quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.3% → 85.2%

Time: 18.3s

Alternatives: 9

Speedup: 11.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.3% 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: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{+112}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 2.25 \cdot 10^{-39}:\\ \;\;\;\;\frac{{\left(\frac{1}{b\_2 \cdot b\_2 - a \cdot c}\right)}^{-0.5} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-0.5 + a \cdot \left(-0.125 \cdot \frac{\frac{c}{b\_2}}{b\_2}\right)\right)}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e+112)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 2.25e-39)
     (/ (- (pow (/ 1.0 (- (* b_2 b_2) (* a c))) -0.5) b_2) a)
     (/ (* c (+ -0.5 (* a (* -0.125 (/ (/ c b_2) b_2))))) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e+112) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 2.25e-39) {
		tmp = (pow((1.0 / ((b_2 * b_2) - (a * c))), -0.5) - b_2) / a;
	} else {
		tmp = (c * (-0.5 + (a * (-0.125 * ((c / b_2) / b_2))))) / b_2;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d+112)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= 2.25d-39) then
        tmp = (((1.0d0 / ((b_2 * b_2) - (a * c))) ** (-0.5d0)) - b_2) / a
    else
        tmp = (c * ((-0.5d0) + (a * ((-0.125d0) * ((c / b_2) / b_2))))) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e+112) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 2.25e-39) {
		tmp = (Math.pow((1.0 / ((b_2 * b_2) - (a * c))), -0.5) - b_2) / a;
	} else {
		tmp = (c * (-0.5 + (a * (-0.125 * ((c / b_2) / b_2))))) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e+112:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= 2.25e-39:
		tmp = (math.pow((1.0 / ((b_2 * b_2) - (a * c))), -0.5) - b_2) / a
	else:
		tmp = (c * (-0.5 + (a * (-0.125 * ((c / b_2) / b_2))))) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e+112)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 2.25e-39)
		tmp = Float64(Float64((Float64(1.0 / Float64(Float64(b_2 * b_2) - Float64(a * c))) ^ -0.5) - b_2) / a);
	else
		tmp = Float64(Float64(c * Float64(-0.5 + Float64(a * Float64(-0.125 * Float64(Float64(c / b_2) / b_2))))) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e+112)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= 2.25e-39)
		tmp = (((1.0 / ((b_2 * b_2) - (a * c))) ^ -0.5) - b_2) / a;
	else
		tmp = (c * (-0.5 + (a * (-0.125 * ((c / b_2) / b_2))))) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e+112], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 2.25e-39], N[(N[(N[Power[N[(1.0 / N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * N[(-0.5 + N[(a * N[(-0.125 * N[(N[(c / b$95$2), $MachinePrecision] / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -5e112

