quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.1% → 85.0%

Time: 11.5s

Alternatives: 8

Speedup: 11.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.1 \cdot 10^{-42}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.75 \cdot 10^{+77}:\\ \;\;\;\;\frac{\left(0 - b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c \cdot 0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5.1e-42)
   (/ (* c -0.5) b_2)
   (if (<= b_2 1.75e+77)
     (/ (- (- 0.0 b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (+ (* -2.0 (/ b_2 a)) (/ (* c 0.5) b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.1e-42) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 1.75e+77) {
		tmp = ((0.0 - b_2) - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5.1d-42)) then
        tmp = (c * (-0.5d0)) / b_2
    else if (b_2 <= 1.75d+77) then
        tmp = ((0.0d0 - b_2) - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((c * 0.5d0) / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.1e-42) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 1.75e+77) {
		tmp = ((0.0 - b_2) - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5.1e-42:
		tmp = (c * -0.5) / b_2
	elif b_2 <= 1.75e+77:
		tmp = ((0.0 - b_2) - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5.1e-42)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 1.75e+77)
		tmp = Float64(Float64(Float64(0.0 - b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c * 0.5) / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5.1e-42)
		tmp = (c * -0.5) / b_2;
	elseif (b_2 <= 1.75e+77)
		tmp = ((0.0 - b_2) - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5.1e-42], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.75e+77], N[(N[(N[(0.0 - b$95$2), $MachinePrecision] - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c * 0.5), $MachinePrecision] / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -5.1e-42

   1. Initial program 12.6%

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

   3. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 89.6%

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

   5. Simplified 89.6%

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

   if -5.1e-42 < b_2 < 1.7500000000000001e77

   1. Initial program 80.4%

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

   if 1.7500000000000001e77 < b_2

   1. Initial program 58.8%

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

   3. Taylor expanded in c around 0

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 97.1%

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

   5. Simplified 97.1%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.1 \cdot 10^{-42}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.75 \cdot 10^{+77}:\\ \;\;\;\;\frac{\left(0 - b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c \cdot 0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.7 \cdot 10^{-44}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.08 \cdot 10^{-80}:\\ \;\;\;\;\frac{\left(0 - b\_2\right) - \sqrt{0 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c \cdot 0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.7e-44)
   (/ (* c -0.5) b_2)
   (if (<= b_2 1.08e-80)
     (/ (- (- 0.0 b_2) (sqrt (- 0.0 (* c a)))) a)
     (+ (* -2.0 (/ b_2 a)) (/ (* c 0.5) b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.7e-44) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 1.08e-80) {
		tmp = ((0.0 - b_2) - sqrt((0.0 - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-3.7d-44)) then
        tmp = (c * (-0.5d0)) / b_2
    else if (b_2 <= 1.08d-80) then
        tmp = ((0.0d0 - b_2) - sqrt((0.0d0 - (c * a)))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((c * 0.5d0) / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.7e-44) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 1.08e-80) {
		tmp = ((0.0 - b_2) - Math.sqrt((0.0 - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.7e-44:
		tmp = (c * -0.5) / b_2
	elif b_2 <= 1.08e-80:
		tmp = ((0.0 - b_2) - math.sqrt((0.0 - (c * a)))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.7e-44)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 1.08e-80)
		tmp = Float64(Float64(Float64(0.0 - b_2) - sqrt(Float64(0.0 - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c * 0.5) / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.7e-44)
		tmp = (c * -0.5) / b_2;
	elseif (b_2 <= 1.08e-80)
		tmp = ((0.0 - b_2) - sqrt((0.0 - (c * a)))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.7e-44], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.08e-80], N[(N[(N[(0.0 - b$95$2), $MachinePrecision] - N[Sqrt[N[(0.0 - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c * 0.5), $MachinePrecision] / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.7e-44

