quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.0% → 85.4%

Time: 12.3s

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: 52.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.6 \cdot 10^{-50}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 4.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7.6e-50)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 4.5e+102)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (/ (* b_2 -2.0) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.6e-50) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 4.5e+102) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-7.6d-50)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 4.5d+102) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.6e-50) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 4.5e+102) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -7.6e-50:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 4.5e+102:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -7.6e-50)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 4.5e+102)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -7.6e-50)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 4.5e+102)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7.6e-50], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 4.5e+102], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -7.5999999999999998e-50

   1. Initial program 15.3%

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

   3. Taylor expanded in b_2 around -inf 87.2%

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

      1. associate-*r/ 87.2%

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

   5. Simplified 87.2%

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

   if -7.5999999999999998e-50 < b_2 < 4.50000000000000021e102

   1. Initial program 84.6%

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

   if 4.50000000000000021e102 < b_2

   1. Initial program 50.5%

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

   3. Taylor expanded in b_2 around inf 98.5%

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

      1. *-commutative 98.5%

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

   5. Simplified 98.5%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.6 \cdot 10^{-50}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 4.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4 \cdot 10^{-50}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.45 \cdot 10^{-85}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e-50)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1.45e-85)
     (/ (- (- b_2) (sqrt (* a (- c)))) a)
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-50) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.45e-85) {
		tmp = (-b_2 - sqrt((a * -c))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-4d-50)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 1.45d-85) then
        tmp = (-b_2 - sqrt((a * -c))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-50) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.45e-85) {
		tmp = (-b_2 - Math.sqrt((a * -c))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4e-50:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 1.45e-85:
		tmp = (-b_2 - math.sqrt((a * -c))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4e-50)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1.45e-85)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(a * Float64(-c)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4e-50)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 1.45e-85)
		tmp = (-b_2 - sqrt((a * -c))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4e-50], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.45e-85], N[(N[((-b$95$2) - N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.00000000000000003e-50

   1. Initial program 15.3%

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

   3. Taylor expanded in b_2 around -inf 87.2%

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

      1. associate-*r/ 87.2%

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

   5. Simplified 87.2%

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

   if -4.00000000000000003e-50 < b_2 < 1.4500000000000001e-85

   1. Initial program 79.5%

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

   3. Taylor expanded in b_2 around 0 74.1%

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

      1. mul-1-neg 74.1%

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

      2. distribute-rgt-neg-out 74.1%

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

   5. Simplified 74.1%

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

   if 1.4500000000000001e-85 < b_2

   1. Initial program 68.0%

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

   3. Taylor expanded in c around 0 85.6%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -4 \cdot 10^{-50}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.45 \cdot 10^{-85}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.6% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 31.7%

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

   3. Taylor expanded in b_2 around -inf 69.0%

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

      1. associate-*r/ 69.0%

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

   5. Simplified 69.0%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 73.0%

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

   3. Taylor expanded in c around 0 64.8%

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

4. Final simplification 66.7%

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

Alternative 4: 67.2% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.6 \cdot 10^{-289}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.6e-289) (* 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 <= -2.6e-289) {
		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 <= (-2.6d-289)) 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 <= -2.6e-289) {
		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 <= -2.6e-289:
		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 <= -2.6e-289)
		tmp = Float64(c * Float64(-0.5 / b_2));
	else
		tmp = Float64(b_2 * Float64(-2.0 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.6e-289)
		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, -2.6e-289], N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.5999999999999999e-289

   1. Initial program 31.1%

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

      1. div-sub 29.8%

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

      2. neg-mul-1 29.8%

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

      3. associate-/l* 29.8%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-prod 26.3%

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

      6. sqr-neg 26.3%

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

      7. sqrt-unprod 26.3%

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

      8. add-sqr-sqrt 26.3%

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

      9. fma-neg 26.3%

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

      10. add-sqr-sqrt 26.3%

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

      11. sqrt-unprod 26.3%

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

      12. sqr-neg 26.3%

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

      13. sqrt-prod 0.0%

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

      14. add-sqr-sqrt 29.8%

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

   4. Applied egg-rr 32.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{b\_2}{a}, -\frac{\mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)}{a}\right)} \]
   5. Step-by-step derivation

      1. fma-undefine 32.8%

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

      2. unsub-neg 32.8%

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

      3. mul-1-neg 32.8%

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

      4. distribute-frac-neg2 32.8%

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

   6. Simplified 32.8%

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

   7. Taylor expanded in b_2 around -inf 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b\_2}} \]
   8. Step-by-step derivation

      1. *-commutative 0.0%

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

      2. associate-/l* 0.0%

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

      3. associate-*r* 0.0%

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

      4. *-commutative 0.0%

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

      5. associate-*r/ 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 69.3%

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

      8. metadata-eval 69.3%

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

   9. Simplified 69.3%

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

   if -2.5999999999999999e-289 < b_2

   1. Initial program 73.2%

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

      1. add-cbrt-cube 57.3%

         \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\sqrt[3]{\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} \cdot \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right) \cdot \sqrt{b\_2 \cdot b\_2 - a \cdot c}}}}{a} \]

