quad2m (problem 3.2.1, negative)

Percentage Accurate: 51.4% → 85.2%

Time: 10.5s

Alternatives: 11

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: 51.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.42 \cdot 10^{-38}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6.4 \cdot 10^{+47}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.42e-38)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 6.4e+47)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.42e-38) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 6.4e+47) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.42d-38)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 6.4d+47) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.42e-38) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 6.4e+47) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.42e-38:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 6.4e+47:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.42e-38)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 6.4e+47)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.42e-38)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 6.4e+47)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.42e-38], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 6.4e+47], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.41999999999999988e-38

   1. Initial program 13.7%

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

   3. Taylor expanded in b_2 around -inf 89.7%

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

      1. associate-*r/ 89.7%

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

   5. Simplified 89.7%

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

   if -1.41999999999999988e-38 < b_2 < 6.4e47

   1. Initial program 79.3%

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

   if 6.4e47 < b_2

   1. Initial program 55.3%

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

   3. Taylor expanded in c around 0 95.7%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.42 \cdot 10^{-38}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6.4 \cdot 10^{+47}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -9 \cdot 10^{-85}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 7 \cdot 10^{-60}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + c \cdot \frac{0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -9e-85)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 7e-60)
     (/ (- (- b_2) (sqrt (* a (- c)))) a)
     (+ (* -2.0 (/ b_2 a)) (* c (/ 0.5 b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9e-85) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 7e-60) {
		tmp = (-b_2 - sqrt((a * -c))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (c * (0.5 / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-9d-85)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 7d-60) then
        tmp = (-b_2 - sqrt((a * -c))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (c * (0.5d0 / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9e-85) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 7e-60) {
		tmp = (-b_2 - Math.sqrt((a * -c))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (c * (0.5 / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -9e-85:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 7e-60:
		tmp = (-b_2 - math.sqrt((a * -c))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + (c * (0.5 / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -9e-85)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 7e-60)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(a * Float64(-c)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + 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 <= -9e-85)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 7e-60)
		tmp = (-b_2 - sqrt((a * -c))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + (c * (0.5 / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -9e-85], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 7e-60], N[(N[((-b$95$2) - N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(c * N[(0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -9.00000000000000008e-85

   1. Initial program 16.4%

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

   3. Taylor expanded in b_2 around -inf 86.4%

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

      1. associate-*r/ 86.4%

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

   5. Simplified 86.4%

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

   if -9.00000000000000008e-85 < b_2 < 6.99999999999999952e-60

   1. Initial program 78.3%

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

   3. Taylor expanded in b_2 around 0 73.2%

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

      1. mul-1-neg 73.2%

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

      2. distribute-rgt-neg-out 73.2%

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

   5. Simplified 73.2%

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

   if 6.99999999999999952e-60 < b_2

   1. Initial program 63.1%

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

   3. Taylor expanded in c around 0 90.1%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. clear-num 90.1%

         \[\leadsto -2 \cdot \frac{b\_2}{a} + 0.5 \cdot \color{blue}{\frac{1}{\frac{b\_2}{c}}} \]

      2. un-div-inv 90.1%

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

   5. Applied egg-rr 90.1%

      \[\leadsto -2 \cdot \frac{b\_2}{a} + \color{blue}{\frac{0.5}{\frac{b\_2}{c}}} \]
   6. Step-by-step derivation

      1. associate-/r/ 90.1%

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

   7. Simplified 90.1%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -9 \cdot 10^{-85}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 7 \cdot 10^{-60}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + c \cdot \frac{0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.65 \cdot 10^{-144}:\\ \;\;\;\;\frac{b\_2}{a} - \sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + c \cdot \frac{0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7.5e-81)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1.65e-144)
     (- (/ b_2 a) (sqrt (/ (- c) a)))
     (+ (* -2.0 (/ b_2 a)) (* c (/ 0.5 b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.5e-81) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.65e-144) {
		tmp = (b_2 / a) - sqrt((-c / a));
	} else {
		tmp = (-2.0 * (b_2 / a)) + (c * (0.5 / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-7.5d-81)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 1.65d-144) then
        tmp = (b_2 / a) - sqrt((-c / a))
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (c * (0.5d0 / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.5e-81) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.65e-144) {
		tmp = (b_2 / a) - Math.sqrt((-c / a));
	} else {
		tmp = (-2.0 * (b_2 / a)) + (c * (0.5 / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -7.5e-81:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 1.65e-144:
		tmp = (b_2 / a) - math.sqrt((-c / a))
	else:
		tmp = (-2.0 * (b_2 / a)) + (c * (0.5 / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -7.5e-81)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1.65e-144)
		tmp = Float64(Float64(b_2 / a) - sqrt(Float64(Float64(-c) / a)));
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + 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 <= -7.5e-81)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 1.65e-144)
		tmp = (b_2 / a) - sqrt((-c / a));
	else
		tmp = (-2.0 * (b_2 / a)) + (c * (0.5 / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7.5e-81], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.65e-144], N[(N[(b$95$2 / a), $MachinePrecision] - N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(c * N[(0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -7.50000000000000018e-81