   1. Initial program 12.6%

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

   3. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 89.6%

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

   5. Simplified 89.6%

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

   if -3.7e-44 < b_2 < 1.07999999999999996e-80

   1. Initial program 76.5%

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

   3. Taylor expanded in b_2 around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-lowering-*.f64 72.2%

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

   5. Simplified 72.2%

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

   if 1.07999999999999996e-80 < b_2

   1. Initial program 69.3%

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

   3. Taylor expanded in c around 0

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 87.8%

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

   5. Simplified 87.8%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.7 \cdot 10^{-44}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.08 \cdot 10^{-80}:\\ \;\;\;\;\frac{\left(0 - b\_2\right) - \sqrt{0 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c \cdot 0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.0% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-310)
   (/ (* c -0.5) b_2)
   (+ (* -2.0 (/ b_2 a)) (/ (* c 0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-310) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1d-310)) then
        tmp = (c * (-0.5d0)) / b_2
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((c * 0.5d0) / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-310) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e-310:
		tmp = (c * -0.5) / b_2
	else:
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-310)
		tmp = Float64(Float64(c * -0.5) / b_2);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c * 0.5) / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e-310)
		tmp = (c * -0.5) / b_2;
	else
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e-310], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c * 0.5), $MachinePrecision] / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -9.999999999999969e-311

   1. Initial program 33.5%

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

   3. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 65.6%

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

   5. Simplified 65.6%

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

   if -9.999999999999969e-311 < b_2

   1. Initial program 71.5%

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

   3. Taylor expanded in c around 0

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 68.5%

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

   5. Simplified 68.5%

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

Alternative 4: 66.9% accurate, 11.2× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-310) (/ (* c -0.5) b_2) (/ (* b_2 -2.0) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-310) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1d-310)) then
        tmp = (c * (-0.5d0)) / b_2
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-310) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e-310:
		tmp = (c * -0.5) / b_2
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-310)
		tmp = Float64(Float64(c * -0.5) / b_2);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e-310)
		tmp = (c * -0.5) / b_2;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e-310], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -9.999999999999969e-311

   1. Initial program 33.5%

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

   3. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 65.6%

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

   5. Simplified 65.6%

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

   if -9.999999999999969e-311 < b_2

   1. Initial program 71.5%

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

   3. Taylor expanded in b_2 around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 68.4%

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

   5. Simplified 68.4%

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

Alternative 5: 66.8% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -6 \cdot 10^{-309}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -6e-309) (* c (/ -0.5 b_2)) (/ (* b_2 -2.0) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6e-309) {
		tmp = c * (-0.5 / b_2);
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-6d-309)) then
        tmp = c * ((-0.5d0) / b_2)
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6e-309) {
		tmp = c * (-0.5 / b_2);
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -6e-309:
		tmp = c * (-0.5 / b_2)
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -6e-309)
		tmp = Float64(c * Float64(-0.5 / b_2));
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -6e-309)
		tmp = c * (-0.5 / b_2);
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -6e-309], N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -6.000000000000001e-309

   1. Initial program 33.5%

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

   3. Taylor expanded in c around 0

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 2.4%

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

   5. Simplified 2.4%

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

   6. Taylor expanded in b_2 around 0

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

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

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

      2. /-lowering-/.f64 19.2%

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

   8. Simplified 19.2%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

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

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

      7. metadata-eval N/A

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

      8. associate-/r* N/A

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

      9. neg-mul-1 N/A

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

      10. neg-sub0 N/A

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

      11. flip3-- N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{0}^{3} - {b\_2}^{3}}{\color{blue}{0 \cdot 0 + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{0 - {b\_2}^{3}}{\color{blue}{0} \cdot 0 + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

      13. sub0-neg N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{\mathsf{neg}\left({b\_2}^{3}\right)}{\color{blue}{0 \cdot 0} + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

      14. cube-neg N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{\left(\mathsf{neg}\left(b\_2\right)\right)}^{3}}{\color{blue}{0 \cdot 0} + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{\left(\mathsf{neg}\left(b\_2\right)\right)}^{\left(2 \cdot \frac{3}{2}\right)}}{0 \cdot \color{blue}{0} + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{\left(\mathsf{neg}\left(b\_2\right)\right)}^{\left(2 \cdot \left(\frac{1}{2} \cdot 3\right)\right)}}{0 \cdot 0 + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

      17. pow-pow N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{\left({\left(\mathsf{neg}\left(b\_2\right)\right)}^{2}\right)}^{\left(\frac{1}{2} \cdot 3\right)}}{\color{blue}{0 \cdot 0} + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

      18. pow2 N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{\left(\left(\mathsf{neg}\left(b\_2\right)\right) \cdot \left(\mathsf{neg}\left(b\_2\right)\right)\right)}^{\left(\frac{1}{2} \cdot 3\right)}}{\color{blue}{0} \cdot 0 + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