      2. pow1/3 54.8%

         \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{{\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} \cdot \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right) \cdot \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right)}^{0.3333333333333333}}}{a} \]

      3. pow3 54.8%

         \[\leadsto \frac{\left(-b\_2\right) - {\color{blue}{\left({\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c}\right)}^{3}\right)}}^{0.3333333333333333}}{a} \]

      4. sqrt-pow2 54.8%

         \[\leadsto \frac{\left(-b\_2\right) - {\color{blue}{\left({\left(b\_2 \cdot b\_2 - a \cdot c\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333}}{a} \]

      5. pow2 54.8%

         \[\leadsto \frac{\left(-b\_2\right) - {\left({\left(\color{blue}{{b\_2}^{2}} - a \cdot c\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333}}{a} \]

      6. metadata-eval 54.8%

         \[\leadsto \frac{\left(-b\_2\right) - {\left({\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}}{a} \]

   4. Applied egg-rr 54.8%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{1.5}\right)}^{0.3333333333333333}}}{a} \]
   5. Step-by-step derivation

      1. unpow1/3 57.3%

         \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\sqrt[3]{{\left({b\_2}^{2} - a \cdot c\right)}^{1.5}}}}{a} \]

   6. Simplified 57.3%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\sqrt[3]{{\left({b\_2}^{2} - a \cdot c\right)}^{1.5}}}}{a} \]

   7. Taylor expanded in b_2 around inf 64.3%

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

      1. metadata-eval 64.3%

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

      2. distribute-lft-neg-in 64.3%

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

      3. associate-*r/ 64.3%

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

      4. *-commutative 64.3%

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

      5. associate-*r/ 64.1%

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

      6. metadata-eval 64.1%

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

      7. associate-*r/ 64.1%

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

      8. distribute-rgt-neg-in 64.1%

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

      9. associate-*r/ 64.1%

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

      10. metadata-eval 64.1%

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

      11. distribute-neg-frac 64.1%

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

      12. metadata-eval 64.1%

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

   9. Simplified 64.1%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.6 \cdot 10^{-289}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.4% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.6 \cdot 10^{-289}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.6e-289) (/ (* -0.5 c) b_2) (* b_2 (/ -2.0 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.6e-289) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = b_2 * (-2.0 / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.6d-289)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = b_2 * ((-2.0d0) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.6e-289) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = b_2 * (-2.0 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.6e-289:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = b_2 * (-2.0 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.6e-289)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(b_2 * Float64(-2.0 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.6e-289)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = b_2 * (-2.0 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.6e-289], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.5999999999999999e-289

   1. Initial program 31.1%

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

   3. Taylor expanded in b_2 around -inf 69.5%

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

      1. associate-*r/ 69.6%

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

   5. Simplified 69.6%

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

   if -2.5999999999999999e-289 < b_2

   1. Initial program 73.2%

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

      1. add-cbrt-cube 57.3%

         \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\sqrt[3]{\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} \cdot \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right) \cdot \sqrt{b\_2 \cdot b\_2 - a \cdot c}}}}{a} \]

      2. pow1/3 54.8%

         \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{{\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} \cdot \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right) \cdot \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right)}^{0.3333333333333333}}}{a} \]

      3. pow3 54.8%

         \[\leadsto \frac{\left(-b\_2\right) - {\color{blue}{\left({\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c}\right)}^{3}\right)}}^{0.3333333333333333}}{a} \]

      4. sqrt-pow2 54.8%

         \[\leadsto \frac{\left(-b\_2\right) - {\color{blue}{\left({\left(b\_2 \cdot b\_2 - a \cdot c\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333}}{a} \]

      5. pow2 54.8%

         \[\leadsto \frac{\left(-b\_2\right) - {\left({\left(\color{blue}{{b\_2}^{2}} - a \cdot c\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333}}{a} \]

      6. metadata-eval 54.8%

         \[\leadsto \frac{\left(-b\_2\right) - {\left({\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}}{a} \]

   4. Applied egg-rr 54.8%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{1.5}\right)}^{0.3333333333333333}}}{a} \]
   5. Step-by-step derivation

      1. unpow1/3 57.3%

         \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\sqrt[3]{{\left({b\_2}^{2} - a \cdot c\right)}^{1.5}}}}{a} \]

   6. Simplified 57.3%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\sqrt[3]{{\left({b\_2}^{2} - a \cdot c\right)}^{1.5}}}}{a} \]

   7. Taylor expanded in b_2 around inf 64.3%

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

      1. metadata-eval 64.3%

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

      2. distribute-lft-neg-in 64.3%

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

      3. associate-*r/ 64.3%

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

      4. *-commutative 64.3%

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

      5. associate-*r/ 64.1%

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

      6. metadata-eval 64.1%

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

      7. associate-*r/ 64.1%

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

      8. distribute-rgt-neg-in 64.1%

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

      9. associate-*r/ 64.1%

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

      10. metadata-eval 64.1%

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

      11. distribute-neg-frac 64.1%

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

      12. metadata-eval 64.1%

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

   9. Simplified 64.1%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.6 \cdot 10^{-289}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.4% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.6 \cdot 10^{-289}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.6e-289) (/ (* -0.5 c) b_2) (/ (* b_2 -2.0) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.6e-289) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.6d-289)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.6e-289) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.6e-289:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.6e-289)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.6e-289)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.6e-289], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.5999999999999999e-289

