2}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{-2 \cdot b\_2}{a}\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{c}{b\_2}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{b\_2 \cdot -2}{a}\right), \left(\frac{1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(b\_2 \cdot \frac{-2}{a}\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{c}{b\_2}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left(b\_2 \cdot \frac{\mathsf{neg}\left(2\right)}{a}\right), \left(\frac{1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\left(b\_2 \cdot \left(\mathsf{neg}\left(\frac{2}{a}\right)\right)\right), \left(\frac{1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left(b\_2 \cdot \left(\mathsf{neg}\left(\frac{2 \cdot 1}{a}\right)\right)\right), \left(\frac{1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(b\_2 \cdot \left(\mathsf{neg}\left(2 \cdot \frac{1}{a}\right)\right)\right), \left(\frac{1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b\_2, \left(\mathsf{neg}\left(2 \cdot \frac{1}{a}\right)\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{c}{b\_2}\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b\_2, \left(\mathsf{neg}\left(\frac{2 \cdot 1}{a}\right)\right)\right), \left(\frac{1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b\_2, \left(\mathsf{neg}\left(\frac{2}{a}\right)\right)\right), \left(\frac{1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b\_2, \left(\frac{\mathsf{neg}\left(2\right)}{a}\right)\right), \left(\frac{1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b\_2, \left(\frac{-2}{a}\right)\right), \left(\frac{1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b\_2, \mathsf{/.f64}\left(-2, a\right)\right), \left(\frac{1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      15. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b\_2, \mathsf{/.f64}\left(-2, a\right)\right), \left(\frac{\frac{1}{2} \cdot c}{\color{blue}{b\_2}}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b\_2, \mathsf{/.f64}\left(-2, a\right)\right), \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot c\right), \color{blue}{b\_2}\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b\_2, \mathsf{/.f64}\left(-2, a\right)\right), \mathsf{/.f64}\left(\left(c \cdot \frac{1}{2}\right), b\_2\right)\right) \]

      18. *-lowering-*.f64 68.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b\_2, \mathsf{/.f64}\left(-2, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{1}{2}\right), b\_2\right)\right) \]

   5. Simplified 68.8%

      \[\leadsto \color{blue}{b\_2 \cdot \frac{-2}{a} + \frac{c \cdot 0.5}{b\_2}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b\_2, \mathsf{/.f64}\left(-2, a\right)\right), \left(c \cdot \color{blue}{\frac{\frac{1}{2}}{b\_2}}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b\_2, \mathsf{/.f64}\left(-2, a\right)\right), \left(\frac{\frac{1}{2}}{b\_2} \cdot \color{blue}{c}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b\_2, \mathsf{/.f64}\left(-2, a\right)\right), \mathsf{*.f64}\left(\left(\frac{\frac{1}{2}}{b\_2}\right), \color{blue}{c}\right)\right) \]

      4. /-lowering-/.f64 68.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b\_2, \mathsf{/.f64}\left(-2, a\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), c\right)\right) \]

   7. Applied egg-rr 68.8%

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

4. Final simplification 68.4%

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

Alternative 6: 67.8% accurate, 11.2× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310) (/ (* -0.5 c) b_2) (/ (* b_2 -2.0) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = (-0.5 * c) / 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 <= (-5d-310)) then
        tmp = ((-0.5d0) * c) / 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 <= -5e-310) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(Float64(-0.5 * c) / 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 <= -5e-310)
		tmp = (-0.5 * c) / 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, -5e-310], N[(N[(-0.5 * c), $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 < -4.999999999999985e-310

   1. Initial program 32.4%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b\_2}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\_2\right) \]

      4. *-lowering-*.f64 67.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\_2\right) \]

   5. Simplified 67.9%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 72.9%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-2 \cdot b\_2\right)}, a\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b\_2 \cdot -2\right), a\right) \]

      2. *-lowering-*.f64 68.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b\_2, -2\right), a\right) \]

   5. Simplified 68.5%

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

4. Final simplification 68.2%

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

Alternative 7: 43.4% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4 \cdot 10^{-224}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e-224) 0.0 (/ (* b_2 -2.0) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-224) {
		tmp = 0.0;
	} 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 <= (-4d-224)) then
        tmp = 0.0d0
    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 <= -4e-224) {
		tmp = 0.0;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4e-224:
		tmp = 0.0
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4e-224)
		tmp = 0.0;
	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 <= -4e-224)
		tmp = 0.0;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4e-224], 0.0, N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.0000000000000001e-224

   1. Initial program 28.6%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \color{blue}{\left(-1 \cdot \left(b\_2 \cdot \left(1 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)}\right), a\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \left(\mathsf{neg}\left(b\_2 \cdot \left(1 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)\right), a\right) \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \left(b\_2 \cdot \left(\mathsf{neg}\left(\left(1 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)\right)\right), a\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \left(\mathsf{neg}\left(\left(1 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)\right)\right), a\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)\right)\right), a\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \left(-1 + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)\right)\right), a\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)\right)\right), a\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)\right), a\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \left(\frac{1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)\right), a\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \left(\frac{\frac{1}{2} \cdot \left(a \cdot c\right)}{{b\_2}^{2}}\right)\right)\right)\right), a\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \left(a \cdot c\right)}{{b\_2}^{2}}\right)\right)\right)\right), a\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \left(a \cdot c\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