   1. Initial program 16.5%

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

   3. Taylor expanded in b_2 around -inf 87.1%

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

      1. associate-*r/ 87.1%

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

   5. Simplified 87.1%

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

   if -7.50000000000000018e-81 < b_2 < 1.64999999999999998e-144

   1. Initial program 76.1%

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

      1. prod-diff 75.5%

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

      2. *-commutative 75.5%

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

      3. fma-neg 75.5%

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

      4. prod-diff 75.5%

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

      5. *-commutative 75.5%

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

      6. fma-neg 75.5%

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

      7. associate-+l+ 75.5%

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

      8. pow2 75.5%

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

      9. *-commutative 75.5%

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

      10. fma-undefine 75.5%

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

      11. distribute-lft-neg-in 75.5%

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

      12. *-commutative 75.5%

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

      13. distribute-rgt-neg-in 75.5%

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

      14. fma-define 75.5%

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

      15. *-commutative 75.5%

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

      16. fma-undefine 75.5%

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

      17. distribute-lft-neg-in 75.5%

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

      18. *-commutative 75.5%

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

      19. distribute-rgt-neg-in 75.5%

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

   4. Applied egg-rr 75.5%

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

      1. count-2 75.5%

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

   6. Simplified 75.5%

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

   7. Taylor expanded in a around inf 39.3%

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

      1. +-commutative 39.3%

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

      2. mul-1-neg 39.3%

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

      3. sub-neg 39.3%

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

      4. associate-*r/ 39.3%

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

      5. mul-1-neg 39.3%

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

      6. *-commutative 39.3%

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

      7. distribute-rgt1-in 39.3%

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

      8. metadata-eval 39.3%

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

   9. Simplified 39.3%

      \[\leadsto \color{blue}{\frac{-b\_2}{a} - \sqrt{\frac{\left(0 \cdot c\right) \cdot 2 - c}{a}}} \]
   10. Step-by-step derivation

       1. sub-neg 39.3%

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

       2. add-sqr-sqrt 23.0%

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

       3. sqrt-unprod 39.5%

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

       4. sqr-neg 39.5%

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

       5. sqrt-unprod 16.3%

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

       6. add-sqr-sqrt 39.4%

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

       7. frac-2neg 39.4%

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

   11. Applied egg-rr 39.4%

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

       1. sub-neg 39.4%

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

       2. distribute-frac-neg2 39.4%

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

       3. distribute-neg-frac 39.4%

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

   13. Simplified 39.4%

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

   if 1.64999999999999998e-144 < b_2

   1. Initial program 65.8%

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

   3. Taylor expanded in c around 0 80.8%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. clear-num 80.8%

         \[\leadsto -2 \cdot \frac{b\_2}{a} + 0.5 \cdot \color{blue}{\frac{1}{\frac{b\_2}{c}}} \]

      2. un-div-inv 80.8%

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

   5. Applied egg-rr 80.8%

      \[\leadsto -2 \cdot \frac{b\_2}{a} + \color{blue}{\frac{0.5}{\frac{b\_2}{c}}} \]
   6. Step-by-step derivation

      1. associate-/r/ 80.8%

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

   7. Simplified 80.8%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.65 \cdot 10^{-144}:\\ \;\;\;\;\frac{b\_2}{a} - \sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + c \cdot \frac{0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 4.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)}}{-a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + c \cdot \frac{0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.2e-85)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 4.2e-60)
     (/ (sqrt (* a (- c))) (- a))
     (+ (* -2.0 (/ b_2 a)) (* c (/ 0.5 b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.2e-85) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 4.2e-60) {
		tmp = sqrt((a * -c)) / -a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (c * (0.5 / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-4.2d-85)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 4.2d-60) then
        tmp = sqrt((a * -c)) / -a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (c * (0.5d0 / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.2e-85) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 4.2e-60) {
		tmp = Math.sqrt((a * -c)) / -a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (c * (0.5 / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4.2e-85:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 4.2e-60:
		tmp = math.sqrt((a * -c)) / -a
	else:
		tmp = (-2.0 * (b_2 / a)) + (c * (0.5 / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4.2e-85)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 4.2e-60)
		tmp = Float64(sqrt(Float64(a * Float64(-c))) / Float64(-a));
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + 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 <= -4.2e-85)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 4.2e-60)
		tmp = sqrt((a * -c)) / -a;
	else
		tmp = (-2.0 * (b_2 / a)) + (c * (0.5 / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4.2e-85], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 4.2e-60], N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] / (-a)), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(c * N[(0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.2e-85

