quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.7% → 84.8%

Time: 10.1s

Alternatives: 7

Speedup: 15.9×

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: 52.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{+150}:\\ \;\;\;\;\frac{b_2}{\frac{a}{-2}}\\ \mathbf{elif}\;b_2 \leq 1.22 \cdot 10^{-148}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e+150)
   (/ b_2 (/ a -2.0))
   (if (<= b_2 1.22e-148)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e+150) {
		tmp = b_2 / (a / -2.0);
	} else if (b_2 <= 1.22e-148) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d+150)) then
        tmp = b_2 / (a / (-2.0d0))
    else if (b_2 <= 1.22d-148) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
    else
        tmp = (c * (-0.5d0)) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e+150) {
		tmp = b_2 / (a / -2.0);
	} else if (b_2 <= 1.22e-148) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e+150:
		tmp = b_2 / (a / -2.0)
	elif b_2 <= 1.22e-148:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e+150)
		tmp = Float64(b_2 / Float64(a / -2.0));
	elseif (b_2 <= 1.22e-148)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e+150)
		tmp = b_2 / (a / -2.0);
	elseif (b_2 <= 1.22e-148)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e+150], N[(b$95$2 / N[(a / -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.22e-148], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -5.00000000000000009e150

   1. Initial program 41.0%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 41.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 41.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 41.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 97.9%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]
   5. Step-by-step derivation

      1. associate-*r/ 95.8%

         \[\leadsto \color{blue}{\frac{-2 \cdot b_2}{a}} \]

      2. *-commutative 95.8%

         \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]

      3. associate-/l* 97.9%

         \[\leadsto \color{blue}{\frac{b_2}{\frac{a}{-2}}} \]

   6. Simplified 97.9%

      \[\leadsto \color{blue}{\frac{b_2}{\frac{a}{-2}}} \]

   if -5.00000000000000009e150 < b_2 < 1.21999999999999992e-148

   1. Initial program 80.4%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 80.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 80.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 80.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   if 1.21999999999999992e-148 < b_2

   1. Initial program 19.5%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 19.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 19.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 19.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 82.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   5. Step-by-step derivation

      1. *-commutative 82.4%

         \[\leadsto \color{blue}{\frac{c}{b_2} \cdot -0.5} \]

      2. associate-*l/ 82.4%

         \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b_2}} \]

   6. Simplified 82.4%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{+150}:\\ \;\;\;\;\frac{b_2}{\frac{a}{-2}}\\ \mathbf{elif}\;b_2 \leq 1.22 \cdot 10^{-148}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 2: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2.05 \cdot 10^{-67}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.22 \cdot 10^{-148}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.05e-67)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 1.22e-148)
     (/ (- (sqrt (* a (- c))) b_2) a)
     (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.05e-67) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 1.22e-148) {
		tmp = (sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.05d-67)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= 1.22d-148) then
        tmp = (sqrt((a * -c)) - b_2) / a
    else
        tmp = (c * (-0.5d0)) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.05e-67) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 1.22e-148) {
		tmp = (Math.sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.05e-67:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 1.22e-148:
		tmp = (math.sqrt((a * -c)) - b_2) / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.05e-67)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 1.22e-148)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-c))) - b_2) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.05e-67)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 1.22e-148)
		tmp = (sqrt((a * -c)) - b_2) / a;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.05e-67], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.22e-148], N[(N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.0499999999999999e-67

   1. Initial program 65.8%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 65.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 65.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 65.8%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 92.4%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]

   if -2.0499999999999999e-67 < b_2 < 1.21999999999999992e-148

   1. Initial program 73.6%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 73.6%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 73.6%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 73.6%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around 0 66.4%

      \[\leadsto \frac{\sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}} - b_2}{a} \]
   5. Step-by-step derivation

      1. associate-*r* 66.4%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}} - b_2}{a} \]

      2. neg-mul-1 66.4%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-a\right)} \cdot c} - b_2}{a} \]

   6. Simplified 66.4%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-a\right) \cdot c}} - b_2}{a} \]

   if 1.21999999999999992e-148 < b_2

   1. Initial program 19.5%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 19.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 19.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 19.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 82.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   5. Step-by-step derivation

      1. *-commutative 82.4%

         \[\leadsto \color{blue}{\frac{c}{b_2} \cdot -0.5} \]

      2. associate-*l/ 82.4%

         \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b_2}} \]

   6. Simplified 82.4%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.05 \cdot 10^{-67}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.22 \cdot 10^{-148}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 3: 67.7% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 70.2%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 70.2%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 70.2%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 70.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 71.3%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]

   if -4.999999999999985e-310 < b_2

   1. Initial program 30.0%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 30.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 30.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 30.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 65.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   5. Step-by-step derivation