      12. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{-1}{2} \cdot \left(a \cdot c\right)\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(a \cdot c\right) \cdot \frac{-1}{2}\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(a \cdot \left(c \cdot \frac{-1}{2}\right)\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(a \cdot \left(\frac{-1}{2} \cdot c\right)\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(a \cdot \left(\mathsf{neg}\left(\frac{-1}{2} \cdot c\right)\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

      17. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(a \cdot \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot c\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(a \cdot \left(\frac{1}{2} \cdot c\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot c\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(c \cdot \frac{1}{2}\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

      21. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \frac{1}{2}\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

      22. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \frac{1}{2}\right)\right), \left(b\_2 \cdot b\_2\right)\right)\right)\right)\right), a\right) \]

      23. *-lowering-*.f64 18.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right)\right)\right)\right), a\right) \]

   5. Simplified 18.1%

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

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \left(b\_2 + -1 \cdot b\_2\right)\right)}, a\right) \]
   7. Step-by-step derivation

      1. distribute-rgt1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(\left(-1 + 1\right) \cdot b\_2\right)\right), a\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(0 \cdot b\_2\right)\right), a\right) \]

      3. mul0-lft N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot 0\right), a\right) \]

      4. metadata-eval 16.5%

         \[\leadsto \mathsf{/.f64}\left(0, a\right) \]

   8. Simplified 16.5%

      \[\leadsto \frac{\color{blue}{0}}{a} \]
   9. Step-by-step derivation

      1. div0 16.5%

         \[\leadsto 0 \]

   10. Applied egg-rr 16.5%

       \[\leadsto \color{blue}{0} \]

   if -4.0000000000000001e-224 < b_2

   1. Initial program 73.2%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-2 \cdot b\_2\right)}, a\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b\_2 \cdot -2\right), a\right) \]

      2. *-lowering-*.f64 64.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b\_2, -2\right), a\right) \]

   5. Simplified 64.5%

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

Alternative 8: 43.3% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4 \cdot 10^{-224}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e-224) 0.0 (* b_2 (/ -2.0 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-224) {
		tmp = 0.0;
	} 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 <= (-4d-224)) then
        tmp = 0.0d0
    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 <= -4e-224) {
		tmp = 0.0;
	} else {
		tmp = b_2 * (-2.0 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4e-224:
		tmp = 0.0
	else:
		tmp = b_2 * (-2.0 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4e-224)
		tmp = 0.0;
	else
		tmp = Float64(b_2 * Float64(-2.0 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4e-224)
		tmp = 0.0;
	else
		tmp = b_2 * (-2.0 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4e-224], 0.0, N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.0000000000000001e-224

   1. Initial program 28.6%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \color{blue}{\left(-1 \cdot \left(b\_2 \cdot \left(1 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)}\right), a\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \left(\mathsf{neg}\left(b\_2 \cdot \left(1 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)\right), a\right) \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \left(b\_2 \cdot \left(\mathsf{neg}\left(\left(1 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)\right)\right), a\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \left(\mathsf{neg}\left(\left(1 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)\right)\right), a\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)\right)\right), a\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \left(-1 + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)\right)\right), a\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)\right)\right), a\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)\right), a\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \left(\frac{1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)\right), a\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \left(\frac{\frac{1}{2} \cdot \left(a \cdot c\right)}{{b\_2}^{2}}\right)\right)\right)\right), a\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \left(a \cdot c\right)}{{b\_2}^{2}}\right)\right)\right)\right), a\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \left(a \cdot c\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

      12. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{-1}{2} \cdot \left(a \cdot c\right)\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(a \cdot c\right) \cdot \frac{-1}{2}\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(a \cdot \left(c \cdot \frac{-1}{2}\right)\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(a \cdot \left(\frac{-1}{2} \cdot c\right)\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(a \cdot \left(\mathsf{neg}\left(\frac{-1}{2} \cdot c\right)\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

      17. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(a \cdot \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot c\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(a \cdot \left(\frac{1}{2} \cdot c\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot c\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(c \cdot \frac{1}{2}\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

      21. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \frac{1}{2}\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

      22. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \frac{1}{2}\right)\right), \left(b\_2 \cdot b\_2\right)\right)\right)\right)\right), a\right) \]

      23. *-lowering-*.f64 18.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right)\right)\right)\right), a\right) \]

   5. Simplified 18.1%

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

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \left(b\_2 + -1 \cdot b\_2\right)\right)}, a\right) \]
   7. Step-by-step derivation

      1. distribute-rgt1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(\left(-1 + 1\right) \cdot b\_2\right)\right), a\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(0 \cdot b\_2\right)\right), a\right) \]

      3. mul0-lft N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot 0\right), a\right) \]

      4. metadata-eval 16.5%

         \[\leadsto \mathsf{/.f64}\left(0, a\right) \]

