quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.2% → 85.4%

Time: 13.1s

Alternatives: 12

Speedup: 11.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.2% 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: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.65 \cdot 10^{+80}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a} + c \cdot \frac{0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.25 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2 \cdot -2 - \left(c \cdot -0.5\right) \cdot \frac{a}{b\_2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.65e+80)
   (+ (/ (* b_2 -2.0) a) (* c (/ 0.5 b_2)))
   (if (<= b_2 2.25e-10)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (/ c (- (* b_2 -2.0) (* (* c -0.5) (/ a b_2)))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.65e+80) {
		tmp = ((b_2 * -2.0) / a) + (c * (0.5 / b_2));
	} else if (b_2 <= 2.25e-10) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = c / ((b_2 * -2.0) - ((c * -0.5) * (a / 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 <= (-1.65d+80)) then
        tmp = ((b_2 * (-2.0d0)) / a) + (c * (0.5d0 / b_2))
    else if (b_2 <= 2.25d-10) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
    else
        tmp = c / ((b_2 * (-2.0d0)) - ((c * (-0.5d0)) * (a / 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 <= -1.65e+80) {
		tmp = ((b_2 * -2.0) / a) + (c * (0.5 / b_2));
	} else if (b_2 <= 2.25e-10) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = c / ((b_2 * -2.0) - ((c * -0.5) * (a / b_2)));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.65e+80:
		tmp = ((b_2 * -2.0) / a) + (c * (0.5 / b_2))
	elif b_2 <= 2.25e-10:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
	else:
		tmp = c / ((b_2 * -2.0) - ((c * -0.5) * (a / b_2)))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.65e+80)
		tmp = Float64(Float64(Float64(b_2 * -2.0) / a) + Float64(c * Float64(0.5 / b_2)));
	elseif (b_2 <= 2.25e-10)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(c / Float64(Float64(b_2 * -2.0) - Float64(Float64(c * -0.5) * Float64(a / b_2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.65e+80)
		tmp = ((b_2 * -2.0) / a) + (c * (0.5 / b_2));
	elseif (b_2 <= 2.25e-10)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	else
		tmp = c / ((b_2 * -2.0) - ((c * -0.5) * (a / b_2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.65e+80], N[(N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision] + N[(c * N[(0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 2.25e-10], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(c / N[(N[(b$95$2 * -2.0), $MachinePrecision] - N[(N[(c * -0.5), $MachinePrecision] * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.64999999999999995e80

