quad2p (problem 3.2.1, positive)

Percentage Accurate: 51.5% → 86.1%

Time: 12.9s

Alternatives: 9

Speedup: 11.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.5% 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: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{+152}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 8.4 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e+152)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 8.4e-60)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e+152) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 8.4e-60) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = -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+152)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= 8.4d-60) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
    else
        tmp = (-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+152) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 8.4e-60) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e+152:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= 8.4e-60:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e+152)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 8.4e-60)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = 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+152)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= 8.4e-60)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e+152], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 8.4e-60], 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[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -5e152

   1. Initial program 28.1%

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

      1. +-commutative 28.1%

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

      2. unsub-neg 28.1%

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

   3. Simplified 28.1%

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

   5. Taylor expanded in b_2 around -inf 92.2%

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

      1. *-commutative 92.2%

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

   7. Simplified 92.2%

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

   if -5e152 < b_2 < 8.39999999999999964e-60

   1. Initial program 82.6%

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

      1. +-commutative 82.6%

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

      2. unsub-neg 82.6%

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

   3. Simplified 82.6%

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

   if 8.39999999999999964e-60 < b_2

   1. Initial program 24.2%

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

      1. +-commutative 24.2%

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

      2. unsub-neg 24.2%

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

   3. Simplified 24.2%

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

   5. Taylor expanded in b_2 around inf 91.5%

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

Alternative 2: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-91}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 7.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-91)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 7.2e-60) (/ (- (sqrt (* a (- c))) b_2) a) (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-91) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 7.2e-60) {
		tmp = (sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = -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 <= (-1d-91)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= 7.2d-60) then
        tmp = (sqrt((a * -c)) - b_2) / a
    else
        tmp = (-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 <= -1e-91) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 7.2e-60) {
		tmp = (Math.sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e-91:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= 7.2e-60:
		tmp = (math.sqrt((a * -c)) - b_2) / a
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-91)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 7.2e-60)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-c))) - b_2) / a);
	else
		tmp = 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 <= -1e-91)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= 7.2e-60)
		tmp = (sqrt((a * -c)) - b_2) / a;
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e-91], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 7.2e-60], N[(N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.00000000000000002e-91

   1. Initial program 59.9%

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

      1. +-commutative 59.9%

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

      2. unsub-neg 59.9%

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

   3. Simplified 59.9%

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

   5. Taylor expanded in b_2 around -inf 85.5%

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

      1. *-commutative 85.5%

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

   7. Simplified 85.5%

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

   if -1.00000000000000002e-91 < b_2 < 7.2e-60

   1. Initial program 77.0%

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

      1. +-commutative 77.0%

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

      2. unsub-neg 77.0%

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

   3. Simplified 77.0%

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

   5. Taylor expanded in b_2 around 0 73.0%

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

      1. associate-*r* 73.0%

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

      2. neg-mul-1 73.0%

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

      3. *-commutative 73.0%

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

   7. Simplified 73.0%

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

   if 7.2e-60 < b_2

   1. Initial program 24.2%

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

      1. +-commutative 24.2%

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

      2. unsub-neg 24.2%

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

   3. Simplified 24.2%

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

   5. Taylor expanded in b_2 around inf 91.5%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-91}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 7.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.15 \cdot 10^{-91}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 5.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.15e-91)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 5.2e-60) (/ (sqrt (* a (- c))) a) (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.15e-91) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 5.2e-60) {
		tmp = sqrt((a * -c)) / a;
	} else {
		tmp = -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 <= (-1.15d-91)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= 5.2d-60) then
        tmp = sqrt((a * -c)) / a
    else
        tmp = (-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 <= -1.15e-91) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 5.2e-60) {
		tmp = Math.sqrt((a * -c)) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.15e-91:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= 5.2e-60:
		tmp = math.sqrt((a * -c)) / a
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.15e-91)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 5.2e-60)
		tmp = Float64(sqrt(Float64(a * Float64(-c))) / a);
	else
		tmp = 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 <= -1.15e-91)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= 5.2e-60)
		tmp = sqrt((a * -c)) / a;
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.15e-91], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 5.2e-60], N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.14999999999999998e-91