   1. Initial program 31.1%

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

   3. Taylor expanded in b_2 around -inf 69.5%

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

      1. associate-*r/ 69.6%

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

   5. Simplified 69.6%

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

   if -2.5999999999999999e-289 < b_2

   1. Initial program 73.2%

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

   3. Taylor expanded in b_2 around inf 64.3%

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

      1. *-commutative 64.3%

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

   5. Simplified 64.3%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.6 \cdot 10^{-289}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.9% 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 54.1%

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

   1. add-cbrt-cube 43.0%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\sqrt[3]{\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} \cdot \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right) \cdot \sqrt{b\_2 \cdot b\_2 - a \cdot c}}}}{a} \]

   2. pow1/3 40.4%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{{\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} \cdot \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right) \cdot \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right)}^{0.3333333333333333}}}{a} \]

   3. pow3 40.4%

      \[\leadsto \frac{\left(-b\_2\right) - {\color{blue}{\left({\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c}\right)}^{3}\right)}}^{0.3333333333333333}}{a} \]

   4. sqrt-pow2 40.4%

      \[\leadsto \frac{\left(-b\_2\right) - {\color{blue}{\left({\left(b\_2 \cdot b\_2 - a \cdot c\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333}}{a} \]

   5. pow2 40.4%

      \[\leadsto \frac{\left(-b\_2\right) - {\left({\left(\color{blue}{{b\_2}^{2}} - a \cdot c\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333}}{a} \]

   6. metadata-eval 40.4%

      \[\leadsto \frac{\left(-b\_2\right) - {\left({\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}}{a} \]

4. Applied egg-rr 40.4%

   \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{1.5}\right)}^{0.3333333333333333}}}{a} \]
5. Step-by-step derivation

   1. unpow1/3 43.0%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\sqrt[3]{{\left({b\_2}^{2} - a \cdot c\right)}^{1.5}}}}{a} \]

6. Simplified 43.0%

   \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\sqrt[3]{{\left({b\_2}^{2} - a \cdot c\right)}^{1.5}}}}{a} \]

7. Taylor expanded in b_2 around inf 36.3%

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

   1. metadata-eval 36.3%

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

   2. distribute-lft-neg-in 36.3%

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

   3. associate-*r/ 36.3%

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

   4. *-commutative 36.3%

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

   5. associate-*r/ 36.2%

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

   6. metadata-eval 36.2%

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

   7. associate-*r/ 36.2%

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

   8. distribute-rgt-neg-in 36.2%

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

   9. associate-*r/ 36.2%

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

   10. metadata-eval 36.2%

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

   11. distribute-neg-frac 36.2%

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

   12. metadata-eval 36.2%

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

9. Simplified 36.2%

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

10. Final simplification 36.2%

    \[\leadsto b\_2 \cdot \frac{-2}{a} \]
11. Add Preprocessing

Alternative 8: 15.2% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ \frac{-b\_2}{a} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.1%

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

   1. prod-diff 53.9%

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

   2. *-commutative 53.9%

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

   3. fma-neg 53.9%

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

   4. prod-diff 53.9%

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

   5. *-commutative 53.9%

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

   6. fma-neg 53.9%

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

   7. associate-+l+ 53.9%

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

   8. pow2 53.9%

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

   9. *-commutative 53.9%

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

   10. fma-undefine 53.9%

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

   11. distribute-lft-neg-in 53.9%

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

   12. *-commutative 53.9%

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

   13. distribute-rgt-neg-in 53.9%

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

   14. fma-define 53.9%

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

   15. *-commutative 53.9%

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

   16. fma-undefine 53.9%

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

   17. distribute-lft-neg-in 53.9%

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

   18. *-commutative 53.9%

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

   19. distribute-rgt-neg-in 53.9%

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

4. Applied egg-rr 53.9%

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

   1. count-2 53.9%

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

6. Simplified 53.9%

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

7. Taylor expanded in c around inf 19.2%

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

   1. mul-1-neg 19.2%

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

   2. *-commutative 19.2%

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

   3. distribute-rgt1-in 19.2%

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

   4. metadata-eval 19.2%

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

9. Simplified 19.2%

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

10. Taylor expanded in b_2 around inf 15.0%

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

    1. associate-*r/ 15.0%

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

    2. mul-1-neg 15.0%

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

12. Simplified 15.0%

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

13. Final simplification 15.0%

    \[\leadsto \frac{-b\_2}{a} \]
14. Add Preprocessing

Developer target: 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 2024077 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  :herbie-expected 10

  :alt
  (if (< b_2 0.0) (/ c (- (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot b_2 (* (sqrt (fabs a)) (sqrt (fabs c))))) b_2)) (/ (+ b_2 (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot b_2 (* (sqrt (fabs a)) (sqrt (fabs c)))))) (- a)))

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