   1. Initial program 63.7%

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

   3. Taylor expanded in b_2 around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 63.5%

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

   5. Simplified 63.5%

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

   6. Taylor expanded in b_2 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

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

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

   8. Simplified 90.3%

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

   9. Taylor expanded in c around 0

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

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

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

   11. Simplified 94.4%

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

   if -1.64999999999999995e80 < b_2 < 2.25e-10

   1. Initial program 83.8%

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

   if 2.25e-10 < b_2

   1. Initial program 5.4%

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

   3. Taylor expanded in b_2 around inf

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 30.8%

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

   5. Simplified 30.8%

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

   7. Applied egg-rr 3.3%

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

   8. Taylor expanded in b_2 around inf

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

   10. Simplified 97.7%

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

   11. Taylor expanded in b_2 around 0

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

       1. *-lowering-*.f64 97.7%

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

   13. Simplified 97.7%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.65 \cdot 10^{+80}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a} + c \cdot \frac{0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.25 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2 \cdot -2 - \left(c \cdot -0.5\right) \cdot \frac{a}{b\_2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -6.2 \cdot 10^{-44}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a} + c \cdot \frac{0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sqrt{0 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2 \cdot -2 - \left(c \cdot -0.5\right) \cdot \frac{a}{b\_2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -6.2e-44)
   (+ (/ (* b_2 -2.0) a) (* c (/ 0.5 b_2)))
   (if (<= b_2 2e-10)
     (/ (- (sqrt (- 0.0 (* a c))) b_2) a)
     (/ c (- (* b_2 -2.0) (* (* c -0.5) (/ a b_2)))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6.2e-44) {
		tmp = ((b_2 * -2.0) / a) + (c * (0.5 / b_2));
	} else if (b_2 <= 2e-10) {
		tmp = (sqrt((0.0 - (a * c))) - b_2) / a;
	} else {
		tmp = c / ((b_2 * -2.0) - ((c * -0.5) * (a / 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 <= (-6.2d-44)) then
        tmp = ((b_2 * (-2.0d0)) / a) + (c * (0.5d0 / b_2))
    else if (b_2 <= 2d-10) then
        tmp = (sqrt((0.0d0 - (a * c))) - b_2) / a
    else
        tmp = c / ((b_2 * (-2.0d0)) - ((c * (-0.5d0)) * (a / 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 <= -6.2e-44) {
		tmp = ((b_2 * -2.0) / a) + (c * (0.5 / b_2));
	} else if (b_2 <= 2e-10) {
		tmp = (Math.sqrt((0.0 - (a * c))) - b_2) / a;
	} else {
		tmp = c / ((b_2 * -2.0) - ((c * -0.5) * (a / b_2)));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -6.2e-44:
		tmp = ((b_2 * -2.0) / a) + (c * (0.5 / b_2))
	elif b_2 <= 2e-10:
		tmp = (math.sqrt((0.0 - (a * c))) - b_2) / a
	else:
		tmp = c / ((b_2 * -2.0) - ((c * -0.5) * (a / b_2)))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -6.2e-44)
		tmp = Float64(Float64(Float64(b_2 * -2.0) / a) + Float64(c * Float64(0.5 / b_2)));
	elseif (b_2 <= 2e-10)
		tmp = Float64(Float64(sqrt(Float64(0.0 - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(c / Float64(Float64(b_2 * -2.0) - Float64(Float64(c * -0.5) * Float64(a / b_2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -6.2e-44)
		tmp = ((b_2 * -2.0) / a) + (c * (0.5 / b_2));
	elseif (b_2 <= 2e-10)
		tmp = (sqrt((0.0 - (a * c))) - b_2) / a;
	else
		tmp = c / ((b_2 * -2.0) - ((c * -0.5) * (a / b_2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -6.2e-44], N[(N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision] + N[(c * N[(0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 2e-10], N[(N[(N[Sqrt[N[(0.0 - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(c / N[(N[(b$95$2 * -2.0), $MachinePrecision] - N[(N[(c * -0.5), $MachinePrecision] * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -6.19999999999999968e-44

   1. Initial program 74.6%

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

   3. Taylor expanded in b_2 around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 74.5%

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

   5. Simplified 74.5%

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

   6. Taylor expanded in b_2 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

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

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

   8. Simplified 81.9%

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

   9. Taylor expanded in c around 0

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

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

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

   11. Simplified 84.4%

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

   if -6.19999999999999968e-44 < b_2 < 2.00000000000000007e-10

   1. Initial program 81.1%

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

   3. Taylor expanded in b_2 around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 21.9%

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

   5. Simplified 21.9%

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

   6. Taylor expanded in b_2 around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot c\right)\right)}\right)\right), a\right) \]
   7. 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 76.4%

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

   8. Simplified 76.4%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

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

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

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

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

      6. *-lowering-*.f64 76.4%

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

   10. Applied egg-rr 76.4%

       \[\leadsto \frac{\color{blue}{\sqrt{0 - c \cdot a} - b\_2}}{a} \]