   1. Initial program 59.9%

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

      1. +-commutative 59.9%

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

      2. unsub-neg 59.9%

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

   3. Simplified 59.9%

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

   5. Taylor expanded in b_2 around -inf 85.5%

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

      1. *-commutative 85.5%

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

   7. Simplified 85.5%

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

   if -1.14999999999999998e-91 < b_2 < 5.1999999999999995e-60

   1. Initial program 77.0%

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

      1. +-commutative 77.0%

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

      2. unsub-neg 77.0%

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

   3. Simplified 77.0%

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

      1. prod-diff 76.8%

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

      2. *-commutative 76.8%

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

      3. fma-neg 76.8%

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

      4. prod-diff 76.8%

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

      5. *-commutative 76.8%

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

      6. fma-neg 76.8%

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

      7. associate-+l+ 76.7%

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

      8. pow2 76.7%

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

      9. *-commutative 76.7%

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

      10. fma-undefine 76.8%

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

      11. distribute-lft-neg-in 76.8%

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

      12. *-commutative 76.8%

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

      13. distribute-rgt-neg-in 76.8%

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

      14. fma-define 76.7%

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

      15. *-commutative 76.7%

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

      16. fma-undefine 76.8%

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

      17. distribute-lft-neg-in 76.8%

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

      18. *-commutative 76.8%

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

      19. distribute-rgt-neg-in 76.8%

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

   6. Applied egg-rr 76.7%

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

      1. *-commutative 76.7%

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

      2. count-2 76.7%

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

      3. *-commutative 76.7%

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

   8. Simplified 76.7%

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

   9. Taylor expanded in c around inf 71.6%

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

       1. fma-neg 71.6%

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

       2. distribute-rgt1-in 71.6%

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

       3. metadata-eval 71.6%

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

       4. mul0-lft 71.6%

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

       5. mul-1-neg 71.6%

          \[\leadsto \frac{\sqrt{c \cdot \mathsf{fma}\left(2, 0, \color{blue}{-1 \cdot a}\right)}}{a} \]

       6. fma-undefine 71.6%

          \[\leadsto \frac{\sqrt{c \cdot \color{blue}{\left(2 \cdot 0 + -1 \cdot a\right)}}}{a} \]

       7. metadata-eval 71.6%

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

       8. mul-1-neg 71.6%

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

       9. sub-neg 71.6%

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

       10. sub0-neg 71.6%

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

   11. Simplified 71.6%

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

   if 5.1999999999999995e-60 < b_2

   1. Initial program 24.2%

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

      1. +-commutative 24.2%

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

      2. unsub-neg 24.2%

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

   3. Simplified 24.2%

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

   5. Taylor expanded in b_2 around inf 91.5%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.15 \cdot 10^{-91}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 5.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.6 \cdot 10^{-184}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 2.7 \cdot 10^{-177}:\\ \;\;\;\;\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.6e-184)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 2.7e-177) (sqrt (/ c (- a))) (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.6e-184) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 2.7e-177) {
		tmp = sqrt((c / -a));
	} else {
		tmp = -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 <= (-2.6d-184)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= 2.7d-177) then
        tmp = sqrt((c / -a))
    else
        tmp = (-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 <= -2.6e-184) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 2.7e-177) {
		tmp = Math.sqrt((c / -a));
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.6e-184:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= 2.7e-177:
		tmp = math.sqrt((c / -a))
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.6e-184)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 2.7e-177)
		tmp = sqrt(Float64(c / Float64(-a)));
	else
		tmp = 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 <= -2.6e-184)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= 2.7e-177)
		tmp = sqrt((c / -a));
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.6e-184], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 2.7e-177], N[Sqrt[N[(c / (-a)), $MachinePrecision]], $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.59999999999999978e-184