   1. Initial program 46.9%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 46.9%

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

   3. Simplified 46.9%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

      7. metadata-eval N/A

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

      8. *-lowering-*.f64 95.8%

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

   7. Simplified 95.8%

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

   if -5e112 < b_2 < 2.25e-39

   1. Initial program 81.9%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 81.9%

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

   3. Simplified 81.9%

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

      1. pow1/2 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

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

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

      9. clear-num N/A

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

      10. flip-- N/A

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

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

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

      12. --lowering--.f64 N/A

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

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

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

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

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

      15. metadata-eval 81.9%

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

   6. Applied egg-rr 81.9%

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

   if 2.25e-39 < b_2

   1. Initial program 12.4%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 12.4%

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

   3. Simplified 12.4%

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

      1. pow1/2 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

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

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

      9. clear-num N/A

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

      10. flip-- N/A

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

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

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

      12. --lowering--.f64 N/A

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

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

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

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

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

      15. metadata-eval 12.3%

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

   6. Applied egg-rr 12.3%

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

   7. Taylor expanded in b_2 around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c + \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}\right), \color{blue}{b\_2}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot c\right), \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}\right)\right), b\_2\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(c \cdot \frac{-1}{2}\right), \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}\right)\right), b\_2\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}\right)\right), b\_2\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \left(\frac{\frac{-1}{8} \cdot \left(a \cdot {c}^{2}\right)}{{b\_2}^{2}}\right)\right), b\_2\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(a \cdot {c}^{2}\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(a \cdot {c}^{2}\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({c}^{2} \cdot a\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(c \cdot c\right) \cdot a\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(c \cdot \left(c \cdot a\right)\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(c \cdot \left(a \cdot c\right)\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, \left(a \cdot c\right)\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, \left(c \cdot a\right)\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, a\right)\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, a\right)\right)\right), \left(b\_2 \cdot b\_2\right)\right)\right), b\_2\right) \]

      16. *-lowering-*.f64 76.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, a\right)\right)\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right)\right), b\_2\right) \]

   9. Simplified 76.0%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5 + \frac{-0.125 \cdot \left(c \cdot \left(c \cdot a\right)\right)}{b\_2 \cdot b\_2}}{b\_2}} \]

   10. Taylor expanded in c around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(c \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}} - \frac{1}{2}\right)\right)}, b\_2\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}} - \frac{1}{2}\right)\right), b\_2\right) \]

       2. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), b\_2\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}} + \frac{-1}{2}\right)\right), b\_2\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \left(\frac{-1}{2} + \frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right), b\_2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right), b\_2\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{a \cdot c}{{b\_2}^{2}} \cdot \frac{-1}{8}\right)\right)\right), b\_2\right) \]

       7. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(a \cdot \frac{c}{{b\_2}^{2}}\right) \cdot \frac{-1}{8}\right)\right)\right), b\_2\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \left(a \cdot \left(\frac{c}{{b\_2}^{2}} \cdot \frac{-1}{8}\right)\right)\right)\right), b\_2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \left(\frac{c}{{b\_2}^{2}} \cdot \frac{-1}{8}\right)\right)\right)\right), b\_2\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{c}{{b\_2}^{2}}\right)\right)\right)\right), b\_2\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{8}, \left(\frac{c}{{b\_2}^{2}}\right)\right)\right)\right)\right), b\_2\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{8}, \left(\frac{c}{b\_2 \cdot b\_2}\right)\right)\right)\right)\right), b\_2\right) \]

       13. associate-/r* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{8}, \left(\frac{\frac{c}{b\_2}}{b\_2}\right)\right)\right)\right)\right), b\_2\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\left(\frac{c}{b\_2}\right), b\_2\right)\right)\right)\right)\right), b\_2\right) \]

       15. /-lowering-/.f64 91.7%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(c, b\_2\right), b\_2\right)\right)\right)\right)\right), b\_2\right) \]

   12. Simplified 91.7%

       \[\leadsto \frac{\color{blue}{c \cdot \left(-0.5 + a \cdot \left(-0.125 \cdot \frac{\frac{c}{b\_2}}{b\_2}\right)\right)}}{b\_2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4 \cdot 10^{+112}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 2.8 \cdot 10^{-39}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-0.5 + a \cdot \left(-0.125 \cdot \frac{\frac{c}{b\_2}}{b\_2}\right)\right)}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e+112)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 2.8e-39)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (/ (* c (+ -0.5 (* a (* -0.125 (/ (/ c b_2) b_2))))) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e+112) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 2.8e-39) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (c * (-0.5 + (a * (-0.125 * ((c / b_2) / b_2))))) / b_2;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-4d+112)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= 2.8d-39) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
    else
        tmp = (c * ((-0.5d0) + (a * ((-0.125d0) * ((c / b_2) / b_2))))) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e+112) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 2.8e-39) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (c * (-0.5 + (a * (-0.125 * ((c / b_2) / b_2))))) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4e+112:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= 2.8e-39:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
	else:
		tmp = (c * (-0.5 + (a * (-0.125 * ((c / b_2) / b_2))))) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4e+112)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 2.8e-39)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(Float64(c * Float64(-0.5 + Float64(a * Float64(-0.125 * Float64(Float64(c / b_2) / b_2))))) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4e+112)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= 2.8e-39)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	else
		tmp = (c * (-0.5 + (a * (-0.125 * ((c / b_2) / b_2))))) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4e+112], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 2.8e-39], 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[(N[(c * N[(-0.5 + N[(a * N[(-0.125 * N[(N[(c / b$95$2), $MachinePrecision] / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.9999999999999997e112