      19. sqr-neg N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{\left(b\_2 \cdot b\_2\right)}^{\left(\frac{1}{2} \cdot 3\right)}}{\color{blue}{0} \cdot 0 + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

      20. pow2 N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{\left({b\_2}^{2}\right)}^{\left(\frac{1}{2} \cdot 3\right)}}{\color{blue}{0} \cdot 0 + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

      21. pow-pow N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{b\_2}^{\left(2 \cdot \left(\frac{1}{2} \cdot 3\right)\right)}}{\color{blue}{0 \cdot 0} + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

      22. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{b\_2}^{\left(2 \cdot \frac{3}{2}\right)}}{0 \cdot 0 + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{b\_2}^{3}}{0 \cdot \color{blue}{0} + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

      24. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{b\_2}^{3}}{0 + \left(\color{blue}{b\_2 \cdot b\_2} + 0 \cdot b\_2\right)}}\right)\right) \]

      25. +-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{b\_2}^{3}}{b\_2 \cdot b\_2 + \color{blue}{0 \cdot b\_2}}}\right)\right) \]

      26. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{b\_2}^{3}}{b\_2 \cdot \color{blue}{\left(b\_2 + 0\right)}}}\right)\right) \]

      27. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{b\_2}^{3}}{b\_2 \cdot \left(0 + \color{blue}{b\_2}\right)}}\right)\right) \]

      28. +-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{b\_2}^{3}}{b\_2 \cdot b\_2}}\right)\right) \]

      29. pow2 N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{b\_2}^{3}}{{b\_2}^{\color{blue}{2}}}}\right)\right) \]

   10. Applied egg-rr 65.4%

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

   if -6.000000000000001e-309 < b_2

   1. Initial program 71.5%

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

   3. Taylor expanded in b_2 around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 68.4%

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

   5. Simplified 68.4%

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

Alternative 6: 66.7% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.4 \cdot 10^{-308}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{a}{b\_2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.4e-308) (* c (/ -0.5 b_2)) (/ -2.0 (/ a b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.4e-308) {
		tmp = c * (-0.5 / b_2);
	} else {
		tmp = -2.0 / (a / b_2);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-4.4d-308)) then
        tmp = c * ((-0.5d0) / b_2)
    else
        tmp = (-2.0d0) / (a / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.4e-308) {
		tmp = c * (-0.5 / b_2);
	} else {
		tmp = -2.0 / (a / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4.4e-308:
		tmp = c * (-0.5 / b_2)
	else:
		tmp = -2.0 / (a / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4.4e-308)
		tmp = Float64(c * Float64(-0.5 / b_2));
	else
		tmp = Float64(-2.0 / Float64(a / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4.4e-308)
		tmp = c * (-0.5 / b_2);
	else
		tmp = -2.0 / (a / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4.4e-308], N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(a / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.3999999999999999e-308

   1. Initial program 33.5%

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

   3. Taylor expanded in c around 0

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 2.4%

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

   5. Simplified 2.4%

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

   6. Taylor expanded in b_2 around 0

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

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

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

      2. /-lowering-/.f64 19.2%

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

   8. Simplified 19.2%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

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

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

      7. metadata-eval N/A

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

      8. associate-/r* N/A

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

      9. neg-mul-1 N/A

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

      10. neg-sub0 N/A

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

      11. flip3-- N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{0}^{3} - {b\_2}^{3}}{\color{blue}{0 \cdot 0 + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{0 - {b\_2}^{3}}{\color{blue}{0} \cdot 0 + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

      13. sub0-neg N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{\mathsf{neg}\left({b\_2}^{3}\right)}{\color{blue}{0 \cdot 0} + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

      14. cube-neg N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{\left(\mathsf{neg}\left(b\_2\right)\right)}^{3}}{\color{blue}{0 \cdot 0} + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{\left(\mathsf{neg}\left(b\_2\right)\right)}^{\left(2 \cdot \frac{3}{2}\right)}}{0 \cdot \color{blue}{0} + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{\left(\mathsf{neg}\left(b\_2\right)\right)}^{\left(2 \cdot \left(\frac{1}{2} \cdot 3\right)\right)}}{0 \cdot 0 + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