   1. Initial program 63.8%

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

      1. +-commutative 63.8%

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

      2. unsub-neg 63.8%

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

   3. Simplified 63.8%

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

   5. Taylor expanded in b_2 around -inf 78.0%

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

      1. *-commutative 78.0%

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

   7. Simplified 78.0%

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

   if -2.59999999999999978e-184 < b_2 < 2.7000000000000002e-177

   1. Initial program 88.4%

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

      1. +-commutative 88.4%

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

      2. unsub-neg 88.4%

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

   3. Simplified 88.4%

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

      1. prod-diff 88.2%

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

      2. *-commutative 88.2%

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

      3. fma-neg 88.2%

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

      4. prod-diff 88.2%

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

      5. *-commutative 88.2%

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

      6. fma-neg 88.2%

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

      7. associate-+l+ 88.1%

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

      8. pow2 88.1%

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

      9. *-commutative 88.1%

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

      10. fma-undefine 88.2%

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

      11. distribute-lft-neg-in 88.2%

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

      12. *-commutative 88.2%

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

      13. distribute-rgt-neg-in 88.2%

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

      14. fma-define 88.1%

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

      15. *-commutative 88.1%

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

      16. fma-undefine 88.2%

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

      17. distribute-lft-neg-in 88.2%

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

      18. *-commutative 88.2%

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

      19. distribute-rgt-neg-in 88.2%

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

   6. Applied egg-rr 88.1%

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

      1. *-commutative 88.1%

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

      2. count-2 88.1%

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

      3. *-commutative 88.1%

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

   8. Simplified 88.1%

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

   9. Taylor expanded in a around inf 43.6%

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

       1. distribute-rgt1-in 43.6%

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

       2. metadata-eval 43.6%

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

   11. Simplified 43.6%

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

   12. Taylor expanded in c around 0 43.6%

       \[\leadsto \sqrt{\frac{\color{blue}{-1 \cdot c}}{a}} \]
   13. Step-by-step derivation

       1. neg-mul-1 43.6%

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

   14. Simplified 43.6%

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

   if 2.7000000000000002e-177 < b_2

   1. Initial program 32.5%

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

      1. +-commutative 32.5%

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

      2. unsub-neg 32.5%

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

   3. Simplified 32.5%

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

   5. Taylor expanded in b_2 around inf 78.0%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.6 \cdot 10^{-184}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 2.7 \cdot 10^{-177}:\\ \;\;\;\;\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.0% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 10^{-307}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 1e-307) (/ (* b_2 -2.0) a) (* -0.5 (/ c b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1e-307) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = -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 <= 1d-307) then
        tmp = (b_2 * (-2.0d0)) / a
    else
        tmp = (-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 <= 1e-307) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 1e-307:
		tmp = (b_2 * -2.0) / a
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 1e-307)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = 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 <= 1e-307)
		tmp = (b_2 * -2.0) / a;
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 1e-307], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 9.99999999999999909e-308

   1. Initial program 65.1%

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

      1. +-commutative 65.1%

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

      2. unsub-neg 65.1%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in b_2 around -inf 68.2%

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

      1. *-commutative 68.2%

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

   7. Simplified 68.2%

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

   if 9.99999999999999909e-308 < b_2

   1. Initial program 42.5%

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

      1. +-commutative 42.5%

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

      2. unsub-neg 42.5%

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

   3. Simplified 42.5%

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

   5. Taylor expanded in b_2 around inf 67.1%

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

Alternative 6: 67.9% accurate, 11.2× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 1e-309) (* b_2 (/ -2.0 a)) (* -0.5 (/ c b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1e-309) {
		tmp = b_2 * (-2.0 / a);
	} else {
		tmp = -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 <= 1d-309) then
        tmp = b_2 * ((-2.0d0) / a)
    else
        tmp = (-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 <= 1e-309) {
		tmp = b_2 * (-2.0 / a);
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 1e-309:
		tmp = b_2 * (-2.0 / a)
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 1e-309)
		tmp = Float64(b_2 * Float64(-2.0 / a));
	else
		tmp = 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 <= 1e-309)
		tmp = b_2 * (-2.0 / a);
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 1e-309], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 1.000000000000002e-309