   1. Initial program 46.9%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 46.9%

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

   3. Simplified 46.9%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

      7. metadata-eval N/A

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

      8. *-lowering-*.f64 95.8%

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

   7. Simplified 95.8%

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

   if -3.9999999999999997e112 < b_2 < 2.8000000000000001e-39

   1. Initial program 81.9%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 81.9%

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

   3. Simplified 81.9%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   if 2.8000000000000001e-39 < b_2

   1. Initial program 12.4%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 12.4%

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

   3. Simplified 12.4%

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

      1. pow1/2 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

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

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

      9. clear-num N/A

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

      10. flip-- N/A

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

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

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

      12. --lowering--.f64 N/A

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

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

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

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

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

      15. metadata-eval 12.3%

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

   6. Applied egg-rr 12.3%

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

   7. Taylor expanded in b_2 around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c + \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}\right), \color{blue}{b\_2}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot c\right), \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}\right)\right), b\_2\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(c \cdot \frac{-1}{2}\right), \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}\right)\right), b\_2\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}\right)\right), b\_2\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \left(\frac{\frac{-1}{8} \cdot \left(a \cdot {c}^{2}\right)}{{b\_2}^{2}}\right)\right), b\_2\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(a \cdot {c}^{2}\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(a \cdot {c}^{2}\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({c}^{2} \cdot a\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(c \cdot c\right) \cdot a\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(c \cdot \left(c \cdot a\right)\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(c \cdot \left(a \cdot c\right)\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, \left(a \cdot c\right)\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, \left(c \cdot a\right)\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, a\right)\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, a\right)\right)\right), \left(b\_2 \cdot b\_2\right)\right)\right), b\_2\right) \]

      16. *-lowering-*.f64 76.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, a\right)\right)\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right)\right), b\_2\right) \]

   9. Simplified 76.0%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5 + \frac{-0.125 \cdot \left(c \cdot \left(c \cdot a\right)\right)}{b\_2 \cdot b\_2}}{b\_2}} \]

   10. Taylor expanded in c around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(c \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}} - \frac{1}{2}\right)\right)}, b\_2\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}} - \frac{1}{2}\right)\right), b\_2\right) \]

       2. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), b\_2\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}} + \frac{-1}{2}\right)\right), b\_2\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \left(\frac{-1}{2} + \frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right), b\_2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right), b\_2\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{a \cdot c}{{b\_2}^{2}} \cdot \frac{-1}{8}\right)\right)\right), b\_2\right) \]

       7. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(a \cdot \frac{c}{{b\_2}^{2}}\right) \cdot \frac{-1}{8}\right)\right)\right), b\_2\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \left(a \cdot \left(\frac{c}{{b\_2}^{2}} \cdot \frac{-1}{8}\right)\right)\right)\right), b\_2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \left(\frac{c}{{b\_2}^{2}} \cdot \frac{-1}{8}\right)\right)\right)\right), b\_2\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{c}{{b\_2}^{2}}\right)\right)\right)\right), b\_2\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{8}, \left(\frac{c}{{b\_2}^{2}}\right)\right)\right)\right)\right), b\_2\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{8}, \left(\frac{c}{b\_2 \cdot b\_2}\right)\right)\right)\right)\right), b\_2\right) \]

       13. associate-/r* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{8}, \left(\frac{\frac{c}{b\_2}}{b\_2}\right)\right)\right)\right)\right), b\_2\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\left(\frac{c}{b\_2}\right), b\_2\right)\right)\right)\right)\right), b\_2\right) \]