   1. Initial program 16.4%

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

   3. Taylor expanded in b_2 around -inf 86.4%

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

      1. associate-*r/ 86.4%

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

   5. Simplified 86.4%

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

   if -4.2e-85 < b_2 < 4.19999999999999982e-60

   1. Initial program 78.3%

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

      1. prod-diff 77.7%

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

      2. *-commutative 77.7%

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

      3. fma-neg 77.7%

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

      4. prod-diff 77.7%

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

      5. *-commutative 77.7%

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

      6. fma-neg 77.7%

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

      7. associate-+l+ 77.6%

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

      8. pow2 77.6%

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

      9. *-commutative 77.6%

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

      10. fma-undefine 77.7%

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

      11. distribute-lft-neg-in 77.7%

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

      12. *-commutative 77.7%

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

      13. distribute-rgt-neg-in 77.7%

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

      14. fma-define 77.6%

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

      15. *-commutative 77.6%

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

      16. fma-undefine 77.7%

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

      17. distribute-lft-neg-in 77.7%

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

      18. *-commutative 77.7%

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

      19. distribute-rgt-neg-in 77.7%

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

   4. Applied egg-rr 77.6%

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

      1. count-2 77.6%

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

   6. Simplified 77.6%

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

   7. Taylor expanded in b_2 around 0 70.7%

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

      1. mul-1-neg 70.7%

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

      2. mul-1-neg 70.7%

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

      3. +-commutative 70.7%

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

      4. sub-neg 70.7%

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

      5. +-inverses 71.3%

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

      6. metadata-eval 71.3%

         \[\leadsto \frac{-\sqrt{\color{blue}{0} - a \cdot c}}{a} \]

   9. Simplified 71.3%

      \[\leadsto \frac{\color{blue}{-\sqrt{0 - a \cdot c}}}{a} \]

   if 4.19999999999999982e-60 < b_2

   1. Initial program 63.1%

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

   3. Taylor expanded in c around 0 90.1%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. clear-num 90.1%

         \[\leadsto -2 \cdot \frac{b\_2}{a} + 0.5 \cdot \color{blue}{\frac{1}{\frac{b\_2}{c}}} \]

      2. un-div-inv 90.1%

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

   5. Applied egg-rr 90.1%

      \[\leadsto -2 \cdot \frac{b\_2}{a} + \color{blue}{\frac{0.5}{\frac{b\_2}{c}}} \]
   6. Step-by-step derivation

      1. associate-/r/ 90.1%

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

   7. Simplified 90.1%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 4.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)}}{-a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + c \cdot \frac{0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.9% accurate, 7.0× speedup

Math

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

FPCore

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

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (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 = ((-0.5d0) * c) / b_2
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (c * (0.5d0 / b_2))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 30.9%

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

   3. Taylor expanded in b_2 around -inf 68.0%

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

      1. associate-*r/ 68.0%

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

   5. Simplified 68.0%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 67.0%

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

   3. Taylor expanded in c around 0 70.4%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. clear-num 70.4%

         \[\leadsto -2 \cdot \frac{b\_2}{a} + 0.5 \cdot \color{blue}{\frac{1}{\frac{b\_2}{c}}} \]

      2. un-div-inv 70.4%

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

   5. Applied egg-rr 70.4%

      \[\leadsto -2 \cdot \frac{b\_2}{a} + \color{blue}{\frac{0.5}{\frac{b\_2}{c}}} \]
   6. Step-by-step derivation

      1. associate-/r/ 70.4%

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

   7. Simplified 70.4%

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

4. Final simplification 69.0%

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

Alternative 6: 68.9% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 30.9%

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

   3. Taylor expanded in b_2 around -inf 68.0%

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

      1. associate-*r/ 68.0%

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

   5. Simplified 68.0%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 67.0%

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

   3. Taylor expanded in c around 0 70.4%

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

Alternative 7: 68.6% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.1 \cdot 10^{-283}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.1e-283) (/ (* -0.5 c) b_2) (/ (* b_2 -2.0) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.1e-283) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.1d-283)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.1e-283) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.1e-283:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.1e-283)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.1e-283)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.1e-283], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.0999999999999999e-283

   1. Initial program 30.2%

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

   3. Taylor expanded in b_2 around -inf 69.3%

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

      1. associate-*r/ 69.3%

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

   5. Simplified 69.3%

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

   if -1.0999999999999999e-283 < b_2

   1. Initial program 67.0%

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

   3. Taylor expanded in b_2 around inf 68.1%

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

      1. *-commutative 68.1%

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

   5. Simplified 68.1%

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

Alternative 8: 68.6% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.8 \cdot 10^{-283}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.8e-283) (/ (* -0.5 c) b_2) (* b_2 (/ -2.0 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.8e-283) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = b_2 * (-2.0 / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.8d-283)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = b_2 * ((-2.0d0) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.8e-283) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = b_2 * (-2.0 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.8e-283:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = b_2 * (-2.0 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.8e-283)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(b_2 * Float64(-2.0 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.8e-283)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = b_2 * (-2.0 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.8e-283], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.8e-283