      1. *-commutative 65.1%

         \[\leadsto \color{blue}{\frac{c}{b_2} \cdot -0.5} \]

      2. associate-*l/ 65.2%

         \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b_2}} \]

   6. Simplified 65.2%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 4: 43.4% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq 7.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{b_2}{\frac{a}{-2}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 7.2e-8) (/ b_2 (/ a -2.0)) (* c (/ 0.5 b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 7.2e-8) {
		tmp = b_2 / (a / -2.0);
	} else {
		tmp = c * (0.5 / b_2);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= 7.2d-8) then
        tmp = b_2 / (a / (-2.0d0))
    else
        tmp = c * (0.5d0 / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 7.2e-8) {
		tmp = b_2 / (a / -2.0);
	} else {
		tmp = c * (0.5 / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 7.2e-8:
		tmp = b_2 / (a / -2.0)
	else:
		tmp = c * (0.5 / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 7.2e-8)
		tmp = Float64(b_2 / Float64(a / -2.0));
	else
		tmp = Float64(c * Float64(0.5 / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 7.2e-8)
		tmp = b_2 / (a / -2.0);
	else
		tmp = c * (0.5 / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 7.2e-8], N[(b$95$2 / N[(a / -2.0), $MachinePrecision]), $MachinePrecision], N[(c * N[(0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 7.19999999999999962e-8

   1. Initial program 65.0%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 65.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 65.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 65.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 50.7%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]
   5. Step-by-step derivation

      1. associate-*r/ 50.2%

         \[\leadsto \color{blue}{\frac{-2 \cdot b_2}{a}} \]

      2. *-commutative 50.2%

         \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]

      3. associate-/l* 50.7%

         \[\leadsto \color{blue}{\frac{b_2}{\frac{a}{-2}}} \]

   6. Simplified 50.7%

      \[\leadsto \color{blue}{\frac{b_2}{\frac{a}{-2}}} \]

   if 7.19999999999999962e-8 < b_2

   1. Initial program 13.1%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 13.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 13.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 13.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 75.2%

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{a \cdot c}{b_2}}}{a} \]
   5. Step-by-step derivation

      1. *-commutative 75.2%

         \[\leadsto \frac{-0.5 \cdot \frac{\color{blue}{c \cdot a}}{b_2}}{a} \]

      2. associate-/l* 65.9%

         \[\leadsto \frac{-0.5 \cdot \color{blue}{\frac{c}{\frac{b_2}{a}}}}{a} \]

   6. Simplified 65.9%

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{c}{\frac{b_2}{a}}}}{a} \]
   7. Step-by-step derivation

      1. clear-num 65.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{-0.5 \cdot \frac{c}{\frac{b_2}{a}}}}} \]

      2. inv-pow 65.8%

         \[\leadsto \color{blue}{{\left(\frac{a}{-0.5 \cdot \frac{c}{\frac{b_2}{a}}}\right)}^{-1}} \]

      3. *-un-lft-identity 65.8%

         \[\leadsto {\left(\frac{\color{blue}{1 \cdot a}}{-0.5 \cdot \frac{c}{\frac{b_2}{a}}}\right)}^{-1} \]

      4. *-commutative 65.8%

         \[\leadsto {\left(\frac{1 \cdot a}{\color{blue}{\frac{c}{\frac{b_2}{a}} \cdot -0.5}}\right)}^{-1} \]

      5. times-frac 65.9%

         \[\leadsto {\color{blue}{\left(\frac{1}{\frac{c}{\frac{b_2}{a}}} \cdot \frac{a}{-0.5}\right)}}^{-1} \]

      6. clear-num 65.9%

         \[\leadsto {\left(\color{blue}{\frac{\frac{b_2}{a}}{c}} \cdot \frac{a}{-0.5}\right)}^{-1} \]

   8. Applied egg-rr 65.9%

      \[\leadsto \color{blue}{{\left(\frac{\frac{b_2}{a}}{c} \cdot \frac{a}{-0.5}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. unpow-1 65.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{b_2}{a}}{c} \cdot \frac{a}{-0.5}}} \]

      2. associate-/l/ 75.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{b_2}{c \cdot a}} \cdot \frac{a}{-0.5}} \]

   10. Simplified 75.0%

       \[\leadsto \color{blue}{\frac{1}{\frac{b_2}{c \cdot a} \cdot \frac{a}{-0.5}}} \]
   11. Step-by-step derivation