       15. /-lowering-/.f64 91.7%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(c, b\_2\right), b\_2\right)\right)\right)\right)\right), b\_2\right) \]

   12. Simplified 91.7%

       \[\leadsto \frac{\color{blue}{c \cdot \left(-0.5 + a \cdot \left(-0.125 \cdot \frac{\frac{c}{b\_2}}{b\_2}\right)\right)}}{b\_2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 80.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-45}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c \cdot 0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.35 \cdot 10^{-50}:\\ \;\;\;\;\frac{{\left(\frac{-1}{a \cdot c}\right)}^{-0.5} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-0.5 + a \cdot \left(-0.125 \cdot \frac{\frac{c}{b\_2}}{b\_2}\right)\right)}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-45)
   (+ (* -2.0 (/ b_2 a)) (/ (* c 0.5) b_2))
   (if (<= b_2 1.35e-50)
     (/ (- (pow (/ -1.0 (* a c)) -0.5) b_2) a)
     (/ (* c (+ -0.5 (* a (* -0.125 (/ (/ c b_2) b_2))))) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-45) {
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	} else if (b_2 <= 1.35e-50) {
		tmp = (pow((-1.0 / (a * c)), -0.5) - b_2) / a;
	} else {
		tmp = (c * (-0.5 + (a * (-0.125 * ((c / b_2) / b_2))))) / 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 <= (-2d-45)) then
        tmp = ((-2.0d0) * (b_2 / a)) + ((c * 0.5d0) / b_2)
    else if (b_2 <= 1.35d-50) then
        tmp = ((((-1.0d0) / (a * c)) ** (-0.5d0)) - b_2) / a
    else
        tmp = (c * ((-0.5d0) + (a * ((-0.125d0) * ((c / b_2) / b_2))))) / 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 <= -2e-45) {
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	} else if (b_2 <= 1.35e-50) {
		tmp = (Math.pow((-1.0 / (a * c)), -0.5) - b_2) / a;
	} else {
		tmp = (c * (-0.5 + (a * (-0.125 * ((c / b_2) / b_2))))) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e-45:
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2)
	elif b_2 <= 1.35e-50:
		tmp = (math.pow((-1.0 / (a * c)), -0.5) - b_2) / a
	else:
		tmp = (c * (-0.5 + (a * (-0.125 * ((c / b_2) / b_2))))) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e-45)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c * 0.5) / b_2));
	elseif (b_2 <= 1.35e-50)
		tmp = Float64(Float64((Float64(-1.0 / Float64(a * c)) ^ -0.5) - b_2) / a);
	else
		tmp = Float64(Float64(c * Float64(-0.5 + Float64(a * Float64(-0.125 * Float64(Float64(c / b_2) / b_2))))) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2e-45)
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	elseif (b_2 <= 1.35e-50)
		tmp = (((-1.0 / (a * c)) ^ -0.5) - b_2) / a;
	else
		tmp = (c * (-0.5 + (a * (-0.125 * ((c / b_2) / b_2))))) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2e-45], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c * 0.5), $MachinePrecision] / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.35e-50], N[(N[(N[Power[N[(-1.0 / N[(a * c), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * N[(-0.5 + N[(a * N[(-0.125 * N[(N[(c / b$95$2), $MachinePrecision] / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.99999999999999997e-45