   8. Simplified 16.5%

      \[\leadsto \frac{\color{blue}{0}}{a} \]
   9. Step-by-step derivation

      1. div0 16.5%

         \[\leadsto 0 \]

   10. Applied egg-rr 16.5%

       \[\leadsto \color{blue}{0} \]

   if -4.0000000000000001e-224 < b_2

   1. Initial program 73.2%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

         \[\leadsto b\_2 \cdot \frac{\mathsf{neg}\left(2\right)}{a} \]

      5. distribute-neg-frac N/A

         \[\leadsto b\_2 \cdot \left(\mathsf{neg}\left(\frac{2}{a}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto b\_2 \cdot \left(\mathsf{neg}\left(\frac{2 \cdot 1}{a}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto b\_2 \cdot \left(\mathsf{neg}\left(2 \cdot \frac{1}{a}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b\_2, \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{1}{a}\right)\right)}\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(b\_2, \left(\mathsf{neg}\left(\frac{2 \cdot 1}{a}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(b\_2, \left(\mathsf{neg}\left(\frac{2}{a}\right)\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(b\_2, \left(\frac{\mathsf{neg}\left(2\right)}{\color{blue}{a}}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(b\_2, \left(\frac{-2}{a}\right)\right) \]

      13. /-lowering-/.f64 64.3%

          \[\leadsto \mathsf{*.f64}\left(b\_2, \mathsf{/.f64}\left(-2, \color{blue}{a}\right)\right) \]

   5. Simplified 64.3%

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

Alternative 9: 11.3% accurate, 112.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 0.0)

C

double code(double a, double b_2, double c) {
	return 0.0;
}

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 = 0.0d0
end function

Java

public static double code(double a, double b_2, double c) {
	return 0.0;
}

Python

def code(a, b_2, c):
	return 0.0

Julia

function code(a, b_2, c)
	return 0.0
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = 0.0;
end

Wolfram

code[a_, b$95$2_, c_] := 0.0

Derivation

1. Initial program 53.9%

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

3. Taylor expanded in b_2 around -inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \color{blue}{\left(-1 \cdot \left(b\_2 \cdot \left(1 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)}\right), a\right) \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \left(\mathsf{neg}\left(b\_2 \cdot \left(1 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)\right), a\right) \]

   2. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \left(b\_2 \cdot \left(\mathsf{neg}\left(\left(1 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)\right)\right), a\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \left(\mathsf{neg}\left(\left(1 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)\right)\right), a\right) \]

   4. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)\right)\right), a\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \left(-1 + \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)\right)\right), a\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)\right)\right), a\right) \]

   7. distribute-lft-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)\right), a\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \left(\frac{1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)\right), a\right) \]

   9. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \left(\frac{\frac{1}{2} \cdot \left(a \cdot c\right)}{{b\_2}^{2}}\right)\right)\right)\right), a\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \left(a \cdot c\right)}{{b\_2}^{2}}\right)\right)\right)\right), a\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \left(a \cdot c\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

   12. distribute-lft-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{-1}{2} \cdot \left(a \cdot c\right)\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(a \cdot c\right) \cdot \frac{-1}{2}\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

   14. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(a \cdot \left(c \cdot \frac{-1}{2}\right)\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(a \cdot \left(\frac{-1}{2} \cdot c\right)\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

   16. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(a \cdot \left(\mathsf{neg}\left(\frac{-1}{2} \cdot c\right)\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

   17. distribute-lft-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(a \cdot \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot c\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(a \cdot \left(\frac{1}{2} \cdot c\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

   19. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{1}{2} \cdot c\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

   20. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \left(c \cdot \frac{1}{2}\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

   21. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \frac{1}{2}\right)\right), \left({b\_2}^{2}\right)\right)\right)\right)\right), a\right) \]

   22. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \frac{1}{2}\right)\right), \left(b\_2 \cdot b\_2\right)\right)\right)\right)\right), a\right) \]

   23. *-lowering-*.f64 8.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{*.f64}\left(b\_2, \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right)\right)\right)\right), a\right) \]

5. Simplified 8.9%

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

6. Taylor expanded in a around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \left(b\_2 + -1 \cdot b\_2\right)\right)}, a\right) \]
7. Step-by-step derivation

   1. distribute-rgt1-in N/A

      \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(\left(-1 + 1\right) \cdot b\_2\right)\right), a\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(0 \cdot b\_2\right)\right), a\right) \]

   3. mul0-lft N/A

      \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot 0\right), a\right) \]

   4. metadata-eval 8.7%

      \[\leadsto \mathsf{/.f64}\left(0, a\right) \]

8. Simplified 8.7%

   \[\leadsto \frac{\color{blue}{0}}{a} \]
9. Step-by-step derivation

   1. div0 8.7%

      \[\leadsto 0 \]

10. Applied egg-rr 8.7%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024191 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) x)) (sqrt (+ (fabs b_2) x))) (hypot b_2 x))))) (if (< b_2 0) (/ c (- sqtD b_2)) (/ (+ b_2 sqtD) (- a)))))

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