       1. *-commutative 75.0%

          \[\leadsto \frac{1}{\color{blue}{\frac{a}{-0.5} \cdot \frac{b_2}{c \cdot a}}} \]

       2. associate-*l/ 75.0%

          \[\leadsto \frac{1}{\color{blue}{\frac{a \cdot \frac{b_2}{c \cdot a}}{-0.5}}} \]

       3. clear-num 75.0%

          \[\leadsto \frac{1}{\frac{a \cdot \color{blue}{\frac{1}{\frac{c \cdot a}{b_2}}}}{-0.5}} \]

       4. associate-*r/ 65.8%

          \[\leadsto \frac{1}{\frac{a \cdot \frac{1}{\color{blue}{c \cdot \frac{a}{b_2}}}}{-0.5}} \]

       5. div-inv 65.8%

          \[\leadsto \frac{1}{\frac{\color{blue}{\frac{a}{c \cdot \frac{a}{b_2}}}}{-0.5}} \]

       6. clear-num 65.8%

          \[\leadsto \color{blue}{\frac{-0.5}{\frac{a}{c \cdot \frac{a}{b_2}}}} \]

       7. frac-2neg 65.8%

          \[\leadsto \color{blue}{\frac{--0.5}{-\frac{a}{c \cdot \frac{a}{b_2}}}} \]

       8. metadata-eval 65.8%

          \[\leadsto \frac{\color{blue}{0.5}}{-\frac{a}{c \cdot \frac{a}{b_2}}} \]

       9. distribute-frac-neg 65.8%

          \[\leadsto \frac{0.5}{\color{blue}{\frac{-a}{c \cdot \frac{a}{b_2}}}} \]

       10. div-inv 65.8%

           \[\leadsto \color{blue}{0.5 \cdot \frac{1}{\frac{-a}{c \cdot \frac{a}{b_2}}}} \]

       11. clear-num 65.9%

           \[\leadsto 0.5 \cdot \color{blue}{\frac{c \cdot \frac{a}{b_2}}{-a}} \]

       12. clear-num 65.9%

           \[\leadsto 0.5 \cdot \frac{c \cdot \color{blue}{\frac{1}{\frac{b_2}{a}}}}{-a} \]

       13. un-div-inv 65.9%

           \[\leadsto 0.5 \cdot \frac{\color{blue}{\frac{c}{\frac{b_2}{a}}}}{-a} \]

       14. add-sqr-sqrt 29.3%

           \[\leadsto 0.5 \cdot \frac{\frac{c}{\frac{b_2}{a}}}{\color{blue}{\sqrt{-a} \cdot \sqrt{-a}}} \]

       15. sqrt-unprod 34.1%

           \[\leadsto 0.5 \cdot \frac{\frac{c}{\frac{b_2}{a}}}{\color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}} \]

       16. sqr-neg 34.1%

           \[\leadsto 0.5 \cdot \frac{\frac{c}{\frac{b_2}{a}}}{\sqrt{\color{blue}{a \cdot a}}} \]

       17. sqrt-unprod 16.5%

           \[\leadsto 0.5 \cdot \frac{\frac{c}{\frac{b_2}{a}}}{\color{blue}{\sqrt{a} \cdot \sqrt{a}}} \]

       18. add-sqr-sqrt 26.6%

           \[\leadsto 0.5 \cdot \frac{\frac{c}{\frac{b_2}{a}}}{\color{blue}{a}} \]

   12. Applied egg-rr 26.6%

       \[\leadsto \color{blue}{0.5 \cdot \frac{\frac{c}{\frac{b_2}{a}}}{a}} \]
   13. Step-by-step derivation

       1. metadata-eval 26.6%

          \[\leadsto \color{blue}{\left(--0.5\right)} \cdot \frac{\frac{c}{\frac{b_2}{a}}}{a} \]

       2. distribute-lft-neg-in 26.6%

          \[\leadsto \color{blue}{--0.5 \cdot \frac{\frac{c}{\frac{b_2}{a}}}{a}} \]

       3. associate-*r/ 26.6%

          \[\leadsto -\color{blue}{\frac{-0.5 \cdot \frac{c}{\frac{b_2}{a}}}{a}} \]

       4. associate-*l/ 26.6%

          \[\leadsto -\color{blue}{\frac{-0.5}{a} \cdot \frac{c}{\frac{b_2}{a}}} \]

       5. *-commutative 26.6%

          \[\leadsto -\color{blue}{\frac{c}{\frac{b_2}{a}} \cdot \frac{-0.5}{a}} \]