   if 2.00000000000000007e-10 < b_2

   1. Initial program 5.4%

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

   3. Taylor expanded in b_2 around inf

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 30.8%

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

   5. Simplified 30.8%

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

   7. Applied egg-rr 3.3%

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

   8. Taylor expanded in b_2 around inf

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

   10. Simplified 97.7%

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

   11. Taylor expanded in b_2 around 0

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

       1. *-lowering-*.f64 97.7%

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

   13. Simplified 97.7%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -6.2 \cdot 10^{-44}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a} + c \cdot \frac{0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sqrt{0 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2 \cdot -2 - \left(c \cdot -0.5\right) \cdot \frac{a}{b\_2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.2% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -6 \cdot 10^{-254}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a} + c \cdot \frac{0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2 \cdot -2 - \left(c \cdot -0.5\right) \cdot \frac{a}{b\_2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -6e-254)
   (+ (/ (* b_2 -2.0) a) (* c (/ 0.5 b_2)))
   (/ c (- (* b_2 -2.0) (* (* c -0.5) (/ a b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6e-254) {
		tmp = ((b_2 * -2.0) / a) + (c * (0.5 / b_2));
	} else {
		tmp = c / ((b_2 * -2.0) - ((c * -0.5) * (a / 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 <= (-6d-254)) then
        tmp = ((b_2 * (-2.0d0)) / a) + (c * (0.5d0 / b_2))
    else
        tmp = c / ((b_2 * (-2.0d0)) - ((c * (-0.5d0)) * (a / 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 <= -6e-254) {
		tmp = ((b_2 * -2.0) / a) + (c * (0.5 / b_2));
	} else {
		tmp = c / ((b_2 * -2.0) - ((c * -0.5) * (a / b_2)));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -6e-254:
		tmp = ((b_2 * -2.0) / a) + (c * (0.5 / b_2))
	else:
		tmp = c / ((b_2 * -2.0) - ((c * -0.5) * (a / b_2)))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -6e-254)
		tmp = Float64(Float64(Float64(b_2 * -2.0) / a) + Float64(c * Float64(0.5 / b_2)));
	else
		tmp = Float64(c / Float64(Float64(b_2 * -2.0) - Float64(Float64(c * -0.5) * Float64(a / b_2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -6e-254)
		tmp = ((b_2 * -2.0) / a) + (c * (0.5 / b_2));
	else
		tmp = c / ((b_2 * -2.0) - ((c * -0.5) * (a / b_2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -6e-254], N[(N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision] + N[(c * N[(0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c / N[(N[(b$95$2 * -2.0), $MachinePrecision] - N[(N[(c * -0.5), $MachinePrecision] * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -6.00000000000000023e-254

   1. Initial program 80.5%

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

   3. Taylor expanded in b_2 around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 64.1%

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

   5. Simplified 64.1%

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

   6. Taylor expanded in b_2 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

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

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

   8. Simplified 63.7%

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

   9. Taylor expanded in c around 0

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

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

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

   11. Simplified 65.6%

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

   if -6.00000000000000023e-254 < b_2

   1. Initial program 33.0%

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

   3. Taylor expanded in b_2 around inf

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 20.0%

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

   5. Simplified 20.0%

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

   7. Applied egg-rr 3.6%

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

   8. Taylor expanded in b_2 around inf

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

   10. Simplified 66.4%

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

   11. Taylor expanded in b_2 around 0

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

       1. *-lowering-*.f64 66.4%

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

   13. Simplified 66.4%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -6 \cdot 10^{-254}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a} + c \cdot \frac{0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2 \cdot -2 - \left(c \cdot -0.5\right) \cdot \frac{a}{b\_2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.1% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-310)
   (+ (/ (* b_2 -2.0) a) (* c (/ 0.5 b_2)))
   (/ c (/ b_2 -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-310) {
		tmp = ((b_2 * -2.0) / a) + (c * (0.5 / b_2));
	} else {
		tmp = c / (b_2 / -0.5);
	}
	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 <= (-2d-310)) then
        tmp = ((b_2 * (-2.0d0)) / a) + (c * (0.5d0 / b_2))
    else
        tmp = c / (b_2 / (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-310) {
		tmp = ((b_2 * -2.0) / a) + (c * (0.5 / b_2));
	} else {
		tmp = c / (b_2 / -0.5);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e-310:
		tmp = ((b_2 * -2.0) / a) + (c * (0.5 / b_2))
	else:
		tmp = c / (b_2 / -0.5)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e-310)
		tmp = Float64(Float64(Float64(b_2 * -2.0) / a) + Float64(c * Float64(0.5 / b_2)));
	else
		tmp = Float64(c / Float64(b_2 / -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2e-310)
		tmp = ((b_2 * -2.0) / a) + (c * (0.5 / b_2));
	else
		tmp = c / (b_2 / -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.999999999999994e-310