   1. Initial program 68.1%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 68.1%

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

   3. Simplified 68.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 N/A

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

      16. neg-sub0 N/A

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

      17. --lowering--.f64 92.3%

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

   7. Simplified 92.3%

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

   8. Taylor expanded in c around 0

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

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

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

      2. *-commutative N/A

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

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

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

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

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

      5. associate-*r/ N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 92.6%

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

   10. Simplified 92.6%

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

   if -1.99999999999999997e-45 < b_2 < 1.35e-50

   1. Initial program 76.5%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 76.5%

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

   3. Simplified 76.5%

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

      1. pow1/2 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

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

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

      9. clear-num N/A

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

      10. flip-- N/A

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

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

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

      12. --lowering--.f64 N/A

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

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

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

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

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

      15. metadata-eval 76.5%

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

   6. Applied egg-rr 76.5%

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

   7. Taylor expanded in b_2 around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 69.6%

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

   9. Simplified 69.6%

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

   if 1.35e-50 < b_2

   1. Initial program 13.2%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 13.2%

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

   3. Simplified 13.2%

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

      1. pow1/2 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

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

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

      9. clear-num N/A

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

      10. flip-- N/A

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

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

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

      12. --lowering--.f64 N/A

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

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

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

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

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

      15. metadata-eval 13.2%

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

   6. Applied egg-rr 13.2%

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

   7. Taylor expanded in b_2 around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c + \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}\right), \color{blue}{b\_2}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot c\right), \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}\right)\right), b\_2\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(c \cdot \frac{-1}{2}\right), \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}\right)\right), b\_2\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}\right)\right), b\_2\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \left(\frac{\frac{-1}{8} \cdot \left(a \cdot {c}^{2}\right)}{{b\_2}^{2}}\right)\right), b\_2\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(a \cdot {c}^{2}\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(a \cdot {c}^{2}\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({c}^{2} \cdot a\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(c \cdot c\right) \cdot a\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(c \cdot \left(c \cdot a\right)\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(c \cdot \left(a \cdot c\right)\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, \left(a \cdot c\right)\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, \left(c \cdot a\right)\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, a\right)\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, a\right)\right)\right), \left(b\_2 \cdot b\_2\right)\right)\right), b\_2\right) \]

      16. *-lowering-*.f64 75.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, a\right)\right)\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right)\right), b\_2\right) \]

   9. Simplified 75.5%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5 + \frac{-0.125 \cdot \left(c \cdot \left(c \cdot a\right)\right)}{b\_2 \cdot b\_2}}{b\_2}} \]

   10. Taylor expanded in c around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(c \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}} - \frac{1}{2}\right)\right)}, b\_2\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}} - \frac{1}{2}\right)\right), b\_2\right) \]

       2. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), b\_2\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}} + \frac{-1}{2}\right)\right), b\_2\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \left(\frac{-1}{2} + \frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right), b\_2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right), b\_2\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{a \cdot c}{{b\_2}^{2}} \cdot \frac{-1}{8}\right)\right)\right), b\_2\right) \]

       7. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(a \cdot \frac{c}{{b\_2}^{2}}\right) \cdot \frac{-1}{8}\right)\right)\right), b\_2\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \left(a \cdot \left(\frac{c}{{b\_2}^{2}} \cdot \frac{-1}{8}\right)\right)\right)\right), b\_2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \left(\frac{c}{{b\_2}^{2}} \cdot \frac{-1}{8}\right)\right)\right)\right), b\_2\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{c}{{b\_2}^{2}}\right)\right)\right)\right), b\_2\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{8}, \left(\frac{c}{{b\_2}^{2}}\right)\right)\right)\right)\right), b\_2\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{8}, \left(\frac{c}{b\_2 \cdot b\_2}\right)\right)\right)\right)\right), b\_2\right) \]

       13. associate-/r* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{8}, \left(\frac{\frac{c}{b\_2}}{b\_2}\right)\right)\right)\right)\right), b\_2\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\left(\frac{c}{b\_2}\right), b\_2\right)\right)\right)\right)\right), b\_2\right) \]

       15. /-lowering-/.f64 90.8%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(c, b\_2\right), b\_2\right)\right)\right)\right)\right), b\_2\right) \]