       6. associate-*l/ 26.2%

          \[\leadsto -\color{blue}{\frac{c \cdot \frac{-0.5}{a}}{\frac{b_2}{a}}} \]

       7. associate-*r/ 26.5%

          \[\leadsto -\color{blue}{c \cdot \frac{\frac{-0.5}{a}}{\frac{b_2}{a}}} \]

       8. distribute-rgt-neg-in 26.5%

          \[\leadsto \color{blue}{c \cdot \left(-\frac{\frac{-0.5}{a}}{\frac{b_2}{a}}\right)} \]

       9. associate-/l/ 26.5%

          \[\leadsto c \cdot \left(-\color{blue}{\frac{-0.5}{\frac{b_2}{a} \cdot a}}\right) \]

       10. distribute-neg-frac 26.5%

           \[\leadsto c \cdot \color{blue}{\frac{--0.5}{\frac{b_2}{a} \cdot a}} \]

       11. metadata-eval 26.5%

           \[\leadsto c \cdot \frac{\color{blue}{0.5}}{\frac{b_2}{a} \cdot a} \]

       12. associate-*l/ 26.4%

           \[\leadsto c \cdot \frac{0.5}{\color{blue}{\frac{b_2 \cdot a}{a}}} \]

       13. *-commutative 26.4%

           \[\leadsto c \cdot \frac{0.5}{\frac{\color{blue}{a \cdot b_2}}{a}} \]

       14. associate-*l/ 26.2%

           \[\leadsto c \cdot \frac{0.5}{\color{blue}{\frac{a}{a} \cdot b_2}} \]

       15. *-inverses 26.2%

           \[\leadsto c \cdot \frac{0.5}{\color{blue}{1} \cdot b_2} \]

       16. *-lft-identity 26.2%

           \[\leadsto c \cdot \frac{0.5}{\color{blue}{b_2}} \]

   14. Simplified 26.2%

       \[\leadsto \color{blue}{c \cdot \frac{0.5}{b_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq 7.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{b_2}{\frac{a}{-2}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b_2}\\ \end{array} \]

Alternative 5: 67.5% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq 3.1 \cdot 10^{-287}:\\ \;\;\;\;\frac{b_2}{\frac{a}{-2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 3.1e-287) (/ b_2 (/ a -2.0)) (/ (* c -0.5) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 3.1e-287) {
		tmp = b_2 / (a / -2.0);
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= 3.1d-287) then
        tmp = b_2 / (a / (-2.0d0))
    else
        tmp = (c * (-0.5d0)) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 3.1e-287) {
		tmp = b_2 / (a / -2.0);
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 3.1e-287:
		tmp = b_2 / (a / -2.0)
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 3.1e-287)
		tmp = Float64(b_2 / Float64(a / -2.0));
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 3.1e-287)
		tmp = b_2 / (a / -2.0);
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 3.1e-287], N[(b$95$2 / N[(a / -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 3.1000000000000001e-287

   1. Initial program 69.7%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 69.7%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 69.7%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 69.7%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 70.1%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]
   5. Step-by-step derivation

      1. associate-*r/ 69.3%

         \[\leadsto \color{blue}{\frac{-2 \cdot b_2}{a}} \]

      2. *-commutative 69.3%

         \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]

      3. associate-/l* 70.1%

         \[\leadsto \color{blue}{\frac{b_2}{\frac{a}{-2}}} \]

   6. Simplified 70.1%

      \[\leadsto \color{blue}{\frac{b_2}{\frac{a}{-2}}} \]

   if 3.1000000000000001e-287 < b_2

   1. Initial program 30.2%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 30.2%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 30.2%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 30.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 65.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   5. Step-by-step derivation

      1. *-commutative 65.6%

         \[\leadsto \color{blue}{\frac{c}{b_2} \cdot -0.5} \]

      2. associate-*l/ 65.7%

         \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b_2}} \]

   6. Simplified 65.7%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq 3.1 \cdot 10^{-287}:\\ \;\;\;\;\frac{b_2}{\frac{a}{-2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 6: 10.5% 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 50.4%

   \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
2. Step-by-step derivation

   1. +-commutative 50.4%

      \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

   2. unsub-neg 50.4%

      \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

3. Simplified 50.4%

   \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

4. Taylor expanded in b_2 around inf 26.1%

   \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{a \cdot c}{b_2}}}{a} \]
5. Step-by-step derivation

   1. *-commutative 26.1%

      \[\leadsto \frac{-0.5 \cdot \frac{\color{blue}{c \cdot a}}{b_2}}{a} \]