   1. Initial program 80.7%

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

   3. Taylor expanded in b_2 around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 60.9%

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

   5. Simplified 60.9%

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

   6. Taylor expanded in b_2 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

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

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

   8. Simplified 60.7%

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

   9. Taylor expanded in c around 0

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

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

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

   11. Simplified 62.5%

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

   if -1.999999999999994e-310 < b_2

   1. Initial program 30.8%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 68.9%

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

   5. Simplified 68.9%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 69.0%

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

   7. Applied egg-rr 69.0%

      \[\leadsto \color{blue}{\frac{c}{\frac{b\_2}{-0.5}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 68.0% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-310)
   (+ (* c (/ 0.5 b_2)) (* b_2 (/ -2.0 a)))
   (/ c (/ b_2 -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-310) {
		tmp = (c * (0.5 / b_2)) + (b_2 * (-2.0 / a));
	} else {
		tmp = c / (b_2 / -0.5);
	}
	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 <= (-2d-310)) then
        tmp = (c * (0.5d0 / b_2)) + (b_2 * ((-2.0d0) / a))
    else
        tmp = c / (b_2 / (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-310) {
		tmp = (c * (0.5 / b_2)) + (b_2 * (-2.0 / a));
	} else {
		tmp = c / (b_2 / -0.5);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e-310:
		tmp = (c * (0.5 / b_2)) + (b_2 * (-2.0 / a))
	else:
		tmp = c / (b_2 / -0.5)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e-310)
		tmp = Float64(Float64(c * Float64(0.5 / b_2)) + Float64(b_2 * Float64(-2.0 / a)));
	else
		tmp = Float64(c / Float64(b_2 / -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2e-310)
		tmp = (c * (0.5 / b_2)) + (b_2 * (-2.0 / a));
	else
		tmp = c / (b_2 / -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.999999999999994e-310

   1. Initial program 80.7%

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

   3. Taylor expanded in b_2 around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 60.9%

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

   5. Simplified 60.9%

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

   6. Taylor expanded in b_2 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

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

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

   8. Simplified 60.7%

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

   9. Taylor expanded in c around 0

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

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

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

   11. Simplified 62.5%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

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

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

       4. /-lowering-/.f64 62.3%

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

   13. Applied egg-rr 62.3%

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

   if -1.999999999999994e-310 < b_2

   1. Initial program 30.8%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 68.9%

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

   5. Simplified 68.9%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 69.0%

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

   7. Applied egg-rr 69.0%

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

4. Final simplification 65.9%

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

Alternative 6: 67.9% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 1.7 \cdot 10^{-299}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{b\_2}{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 1.7e-299) (/ (* b_2 -2.0) a) (/ c (/ b_2 -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1.7e-299) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = c / (b_2 / -0.5);
	}
	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 <= 1.7d-299) then
        tmp = (b_2 * (-2.0d0)) / a
    else
        tmp = c / (b_2 / (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1.7e-299) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = c / (b_2 / -0.5);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 1.7e-299:
		tmp = (b_2 * -2.0) / a
	else:
		tmp = c / (b_2 / -0.5)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 1.7e-299)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = Float64(c / Float64(b_2 / -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 1.7e-299)
		tmp = (b_2 * -2.0) / a;
	else
		tmp = c / (b_2 / -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 1.7e-299], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(c / N[(b$95$2 / -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 1.6999999999999999e-299