   12. Simplified 90.8%

       \[\leadsto \frac{\color{blue}{c \cdot \left(-0.5 + a \cdot \left(-0.125 \cdot \frac{\frac{c}{b\_2}}{b\_2}\right)\right)}}{b\_2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-45}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c \cdot 0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.35 \cdot 10^{-50}:\\ \;\;\;\;\frac{{\left(\frac{-1}{a \cdot c}\right)}^{-0.5} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-0.5 + a \cdot \left(-0.125 \cdot \frac{\frac{c}{b\_2}}{b\_2}\right)\right)}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.7 \cdot 10^{-34}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c \cdot 0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 3.8 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{0 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-0.5 + a \cdot \left(-0.125 \cdot \frac{\frac{c}{b\_2}}{b\_2}\right)\right)}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.7e-34)
   (+ (* -2.0 (/ b_2 a)) (/ (* c 0.5) b_2))
   (if (<= b_2 3.8e-49)
     (/ (- (sqrt (- 0.0 (* a c))) b_2) a)
     (/ (* c (+ -0.5 (* a (* -0.125 (/ (/ c b_2) b_2))))) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.7e-34) {
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	} else if (b_2 <= 3.8e-49) {
		tmp = (sqrt((0.0 - (a * c))) - b_2) / a;
	} else {
		tmp = (c * (-0.5 + (a * (-0.125 * ((c / b_2) / b_2))))) / 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 <= (-1.7d-34)) then
        tmp = ((-2.0d0) * (b_2 / a)) + ((c * 0.5d0) / b_2)
    else if (b_2 <= 3.8d-49) then
        tmp = (sqrt((0.0d0 - (a * c))) - b_2) / a
    else
        tmp = (c * ((-0.5d0) + (a * ((-0.125d0) * ((c / b_2) / b_2))))) / 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 <= -1.7e-34) {
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	} else if (b_2 <= 3.8e-49) {
		tmp = (Math.sqrt((0.0 - (a * c))) - b_2) / a;
	} else {
		tmp = (c * (-0.5 + (a * (-0.125 * ((c / b_2) / b_2))))) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.7e-34:
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2)
	elif b_2 <= 3.8e-49:
		tmp = (math.sqrt((0.0 - (a * c))) - b_2) / a
	else:
		tmp = (c * (-0.5 + (a * (-0.125 * ((c / b_2) / b_2))))) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.7e-34)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c * 0.5) / b_2));
	elseif (b_2 <= 3.8e-49)
		tmp = Float64(Float64(sqrt(Float64(0.0 - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(Float64(c * Float64(-0.5 + Float64(a * Float64(-0.125 * Float64(Float64(c / b_2) / b_2))))) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.7e-34)
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	elseif (b_2 <= 3.8e-49)
		tmp = (sqrt((0.0 - (a * c))) - b_2) / a;
	else
		tmp = (c * (-0.5 + (a * (-0.125 * ((c / b_2) / b_2))))) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.7e-34], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c * 0.5), $MachinePrecision] / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 3.8e-49], N[(N[(N[Sqrt[N[(0.0 - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * N[(-0.5 + N[(a * N[(-0.125 * N[(N[(c / b$95$2), $MachinePrecision] / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.7e-34

   1. Initial program 68.1%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 68.1%

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

   3. Simplified 68.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 N/A

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

      16. neg-sub0 N/A

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

      17. --lowering--.f64 92.3%

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

   7. Simplified 92.3%

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

   8. Taylor expanded in c around 0

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

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

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

      2. *-commutative N/A

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

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

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

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

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

      5. associate-*r/ N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 92.6%

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

   10. Simplified 92.6%

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

   if -1.7e-34 < b_2 < 3.7999999999999997e-49

   1. Initial program 76.5%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 76.5%

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

   3. Simplified 76.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 69.6%

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

   7. Simplified 69.6%

      \[\leadsto \frac{\sqrt{\color{blue}{0 - c \cdot a}} - b\_2}{a} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 69.6%