   2. associate-/l* 26.0%

      \[\leadsto \frac{-0.5 \cdot \color{blue}{\frac{c}{\frac{b_2}{a}}}}{a} \]

6. Simplified 26.0%

   \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{c}{\frac{b_2}{a}}}}{a} \]
7. Step-by-step derivation

   1. clear-num 26.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{-0.5 \cdot \frac{c}{\frac{b_2}{a}}}}} \]

   2. inv-pow 26.0%

      \[\leadsto \color{blue}{{\left(\frac{a}{-0.5 \cdot \frac{c}{\frac{b_2}{a}}}\right)}^{-1}} \]

   3. *-un-lft-identity 26.0%

      \[\leadsto {\left(\frac{\color{blue}{1 \cdot a}}{-0.5 \cdot \frac{c}{\frac{b_2}{a}}}\right)}^{-1} \]

   4. *-commutative 26.0%

      \[\leadsto {\left(\frac{1 \cdot a}{\color{blue}{\frac{c}{\frac{b_2}{a}} \cdot -0.5}}\right)}^{-1} \]

   5. times-frac 26.0%

      \[\leadsto {\color{blue}{\left(\frac{1}{\frac{c}{\frac{b_2}{a}}} \cdot \frac{a}{-0.5}\right)}}^{-1} \]

   6. clear-num 26.0%

      \[\leadsto {\left(\color{blue}{\frac{\frac{b_2}{a}}{c}} \cdot \frac{a}{-0.5}\right)}^{-1} \]

8. Applied egg-rr 26.0%

   \[\leadsto \color{blue}{{\left(\frac{\frac{b_2}{a}}{c} \cdot \frac{a}{-0.5}\right)}^{-1}} \]
9. Step-by-step derivation

   1. unpow-1 26.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{b_2}{a}}{c} \cdot \frac{a}{-0.5}}} \]

   2. associate-/l/ 26.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{b_2}{c \cdot a}} \cdot \frac{a}{-0.5}} \]

10. Simplified 26.0%

    \[\leadsto \color{blue}{\frac{1}{\frac{b_2}{c \cdot a} \cdot \frac{a}{-0.5}}} \]
11. Step-by-step derivation

    1. *-commutative 26.0%

       \[\leadsto \frac{1}{\color{blue}{\frac{a}{-0.5} \cdot \frac{b_2}{c \cdot a}}} \]

    2. associate-*l/ 26.0%

       \[\leadsto \frac{1}{\color{blue}{\frac{a \cdot \frac{b_2}{c \cdot a}}{-0.5}}} \]

    3. clear-num 26.0%

       \[\leadsto \frac{1}{\frac{a \cdot \color{blue}{\frac{1}{\frac{c \cdot a}{b_2}}}}{-0.5}} \]

    4. associate-*r/ 26.0%

       \[\leadsto \frac{1}{\frac{a \cdot \frac{1}{\color{blue}{c \cdot \frac{a}{b_2}}}}{-0.5}} \]

    5. div-inv 26.0%

       \[\leadsto \frac{1}{\frac{\color{blue}{\frac{a}{c \cdot \frac{a}{b_2}}}}{-0.5}} \]

    6. clear-num 26.0%

       \[\leadsto \color{blue}{\frac{-0.5}{\frac{a}{c \cdot \frac{a}{b_2}}}} \]

    7. frac-2neg 26.0%

       \[\leadsto \color{blue}{\frac{--0.5}{-\frac{a}{c \cdot \frac{a}{b_2}}}} \]

    8. metadata-eval 26.0%

       \[\leadsto \frac{\color{blue}{0.5}}{-\frac{a}{c \cdot \frac{a}{b_2}}} \]

    9. distribute-frac-neg 26.0%

       \[\leadsto \frac{0.5}{\color{blue}{\frac{-a}{c \cdot \frac{a}{b_2}}}} \]

    10. div-inv 26.0%

        \[\leadsto \color{blue}{0.5 \cdot \frac{1}{\frac{-a}{c \cdot \frac{a}{b_2}}}} \]

    11. clear-num 26.0%

        \[\leadsto 0.5 \cdot \color{blue}{\frac{c \cdot \frac{a}{b_2}}{-a}} \]

    12. clear-num 26.0%

        \[\leadsto 0.5 \cdot \frac{c \cdot \color{blue}{\frac{1}{\frac{b_2}{a}}}}{-a} \]

    13. un-div-inv 26.0%

        \[\leadsto 0.5 \cdot \frac{\color{blue}{\frac{c}{\frac{b_2}{a}}}}{-a} \]