      17. pow-pow N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{\left({\left(\mathsf{neg}\left(b\_2\right)\right)}^{2}\right)}^{\left(\frac{1}{2} \cdot 3\right)}}{\color{blue}{0 \cdot 0} + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

      18. pow2 N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{\left(\left(\mathsf{neg}\left(b\_2\right)\right) \cdot \left(\mathsf{neg}\left(b\_2\right)\right)\right)}^{\left(\frac{1}{2} \cdot 3\right)}}{\color{blue}{0} \cdot 0 + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

      19. sqr-neg N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{\left(b\_2 \cdot b\_2\right)}^{\left(\frac{1}{2} \cdot 3\right)}}{\color{blue}{0} \cdot 0 + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

      20. pow2 N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{\left({b\_2}^{2}\right)}^{\left(\frac{1}{2} \cdot 3\right)}}{\color{blue}{0} \cdot 0 + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

      21. pow-pow N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{b\_2}^{\left(2 \cdot \left(\frac{1}{2} \cdot 3\right)\right)}}{\color{blue}{0 \cdot 0} + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

      22. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{b\_2}^{\left(2 \cdot \frac{3}{2}\right)}}{0 \cdot 0 + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{b\_2}^{3}}{0 \cdot \color{blue}{0} + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

      24. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{b\_2}^{3}}{0 + \left(\color{blue}{b\_2 \cdot b\_2} + 0 \cdot b\_2\right)}}\right)\right) \]

      25. +-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{b\_2}^{3}}{b\_2 \cdot b\_2 + \color{blue}{0 \cdot b\_2}}}\right)\right) \]

      26. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{b\_2}^{3}}{b\_2 \cdot \color{blue}{\left(b\_2 + 0\right)}}}\right)\right) \]

      27. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{b\_2}^{3}}{b\_2 \cdot \left(0 + \color{blue}{b\_2}\right)}}\right)\right) \]

      28. +-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{b\_2}^{3}}{b\_2 \cdot b\_2}}\right)\right) \]

      29. pow2 N/A

          \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{b\_2}^{3}}{{b\_2}^{\color{blue}{2}}}}\right)\right) \]

   10. Applied egg-rr 65.4%

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

   if -4.3999999999999999e-308 < b_2

   1. Initial program 71.5%

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

   3. Taylor expanded in b_2 around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 68.4%

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

   5. Simplified 68.4%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

      6. /-lowering-/.f64 68.1%

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

   7. Applied egg-rr 68.1%

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

Alternative 7: 34.6% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{-0.5}{b\_2} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* c (/ -0.5 b_2)))

C

double code(double a, double b_2, double c) {
	return c * (-0.5 / b_2);
}

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 = c * ((-0.5d0) / b_2)
end function

Java

public static double code(double a, double b_2, double c) {
	return c * (-0.5 / b_2);
}

Python

def code(a, b_2, c):
	return c * (-0.5 / b_2)

Julia

function code(a, b_2, c)
	return Float64(c * Float64(-0.5 / b_2))
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = c * (-0.5 / b_2);
end

Wolfram

code[a_, b$95$2_, c_] := N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.5%

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

3. Taylor expanded in c around 0

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

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

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

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

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

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

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

   4. associate-*r/ N/A

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

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

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

   6. *-commutative N/A

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

   7. *-lowering-*.f64 37.3%

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

5. Simplified 37.3%

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

6. Taylor expanded in b_2 around 0

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

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

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

   2. /-lowering-/.f64 11.0%

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

8. Simplified 11.0%

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

   1. *-commutative N/A

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

   2. div-inv N/A

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

   3. associate-*l* N/A

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

   4. associate-/r/ N/A

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

   5. clear-num N/A

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

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

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

   7. metadata-eval N/A

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

   8. associate-/r* N/A

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

   9. neg-mul-1 N/A

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

   10. neg-sub0 N/A

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

   11. flip3-- N/A

       \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{0}^{3} - {b\_2}^{3}}{\color{blue}{0 \cdot 0 + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{0 - {b\_2}^{3}}{\color{blue}{0} \cdot 0 + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

   13. sub0-neg N/A

       \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{\mathsf{neg}\left({b\_2}^{3}\right)}{\color{blue}{0 \cdot 0} + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