   1. Initial program 81.3%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 60.3%

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

   5. Simplified 60.3%

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

   if 1.6999999999999999e-299 < b_2

   1. Initial program 28.7%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   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 70.9%

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

   5. Simplified 70.9%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 71.0%

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

   7. Applied egg-rr 71.0%

      \[\leadsto \color{blue}{\frac{c}{\frac{b\_2}{-0.5}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 67.8% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 1.4 \cdot 10^{-299}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{b\_2}{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 1.4e-299) (* b_2 (/ -2.0 a)) (/ c (/ b_2 -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1.4e-299) {
		tmp = b_2 * (-2.0 / a);
	} else {
		tmp = c / (b_2 / -0.5);
	}
	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 <= 1.4d-299) then
        tmp = b_2 * ((-2.0d0) / a)
    else
        tmp = c / (b_2 / (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1.4e-299) {
		tmp = b_2 * (-2.0 / a);
	} else {
		tmp = c / (b_2 / -0.5);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 1.4e-299:
		tmp = b_2 * (-2.0 / a)
	else:
		tmp = c / (b_2 / -0.5)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 1.4e-299)
		tmp = Float64(b_2 * Float64(-2.0 / a));
	else
		tmp = Float64(c / Float64(b_2 / -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 1.4e-299)
		tmp = b_2 * (-2.0 / a);
	else
		tmp = c / (b_2 / -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 1.4e-299], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision], N[(c / N[(b$95$2 / -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 1.4000000000000001e-299

   1. Initial program 81.3%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 60.3%

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

   5. Simplified 60.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 60.1%

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

   7. Applied egg-rr 60.1%

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

   if 1.4000000000000001e-299 < b_2

   1. Initial program 28.7%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   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 70.9%

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

   5. Simplified 70.9%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 71.0%

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

   7. Applied egg-rr 71.0%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 1.4 \cdot 10^{-299}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{b\_2}{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.0% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.7 \cdot 10^{-300}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.7e-300) (* b_2 (/ -2.0 a)) (/ 0.0 a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.7e-300) {
		tmp = b_2 * (-2.0 / a);
	} else {
		tmp = 0.0 / 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 <= (-3.7d-300)) then
        tmp = b_2 * ((-2.0d0) / a)
    else
        tmp = 0.0d0 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.7e-300) {
		tmp = b_2 * (-2.0 / a);
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.7e-300:
		tmp = b_2 * (-2.0 / a)
	else:
		tmp = 0.0 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.7e-300)
		tmp = Float64(b_2 * Float64(-2.0 / a));
	else
		tmp = Float64(0.0 / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.7e-300)
		tmp = b_2 * (-2.0 / a);
	else
		tmp = 0.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.7e-300], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision], N[(0.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.7000000000000001e-300

   1. Initial program 80.5%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 62.6%

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

   5. Simplified 62.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 62.4%

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

   7. Applied egg-rr 62.4%

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

   if -3.7000000000000001e-300 < b_2

   1. Initial program 31.3%

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

   3. Taylor expanded in b_2 around inf

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 20.7%

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

   5. Simplified 20.7%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   7. Applied egg-rr 21.1%

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

   9. Applied egg-rr 3.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{b\_2 + \frac{c \cdot -0.5}{\frac{b\_2}{a}}}{b\_2 \cdot b\_2 + 0.25 \cdot \left(\left(c \cdot \frac{a}{b\_2}\right) \cdot \left(c \cdot \frac{a}{b\_2}\right)\right)}}} - b\_2}{a} \]