    14. add-sqr-sqrt 11.8%

        \[\leadsto 0.5 \cdot \frac{\frac{c}{\frac{b_2}{a}}}{\color{blue}{\sqrt{-a} \cdot \sqrt{-a}}} \]

    15. sqrt-unprod 12.1%

        \[\leadsto 0.5 \cdot \frac{\frac{c}{\frac{b_2}{a}}}{\color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}} \]

    16. sqr-neg 12.1%

        \[\leadsto 0.5 \cdot \frac{\frac{c}{\frac{b_2}{a}}}{\sqrt{\color{blue}{a \cdot a}}} \]

    17. sqrt-unprod 5.6%

        \[\leadsto 0.5 \cdot \frac{\frac{c}{\frac{b_2}{a}}}{\color{blue}{\sqrt{a} \cdot \sqrt{a}}} \]

    18. add-sqr-sqrt 9.3%

        \[\leadsto 0.5 \cdot \frac{\frac{c}{\frac{b_2}{a}}}{\color{blue}{a}} \]

12. Applied egg-rr 9.3%

    \[\leadsto \color{blue}{0.5 \cdot \frac{\frac{c}{\frac{b_2}{a}}}{a}} \]
13. Step-by-step derivation

    1. metadata-eval 9.3%

       \[\leadsto \color{blue}{\left(--0.5\right)} \cdot \frac{\frac{c}{\frac{b_2}{a}}}{a} \]

    2. distribute-lft-neg-in 9.3%

       \[\leadsto \color{blue}{--0.5 \cdot \frac{\frac{c}{\frac{b_2}{a}}}{a}} \]

    3. associate-*r/ 9.3%

       \[\leadsto -\color{blue}{\frac{-0.5 \cdot \frac{c}{\frac{b_2}{a}}}{a}} \]

    4. associate-*l/ 9.3%

       \[\leadsto -\color{blue}{\frac{-0.5}{a} \cdot \frac{c}{\frac{b_2}{a}}} \]

    5. *-commutative 9.3%

       \[\leadsto -\color{blue}{\frac{c}{\frac{b_2}{a}} \cdot \frac{-0.5}{a}} \]

    6. associate-*l/ 9.1%

       \[\leadsto -\color{blue}{\frac{c \cdot \frac{-0.5}{a}}{\frac{b_2}{a}}} \]

    7. associate-*r/ 9.3%

       \[\leadsto -\color{blue}{c \cdot \frac{\frac{-0.5}{a}}{\frac{b_2}{a}}} \]

    8. distribute-rgt-neg-in 9.3%

       \[\leadsto \color{blue}{c \cdot \left(-\frac{\frac{-0.5}{a}}{\frac{b_2}{a}}\right)} \]

    9. associate-/l/ 9.3%

       \[\leadsto c \cdot \left(-\color{blue}{\frac{-0.5}{\frac{b_2}{a} \cdot a}}\right) \]

    10. distribute-neg-frac 9.3%

        \[\leadsto c \cdot \color{blue}{\frac{--0.5}{\frac{b_2}{a} \cdot a}} \]

    11. metadata-eval 9.3%

        \[\leadsto c \cdot \frac{\color{blue}{0.5}}{\frac{b_2}{a} \cdot a} \]

    12. associate-*l/ 9.2%

        \[\leadsto c \cdot \frac{0.5}{\color{blue}{\frac{b_2 \cdot a}{a}}} \]

    13. *-commutative 9.2%

        \[\leadsto c \cdot \frac{0.5}{\frac{\color{blue}{a \cdot b_2}}{a}} \]

    14. associate-*l/ 9.3%

        \[\leadsto c \cdot \frac{0.5}{\color{blue}{\frac{a}{a} \cdot b_2}} \]

    15. *-inverses 9.3%

        \[\leadsto c \cdot \frac{0.5}{\color{blue}{1} \cdot b_2} \]

    16. *-lft-identity 9.3%

        \[\leadsto c \cdot \frac{0.5}{\color{blue}{b_2}} \]

14. Simplified 9.3%

    \[\leadsto \color{blue}{c \cdot \frac{0.5}{b_2}} \]

15. Final simplification 9.3%

    \[\leadsto c \cdot \frac{0.5}{b_2} \]

Alternative 7: 10.5% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\frac{b_2}{c}} \end{array} \]

FPCore

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

C

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

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 = 0.5d0 / (b_2 / c)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.4%

   \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
2. Step-by-step derivation

   1. +-commutative 50.4%

      \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

   2. unsub-neg 50.4%

      \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

3. Simplified 50.4%

   \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

4. Taylor expanded in b_2 around inf 26.1%

   \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{a \cdot c}{b_2}}}{a} \]
5. Step-by-step derivation