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

   9. Applied egg-rr 69.6%

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

   if 3.7999999999999997e-49 < b_2

   1. Initial program 13.2%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 13.2%

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

   3. Simplified 13.2%

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

      1. pow1/2 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

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

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

      9. clear-num N/A

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

      10. flip-- N/A

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

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

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

      12. --lowering--.f64 N/A

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

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

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

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

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

      15. metadata-eval 13.2%

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

   6. Applied egg-rr 13.2%

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

   7. Taylor expanded in b_2 around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c + \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}\right), \color{blue}{b\_2}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot c\right), \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}\right)\right), b\_2\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(c \cdot \frac{-1}{2}\right), \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}\right)\right), b\_2\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \left(\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}\right)\right), b\_2\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \left(\frac{\frac{-1}{8} \cdot \left(a \cdot {c}^{2}\right)}{{b\_2}^{2}}\right)\right), b\_2\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left(a \cdot {c}^{2}\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(a \cdot {c}^{2}\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left({c}^{2} \cdot a\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\left(c \cdot c\right) \cdot a\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(c \cdot \left(c \cdot a\right)\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(c \cdot \left(a \cdot c\right)\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, \left(a \cdot c\right)\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, \left(c \cdot a\right)\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, a\right)\right)\right), \left({b\_2}^{2}\right)\right)\right), b\_2\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, a\right)\right)\right), \left(b\_2 \cdot b\_2\right)\right)\right), b\_2\right) \]

      16. *-lowering-*.f64 75.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, a\right)\right)\right), \mathsf{*.f64}\left(b\_2, b\_2\right)\right)\right), b\_2\right) \]

   9. Simplified 75.5%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5 + \frac{-0.125 \cdot \left(c \cdot \left(c \cdot a\right)\right)}{b\_2 \cdot b\_2}}{b\_2}} \]

   10. Taylor expanded in c around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(c \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}} - \frac{1}{2}\right)\right)}, b\_2\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}} - \frac{1}{2}\right)\right), b\_2\right) \]

       2. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), b\_2\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}} + \frac{-1}{2}\right)\right), b\_2\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \left(\frac{-1}{2} + \frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right), b\_2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right), b\_2\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{a \cdot c}{{b\_2}^{2}} \cdot \frac{-1}{8}\right)\right)\right), b\_2\right) \]

       7. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\left(a \cdot \frac{c}{{b\_2}^{2}}\right) \cdot \frac{-1}{8}\right)\right)\right), b\_2\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \left(a \cdot \left(\frac{c}{{b\_2}^{2}} \cdot \frac{-1}{8}\right)\right)\right)\right), b\_2\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \left(\frac{c}{{b\_2}^{2}} \cdot \frac{-1}{8}\right)\right)\right)\right), b\_2\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{c}{{b\_2}^{2}}\right)\right)\right)\right), b\_2\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{8}, \left(\frac{c}{{b\_2}^{2}}\right)\right)\right)\right)\right), b\_2\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{8}, \left(\frac{c}{b\_2 \cdot b\_2}\right)\right)\right)\right)\right), b\_2\right) \]

       13. associate-/r* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{8}, \left(\frac{\frac{c}{b\_2}}{b\_2}\right)\right)\right)\right)\right), b\_2\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\left(\frac{c}{b\_2}\right), b\_2\right)\right)\right)\right)\right), b\_2\right) \]

       15. /-lowering-/.f64 90.8%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(c, b\_2\right), b\_2\right)\right)\right)\right)\right), b\_2\right) \]

   12. Simplified 90.8%

       \[\leadsto \frac{\color{blue}{c \cdot \left(-0.5 + a \cdot \left(-0.125 \cdot \frac{\frac{c}{b\_2}}{b\_2}\right)\right)}}{b\_2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.7 \cdot 10^{-34}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c \cdot 0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 3.8 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{0 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-0.5 + a \cdot \left(-0.125 \cdot \frac{\frac{c}{b\_2}}{b\_2}\right)\right)}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.0% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d-310)) then
        tmp = ((-2.0d0) * (b_2 / a)) + ((c * 0.5d0) / b_2)
    else
        tmp = (c * (-0.5d0)) / b_2
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 73.2%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 73.2%