   11. Applied egg-rr 20.1%

       \[\leadsto \color{blue}{\frac{0}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.7 \cdot 10^{-300}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 24.0% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.3 \cdot 10^{-307}:\\ \;\;\;\;\frac{b\_2}{0 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 2.3e-307) (/ b_2 (- 0.0 a)) (/ 0.0 a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 2.3e-307) {
		tmp = b_2 / (0.0 - a);
	} else {
		tmp = 0.0 / 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 <= 2.3d-307) then
        tmp = b_2 / (0.0d0 - a)
    else
        tmp = 0.0d0 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 2.3e-307) {
		tmp = b_2 / (0.0 - a);
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 2.3e-307:
		tmp = b_2 / (0.0 - a)
	else:
		tmp = 0.0 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 2.3e-307)
		tmp = Float64(b_2 / Float64(0.0 - a));
	else
		tmp = Float64(0.0 / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 2.3e-307)
		tmp = b_2 / (0.0 - a);
	else
		tmp = 0.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 2.3e-307], N[(b$95$2 / N[(0.0 - a), $MachinePrecision]), $MachinePrecision], N[(0.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 2.2999999999999999e-307

   1. Initial program 80.8%

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

   3. Taylor expanded in b_2 around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 60.4%

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

   5. Simplified 60.4%

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

   6. Taylor expanded in b_2 around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot c\right)\right)}\right)\right), a\right) \]
   7. 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 44.7%

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

   8. Simplified 44.7%

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

   9. Taylor expanded in b_2 around inf

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

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

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

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

       5. mul-1-neg N/A

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

       6. neg-lowering-neg.f64 26.3%

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

   11. Simplified 26.3%

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

   if 2.2999999999999999e-307 < b_2

   1. Initial program 30.2%

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

   3. Taylor expanded in b_2 around inf

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 21.0%

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

   5. Simplified 21.0%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   7. Applied egg-rr 21.4%

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

   9. Applied egg-rr 3.8%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{b\_2 + \frac{c \cdot -0.5}{\frac{b\_2}{a}}}{b\_2 \cdot b\_2 + 0.25 \cdot \left(\left(c \cdot \frac{a}{b\_2}\right) \cdot \left(c \cdot \frac{a}{b\_2}\right)\right)}}} - b\_2}{a} \]

   11. Applied egg-rr 20.4%

       \[\leadsto \color{blue}{\frac{0}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.3 \cdot 10^{-307}:\\ \;\;\;\;\frac{b\_2}{0 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 15.4% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.2:\\ \;\;\;\;\frac{c}{0}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (if (<= b_2 -7.2) (/ c 0.0) (/ 0.0 a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.2) {
		tmp = c / 0.0;
	} else {
		tmp = 0.0 / 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 <= (-7.2d0)) then
        tmp = c / 0.0d0
    else
        tmp = 0.0d0 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.2) {
		tmp = c / 0.0;
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -7.2:
		tmp = c / 0.0
	else:
		tmp = 0.0 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -7.2)
		tmp = Float64(c / 0.0);
	else
		tmp = Float64(0.0 / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -7.2)
		tmp = c / 0.0;
	else
		tmp = 0.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7.2], N[(c / 0.0), $MachinePrecision], N[(0.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -7.20000000000000018

   1. Initial program 71.7%

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

   3. Taylor expanded in b_2 around inf

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 2.0%

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

   5. Simplified 2.0%

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

   7. Applied egg-rr 1.3%

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

   8. Taylor expanded in b_2 around inf

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

   10. Simplified 2.8%

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

   12. Applied egg-rr 15.9%

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

   if -7.20000000000000018 < b_2

   1. Initial program 47.4%

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

   3. Taylor expanded in b_2 around inf

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 15.8%

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

   5. Simplified 15.8%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   7. Applied egg-rr 16.1%

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

   9. Applied egg-rr 3.2%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{b\_2 + \frac{c \cdot -0.5}{\frac{b\_2}{a}}}{b\_2 \cdot b\_2 + 0.25 \cdot \left(\left(c \cdot \frac{a}{b\_2}\right) \cdot \left(c \cdot \frac{a}{b\_2}\right)\right)}}} - b\_2}{a} \]