   1. *-commutative 26.1%

      \[\leadsto \frac{-0.5 \cdot \frac{\color{blue}{c \cdot a}}{b_2}}{a} \]

   2. associate-/l* 26.0%

      \[\leadsto \frac{-0.5 \cdot \color{blue}{\frac{c}{\frac{b_2}{a}}}}{a} \]

6. Simplified 26.0%

   \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{c}{\frac{b_2}{a}}}}{a} \]
7. Step-by-step derivation

   1. frac-2neg 26.0%

      \[\leadsto \color{blue}{\frac{--0.5 \cdot \frac{c}{\frac{b_2}{a}}}{-a}} \]

   2. div-inv 26.0%

      \[\leadsto \color{blue}{\left(--0.5 \cdot \frac{c}{\frac{b_2}{a}}\right) \cdot \frac{1}{-a}} \]

   3. *-commutative 26.0%

      \[\leadsto \left(-\color{blue}{\frac{c}{\frac{b_2}{a}} \cdot -0.5}\right) \cdot \frac{1}{-a} \]

   4. distribute-rgt-neg-in 26.0%

      \[\leadsto \color{blue}{\left(\frac{c}{\frac{b_2}{a}} \cdot \left(--0.5\right)\right)} \cdot \frac{1}{-a} \]

   5. div-inv 26.0%

      \[\leadsto \left(\color{blue}{\left(c \cdot \frac{1}{\frac{b_2}{a}}\right)} \cdot \left(--0.5\right)\right) \cdot \frac{1}{-a} \]

   6. clear-num 26.0%

      \[\leadsto \left(\left(c \cdot \color{blue}{\frac{a}{b_2}}\right) \cdot \left(--0.5\right)\right) \cdot \frac{1}{-a} \]

   7. metadata-eval 26.0%

      \[\leadsto \left(\left(c \cdot \frac{a}{b_2}\right) \cdot \color{blue}{0.5}\right) \cdot \frac{1}{-a} \]

8. Applied egg-rr 26.0%

   \[\leadsto \color{blue}{\left(\left(c \cdot \frac{a}{b_2}\right) \cdot 0.5\right) \cdot \frac{1}{-a}} \]
9. Step-by-step derivation

   1. associate-*r/ 26.0%

      \[\leadsto \color{blue}{\frac{\left(\left(c \cdot \frac{a}{b_2}\right) \cdot 0.5\right) \cdot 1}{-a}} \]

   2. *-commutative 26.0%

      \[\leadsto \frac{\color{blue}{\left(0.5 \cdot \left(c \cdot \frac{a}{b_2}\right)\right)} \cdot 1}{-a} \]

   3. metadata-eval 26.0%

      \[\leadsto \frac{\left(\color{blue}{\left(--0.5\right)} \cdot \left(c \cdot \frac{a}{b_2}\right)\right) \cdot 1}{-a} \]

   4. distribute-lft-neg-in 26.0%

      \[\leadsto \frac{\color{blue}{\left(--0.5 \cdot \left(c \cdot \frac{a}{b_2}\right)\right)} \cdot 1}{-a} \]

   5. distribute-lft-neg-in 26.0%

      \[\leadsto \frac{\color{blue}{-\left(-0.5 \cdot \left(c \cdot \frac{a}{b_2}\right)\right) \cdot 1}}{-a} \]

   6. *-rgt-identity 26.0%

      \[\leadsto \frac{-\color{blue}{-0.5 \cdot \left(c \cdot \frac{a}{b_2}\right)}}{-a} \]

   7. distribute-neg-frac 26.0%

      \[\leadsto \color{blue}{-\frac{-0.5 \cdot \left(c \cdot \frac{a}{b_2}\right)}{-a}} \]

   8. associate-/l* 26.0%

      \[\leadsto -\color{blue}{\frac{-0.5}{\frac{-a}{c \cdot \frac{a}{b_2}}}} \]

   9. distribute-neg-frac 26.0%

      \[\leadsto \color{blue}{\frac{--0.5}{\frac{-a}{c \cdot \frac{a}{b_2}}}} \]

   10. metadata-eval 26.0%

       \[\leadsto \frac{\color{blue}{0.5}}{\frac{-a}{c \cdot \frac{a}{b_2}}} \]

10. Simplified 26.0%

    \[\leadsto \color{blue}{\frac{0.5}{\frac{-a}{c \cdot \frac{a}{b_2}}}} \]
11. Step-by-step derivation