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

   3. Simplified 73.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 N/A

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

      16. neg-sub0 N/A

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

      17. --lowering--.f64 65.8%

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

   7. Simplified 65.8%

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

   8. Taylor expanded in c around 0

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

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

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

      2. *-commutative N/A

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

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

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

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

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

      5. associate-*r/ N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 68.8%

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

   10. Simplified 68.8%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 29.4%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 29.4%

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

   3. Simplified 29.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   6. 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 68.5%

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

   7. Simplified 68.5%

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

4. Final simplification 68.6%

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

Alternative 6: 67.8% accurate, 11.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d-310)) then
        tmp = (b_2 * (-2.0d0)) / a
    else
        tmp = (c * (-0.5d0)) / b_2
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 73.2%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 73.2%

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

   3. Simplified 73.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

      7. metadata-eval N/A

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

      8. *-lowering-*.f64 67.9%

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

   7. Simplified 67.9%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 29.4%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 29.4%

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

   3. Simplified 29.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   6. 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 68.5%

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

   7. Simplified 68.5%

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

Alternative 7: 67.7% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 2e-309) (/ (* b_2 -2.0) a) (* c (/ -0.5 b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 2e-309) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = c * (-0.5 / b_2);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= 2d-309) then
        tmp = (b_2 * (-2.0d0)) / a
    else
        tmp = c * ((-0.5d0) / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 2e-309) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = c * (-0.5 / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 2e-309:
		tmp = (b_2 * -2.0) / a
	else:
		tmp = c * (-0.5 / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 2e-309)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = Float64(c * Float64(-0.5 / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 2e-309)
		tmp = (b_2 * -2.0) / a;
	else
		tmp = c * (-0.5 / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < 1.9999999999999988e-309

   1. Initial program 73.2%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 73.2%

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

   3. Simplified 73.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

      7. metadata-eval N/A

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

      8. *-lowering-*.f64 67.9%

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

   7. Simplified 67.9%

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

   if 1.9999999999999988e-309 < b_2

   1. Initial program 29.4%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 29.4%

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

   3. Simplified 29.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   6. 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 68.5%

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

   7. Simplified 68.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 68.3%

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

   9. Applied egg-rr 68.3%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.6% accurate, 11.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d-310)) then
        tmp = b_2 * ((-2.0d0) / a)
    else
        tmp = c * ((-0.5d0) / b_2)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 73.2%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 73.2%

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

   3. Simplified 73.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

      7. metadata-eval N/A

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

      8. *-lowering-*.f64 67.9%

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

   7. Simplified 67.9%

      \[\leadsto \color{blue}{\frac{b\_2 \cdot -2}{a}} \]
   8. 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 67.7%

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

   9. Applied egg-rr 67.7%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 29.4%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 29.4%

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

   3. Simplified 29.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   6. 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 68.5%

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

   7. Simplified 68.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 68.3%

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

   9. Applied egg-rr 68.3%

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

4. Final simplification 68.0%

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

Alternative 9: 35.6% accurate, 22.4× speedup

Math

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

FPCore

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

C

double code(double a, double b_2, double c) {
	return b_2 * (-2.0 / 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 = b_2 * ((-2.0d0) / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.9%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

   8. *-lowering-*.f64 49.9%

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

3. Simplified 49.9%

   \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
4. Add Preprocessing

5. Taylor expanded in b_2 around -inf

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

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

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

   4. *-commutative N/A

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

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

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

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

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

   7. metadata-eval N/A

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

   8. *-lowering-*.f64 33.2%

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

7. Simplified 33.2%

   \[\leadsto \color{blue}{\frac{b\_2 \cdot -2}{a}} \]
8. 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 33.1%

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

9. Applied egg-rr 33.1%

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

10. Final simplification 33.1%

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

Developer Target 1: 99.7% accurate, 0.1× speedup

Math

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

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 2024191 
(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))