   1. Initial program 65.1%

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

      1. +-commutative 65.1%

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

      2. unsub-neg 65.1%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in b_2 around -inf 68.2%

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

      1. *-commutative 68.2%

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

   7. Simplified 68.2%

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

      1. associate-/l* 67.9%

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

      2. *-commutative 67.9%

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

   9. Applied egg-rr 67.9%

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

   if 1.000000000000002e-309 < b_2

   1. Initial program 42.5%

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

      1. +-commutative 42.5%

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

      2. unsub-neg 42.5%

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

   3. Simplified 42.5%

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

   5. Taylor expanded in b_2 around inf 67.1%

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

4. Final simplification 67.5%

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

Alternative 7: 47.9% accurate, 11.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-310)
		tmp = Float64(b_2 / Float64(-a));
	else
		tmp = 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 <= -1e-310)
		tmp = b_2 / -a;
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -9.999999999999969e-311

   1. Initial program 65.1%

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

      1. +-commutative 65.1%

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

      2. unsub-neg 65.1%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in b_2 around 0 38.8%

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

      1. associate-*r* 38.8%

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

      2. neg-mul-1 38.8%

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

      3. *-commutative 38.8%

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

   7. Simplified 38.8%

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

   8. Taylor expanded in b_2 around inf 24.5%

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

      1. neg-mul-1 24.5%

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

      2. distribute-neg-frac2 24.5%

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

   10. Simplified 24.5%

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

   if -9.999999999999969e-311 < b_2

   1. Initial program 42.5%

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

      1. +-commutative 42.5%

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

      2. unsub-neg 42.5%

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

   3. Simplified 42.5%

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

   5. Taylor expanded in b_2 around inf 67.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
3. Recombined 2 regimes into one program.
4. 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(b_2 / Float64(-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 53.2%

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

   1. +-commutative 53.2%

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

   2. unsub-neg 53.2%

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

3. Simplified 53.2%

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

5. Taylor expanded in b_2 around 0 34.9%

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

   1. associate-*r* 34.9%

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

   2. neg-mul-1 34.9%

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

   3. *-commutative 34.9%

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

7. Simplified 34.9%

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

8. Taylor expanded in b_2 around inf 13.0%

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

   1. neg-mul-1 13.0%

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

   2. distribute-neg-frac2 13.0%

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

10. Simplified 13.0%

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

Alternative 9: 2.5% accurate, 37.3× 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(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 53.2%

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

   1. +-commutative 53.2%

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

   2. unsub-neg 53.2%

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

3. Simplified 53.2%

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

5. Taylor expanded in b_2 around 0 34.9%

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

   1. associate-*r* 34.9%

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

   2. neg-mul-1 34.9%

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

   3. *-commutative 34.9%

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

7. Simplified 34.9%

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

8. Taylor expanded in b_2 around inf 13.0%

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

   1. neg-mul-1 13.0%

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

   2. distribute-neg-frac2 13.0%

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

10. Simplified 13.0%

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

    1. div-inv 13.0%

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

    2. add-sqr-sqrt 6.5%

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

    3. sqrt-unprod 8.6%

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

    4. sqr-neg 8.6%

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

    5. sqrt-unprod 1.1%

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

    6. add-sqr-sqrt 2.6%

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

12. Applied egg-rr 2.6%

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

    1. associate-*r/ 2.6%

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

    2. *-rgt-identity 2.6%

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

14. Simplified 2.6%

    \[\leadsto \color{blue}{\frac{b\_2}{a}} \]
15. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024137 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  :herbie-expected 10

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

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