   14. cube-neg N/A

       \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{\left(\mathsf{neg}\left(b\_2\right)\right)}^{3}}{\color{blue}{0 \cdot 0} + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{\left(\mathsf{neg}\left(b\_2\right)\right)}^{\left(2 \cdot \frac{3}{2}\right)}}{0 \cdot \color{blue}{0} + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{\left(\mathsf{neg}\left(b\_2\right)\right)}^{\left(2 \cdot \left(\frac{1}{2} \cdot 3\right)\right)}}{0 \cdot 0 + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

   17. pow-pow N/A

       \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{\left({\left(\mathsf{neg}\left(b\_2\right)\right)}^{2}\right)}^{\left(\frac{1}{2} \cdot 3\right)}}{\color{blue}{0 \cdot 0} + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

   18. pow2 N/A

       \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{\left(\left(\mathsf{neg}\left(b\_2\right)\right) \cdot \left(\mathsf{neg}\left(b\_2\right)\right)\right)}^{\left(\frac{1}{2} \cdot 3\right)}}{\color{blue}{0} \cdot 0 + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

   19. sqr-neg N/A

       \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{\left(b\_2 \cdot b\_2\right)}^{\left(\frac{1}{2} \cdot 3\right)}}{\color{blue}{0} \cdot 0 + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

   20. pow2 N/A

       \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{\left({b\_2}^{2}\right)}^{\left(\frac{1}{2} \cdot 3\right)}}{\color{blue}{0} \cdot 0 + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

   21. pow-pow N/A

       \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{b\_2}^{\left(2 \cdot \left(\frac{1}{2} \cdot 3\right)\right)}}{\color{blue}{0 \cdot 0} + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

   22. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{b\_2}^{\left(2 \cdot \frac{3}{2}\right)}}{0 \cdot 0 + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

   23. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{b\_2}^{3}}{0 \cdot \color{blue}{0} + \left(b\_2 \cdot b\_2 + 0 \cdot b\_2\right)}}\right)\right) \]

   24. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{b\_2}^{3}}{0 + \left(\color{blue}{b\_2 \cdot b\_2} + 0 \cdot b\_2\right)}}\right)\right) \]

   25. +-lft-identity N/A

       \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{b\_2}^{3}}{b\_2 \cdot b\_2 + \color{blue}{0 \cdot b\_2}}}\right)\right) \]

   26. distribute-rgt-out N/A

       \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{b\_2}^{3}}{b\_2 \cdot \color{blue}{\left(b\_2 + 0\right)}}}\right)\right) \]

   27. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{b\_2}^{3}}{b\_2 \cdot \left(0 + \color{blue}{b\_2}\right)}}\right)\right) \]

   28. +-lft-identity N/A

       \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{b\_2}^{3}}{b\_2 \cdot b\_2}}\right)\right) \]

   29. pow2 N/A

       \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{\frac{-1}{2}}{\frac{{b\_2}^{3}}{{b\_2}^{\color{blue}{2}}}}\right)\right) \]

10. Applied egg-rr 32.0%

    \[\leadsto \color{blue}{c \cdot \frac{-0.5}{b\_2}} \]
11. Add Preprocessing

Alternative 8: 10.8% accurate, 112.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.5%

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

   1. flip-- N/A

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

   2. clear-num N/A

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

   3. sqrt-div N/A

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

   4. metadata-eval N/A

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

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

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

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

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

   7. clear-num N/A

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

   8. flip-- N/A

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

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

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

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

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

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

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

   12. *-lowering-*.f64 53.0%

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

4. Applied egg-rr 53.0%

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

5. Taylor expanded in b_2 around inf

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

   1. /-lowering-/.f64 37.4%

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

7. Simplified 37.4%

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

   1. div-sub N/A

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

   2. frac-2neg N/A

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

   3. neg-mul-1 N/A

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

   4. remove-double-div N/A

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

   5. un-div-inv N/A

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

   6. distribute-frac-neg2 N/A

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

   7. metadata-eval N/A

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

   8. frac-2neg N/A

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

   9. remove-double-div N/A

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

   10. remove-double-div N/A

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

   11. frac-2neg N/A

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

   12. sub-div N/A

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

9. Applied egg-rr 11.2%

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

    1. div0 11.2%

       \[\leadsto 0 \]

11. Applied egg-rr 11.2%

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

Developer Target 1: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024140 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  :herbie-expected 10

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

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