    1. expm1-log1p-u 14.8%

       \[\leadsto \frac{0.5}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-a}{c \cdot \frac{a}{b_2}}\right)\right)}} \]

    2. expm1-udef 10.4%

       \[\leadsto \frac{0.5}{\color{blue}{e^{\mathsf{log1p}\left(\frac{-a}{c \cdot \frac{a}{b_2}}\right)} - 1}} \]

    3. add-sqr-sqrt 4.4%

       \[\leadsto \frac{0.5}{e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-a} \cdot \sqrt{-a}}}{c \cdot \frac{a}{b_2}}\right)} - 1} \]

    4. sqrt-unprod 7.3%

       \[\leadsto \frac{0.5}{e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}}{c \cdot \frac{a}{b_2}}\right)} - 1} \]

    5. sqr-neg 7.3%

       \[\leadsto \frac{0.5}{e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{a \cdot a}}}{c \cdot \frac{a}{b_2}}\right)} - 1} \]

    6. sqrt-unprod 4.3%

       \[\leadsto \frac{0.5}{e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{a} \cdot \sqrt{a}}}{c \cdot \frac{a}{b_2}}\right)} - 1} \]

    7. add-sqr-sqrt 6.1%

       \[\leadsto \frac{0.5}{e^{\mathsf{log1p}\left(\frac{\color{blue}{a}}{c \cdot \frac{a}{b_2}}\right)} - 1} \]

    8. *-un-lft-identity 6.1%

       \[\leadsto \frac{0.5}{e^{\mathsf{log1p}\left(\frac{\color{blue}{1 \cdot a}}{c \cdot \frac{a}{b_2}}\right)} - 1} \]

    9. *-commutative 6.1%

       \[\leadsto \frac{0.5}{e^{\mathsf{log1p}\left(\frac{1 \cdot a}{\color{blue}{\frac{a}{b_2} \cdot c}}\right)} - 1} \]

    10. times-frac 5.6%

        \[\leadsto \frac{0.5}{e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{a}{b_2}} \cdot \frac{a}{c}}\right)} - 1} \]

    11. clear-num 5.6%

        \[\leadsto \frac{0.5}{e^{\mathsf{log1p}\left(\color{blue}{\frac{b_2}{a}} \cdot \frac{a}{c}\right)} - 1} \]

12. Applied egg-rr 5.6%

    \[\leadsto \frac{0.5}{\color{blue}{e^{\mathsf{log1p}\left(\frac{b_2}{a} \cdot \frac{a}{c}\right)} - 1}} \]
13. Step-by-step derivation

    1. expm1-def 5.8%

       \[\leadsto \frac{0.5}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b_2}{a} \cdot \frac{a}{c}\right)\right)}} \]

    2. expm1-log1p 9.2%

       \[\leadsto \frac{0.5}{\color{blue}{\frac{b_2}{a} \cdot \frac{a}{c}}} \]

    3. *-commutative 9.2%

       \[\leadsto \frac{0.5}{\color{blue}{\frac{a}{c} \cdot \frac{b_2}{a}}} \]

    4. associate-*l/ 9.3%

       \[\leadsto \frac{0.5}{\color{blue}{\frac{a \cdot \frac{b_2}{a}}{c}}} \]

    5. associate-*r/ 9.2%

       \[\leadsto \frac{0.5}{\frac{\color{blue}{\frac{a \cdot b_2}{a}}}{c}} \]

    6. associate-*l/ 9.3%

       \[\leadsto \frac{0.5}{\frac{\color{blue}{\frac{a}{a} \cdot b_2}}{c}} \]

    7. *-inverses 9.3%

       \[\leadsto \frac{0.5}{\frac{\color{blue}{1} \cdot b_2}{c}} \]

    8. *-lft-identity 9.3%

       \[\leadsto \frac{0.5}{\frac{\color{blue}{b_2}}{c}} \]

14. Simplified 9.3%

    \[\leadsto \frac{0.5}{\color{blue}{\frac{b_2}{c}}} \]

15. Final simplification 9.3%

    \[\leadsto \frac{0.5}{\frac{b_2}{c}} \]

Developer target: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_1 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{\left|b_2\right| - t_0} \cdot \sqrt{\left|b_2\right| + t_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b_2, t_0\right)\\ \end{array}\\ \mathbf{if}\;b_2 < 0:\\ \;\;\;\;\frac{t_1 - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b_2 + t_1}\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023331 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  :herbie-expected 10

  :herbie-target
  (if (< b_2 0.0) (/ (- (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) a) (/ (- c) (+ 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))))))))

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