   1. Initial program 30.2%

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

   3. Taylor expanded in b_2 around -inf 69.3%

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

      1. associate-*r/ 69.3%

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

   5. Simplified 69.3%

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

   if -1.8e-283 < b_2

   1. Initial program 67.0%

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

      1. frac-2neg 67.0%

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

      2. div-inv 66.9%

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

   4. Applied egg-rr 61.2%

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

   5. Taylor expanded in b_2 around inf 68.1%

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

      1. associate-*r/ 68.1%

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

      2. *-commutative 68.1%

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

      3. associate-/l* 67.9%

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

   7. Simplified 67.9%

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

Alternative 9: 68.4% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.2 \cdot 10^{-284}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7.2e-284) (* c (/ -0.5 b_2)) (* b_2 (/ -2.0 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.2e-284) {
		tmp = c * (-0.5 / b_2);
	} else {
		tmp = b_2 * (-2.0 / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-7.2d-284)) then
        tmp = c * ((-0.5d0) / b_2)
    else
        tmp = b_2 * ((-2.0d0) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.2e-284) {
		tmp = c * (-0.5 / b_2);
	} else {
		tmp = b_2 * (-2.0 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -7.2e-284:
		tmp = c * (-0.5 / b_2)
	else:
		tmp = b_2 * (-2.0 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -7.2e-284)
		tmp = Float64(c * Float64(-0.5 / b_2));
	else
		tmp = Float64(b_2 * Float64(-2.0 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -7.2e-284)
		tmp = c * (-0.5 / b_2);
	else
		tmp = b_2 * (-2.0 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7.2e-284], N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -7.2000000000000004e-284

   1. Initial program 30.2%

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

      1. frac-2neg 30.2%

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

      2. div-inv 30.1%

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

   4. Applied egg-rr 36.8%

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

   5. Taylor expanded in b_2 around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 69.3%

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

      4. neg-mul-1 69.3%

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

      5. distribute-neg-frac 69.3%

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

      6. distribute-rgt-neg-in 69.3%

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

      7. associate-*r/ 69.3%

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

      8. *-commutative 69.3%

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

      9. distribute-frac-neg2 69.3%

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

      10. associate-/l* 69.1%

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

      11. distribute-frac-neg2 69.1%

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

      12. distribute-neg-frac 69.1%

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

      13. metadata-eval 69.1%

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

   7. Simplified 69.1%

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

   if -7.2000000000000004e-284 < b_2

   1. Initial program 67.0%

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

      1. frac-2neg 67.0%

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

      2. div-inv 66.9%

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

   4. Applied egg-rr 61.2%

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

   5. Taylor expanded in b_2 around inf 68.1%

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

      1. associate-*r/ 68.1%

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

      2. *-commutative 68.1%

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

      3. associate-/l* 67.9%

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

   7. Simplified 67.9%

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

Alternative 10: 35.9% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ b\_2 \cdot \frac{-2}{a} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* b_2 (/ -2.0 a)))

C

double code(double a, double b_2, double c) {
	return b_2 * (-2.0 / a);
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = b_2 * ((-2.0d0) / a)
end function

Java

public static double code(double a, double b_2, double c) {
	return b_2 * (-2.0 / a);
}

Python

def code(a, b_2, c):
	return b_2 * (-2.0 / a)

Julia

function code(a, b_2, c)
	return Float64(b_2 * Float64(-2.0 / a))
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = b_2 * (-2.0 / a);
end

Wolfram

code[a_, b$95$2_, c_] := N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.4%

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

   1. frac-2neg 46.4%

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

   2. div-inv 46.3%

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

4. Applied egg-rr 47.6%

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

5. Taylor expanded in b_2 around inf 31.5%

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

   1. associate-*r/ 31.5%

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

   2. *-commutative 31.5%

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

   3. associate-/l* 31.4%

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

7. Simplified 31.4%

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

Alternative 11: 15.4% accurate, 28.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.4%

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

3. Taylor expanded in b_2 around 0 29.6%

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

   1. mul-1-neg 29.6%

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

   2. distribute-rgt-neg-out 29.6%

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

5. Simplified 29.6%

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

6. Taylor expanded in b_2 around inf 12.8%

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

   1. associate-*r/ 12.8%

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

   2. mul-1-neg 12.8%

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

8. Simplified 12.8%

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

9. Final simplification 12.8%

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

Developer target: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024103 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  :herbie-expected 10

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

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