   11. Applied egg-rr 15.8%

       \[\leadsto \color{blue}{\frac{0}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 15.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.2:\\ \;\;\;\;\frac{c}{0}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 15.3% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.15 \cdot 10^{+33}:\\ \;\;\;\;\frac{c}{0}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.15e+33) (/ c 0.0) (/ 0.0 a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.15e+33) {
		tmp = c / 0.0;
	} else {
		tmp = 0.0 / 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 <= (-2.15d+33)) then
        tmp = c / 0.0d0
    else
        tmp = 0.0d0 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.15e+33) {
		tmp = c / 0.0;
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.15e+33:
		tmp = c / 0.0
	else:
		tmp = 0.0 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.15e+33)
		tmp = Float64(c / 0.0);
	else
		tmp = Float64(0.0 / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.15e+33)
		tmp = c / 0.0;
	else
		tmp = 0.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.15e+33], N[(c / 0.0), $MachinePrecision], N[(0.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.15000000000000014e33

   1. Initial program 69.7%

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

   3. Taylor expanded in b_2 around inf

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 2.0%

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

   5. Simplified 2.0%

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

   7. Applied egg-rr 1.2%

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

   8. Taylor expanded in b_2 around inf

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

   10. Simplified 2.8%

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

   12. Applied egg-rr 16.3%

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

   if -2.15000000000000014e33 < b_2

   1. Initial program 49.1%

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

   3. Taylor expanded in b_2 around inf

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 15.2%

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

   5. Simplified 15.2%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

   7. Applied egg-rr 15.5%

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

   9. Applied egg-rr 3.2%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{b\_2 + \frac{c \cdot -0.5}{\frac{b\_2}{a}}}{b\_2 \cdot b\_2 + 0.25 \cdot \left(\left(c \cdot \frac{a}{b\_2}\right) \cdot \left(c \cdot \frac{a}{b\_2}\right)\right)}}} - b\_2}{a} \]

   11. Applied egg-rr 15.3%

       \[\leadsto \color{blue}{\frac{0}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 11.4% accurate, 37.3× speedup

Math

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

FPCore

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

C

double code(double a, double b_2, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

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

Python

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

Julia

function code(a, b_2, c)
	return Float64(0.0 / a)
end

MATLAB

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

Wolfram

code[a_, b$95$2_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 54.2%

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

3. Taylor expanded in b_2 around inf

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

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

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

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

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

   3. associate-*r/ N/A

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

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

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

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

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

   9. *-commutative N/A

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

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

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

   11. unpow2 N/A

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

   12. *-lowering-*.f64 12.0%

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

5. Simplified 12.0%

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

   1. +-commutative N/A

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

   2. unsub-neg N/A

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

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

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

7. Applied egg-rr 12.2%

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

9. Applied egg-rr 3.5%

   \[\leadsto \frac{\color{blue}{\frac{1}{\frac{b\_2 + \frac{c \cdot -0.5}{\frac{b\_2}{a}}}{b\_2 \cdot b\_2 + 0.25 \cdot \left(\left(c \cdot \frac{a}{b\_2}\right) \cdot \left(c \cdot \frac{a}{b\_2}\right)\right)}}} - b\_2}{a} \]

11. Applied egg-rr 12.1%

    \[\leadsto \color{blue}{\frac{0}{a}} \]
12. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.1× speedup

Math

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

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) (/ (- t_1 b_2) a) (/ (- c) (+ b_2 t_1)))))

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 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	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 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	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 = (t_1 - b_2) / a
	else:
		tmp_1 = -c / (b_2 + t_1)
	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(Float64(t_1 - b_2) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(b_2 + t_1));
	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 = (t_1 - b_2) / a;
	else
		tmp_2 = -c / (b_2 + t_1);
	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[(N[(t$95$1 - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(b$95$2 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2024192 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :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) (/ (- sqtD b_2) a) (/ (- c) (+ b_2 sqtD)))))

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