quad2m (problem 3.2.1, negative)

Percentage Accurate: 51.6% → 85.5%

Time: 12.5s

Alternatives: 10

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: 51.6% 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: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.45 \cdot 10^{-125}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.55 \cdot 10^{+107}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.45e-125)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 2.55e+107)
     (/ (- (- 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 <= -4.45e-125) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 2.55e+107) {
		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 <= (-4.45d-125)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 2.55d+107) 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 <= -4.45e-125) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 2.55e+107) {
		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 <= -4.45e-125:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 2.55e+107:
		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 <= -4.45e-125)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 2.55e+107)
		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 <= -4.45e-125)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 2.55e+107)
		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, -4.45e-125], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 2.55e+107], 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 < -4.45000000000000001e-125

   1. Initial program 19.1%

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

   3. Taylor expanded in b_2 around -inf 86.2%

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

      1. associate-*r/ 86.2%

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

   5. Simplified 86.2%

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

   if -4.45000000000000001e-125 < b_2 < 2.5500000000000001e107

   1. Initial program 84.8%

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

   if 2.5500000000000001e107 < b_2

   1. Initial program 58.4%

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

   3. Taylor expanded in c around 0 97.3%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.45 \cdot 10^{-125}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.55 \cdot 10^{+107}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.4 \cdot 10^{-125}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.9 \cdot 10^{-49}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.4e-125)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 2.9e-49)
     (/ (- (- b_2) (sqrt (* 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 <= -4.4e-125) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 2.9e-49) {
		tmp = (-b_2 - sqrt((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 <= (-4.4d-125)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 2.9d-49) then
        tmp = (-b_2 - sqrt((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 <= -4.4e-125) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 2.9e-49) {
		tmp = (-b_2 - Math.sqrt((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 <= -4.4e-125:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 2.9e-49:
		tmp = (-b_2 - math.sqrt((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 <= -4.4e-125)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 2.9e-49)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(c * Float64(-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 <= -4.4e-125)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 2.9e-49)
		tmp = (-b_2 - sqrt((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, -4.4e-125], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 2.9e-49], N[(N[((-b$95$2) - N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.3999999999999999e-125

   1. Initial program 19.1%

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

   3. Taylor expanded in b_2 around -inf 86.2%

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

      1. associate-*r/ 86.2%

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

   5. Simplified 86.2%

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

   if -4.3999999999999999e-125 < b_2 < 2.9e-49

   1. Initial program 80.1%

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

   3. Taylor expanded in b_2 around 0 65.9%

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

      1. associate-*r* 65.9%

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

      2. neg-mul-1 65.9%

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

   5. Simplified 65.9%

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

   if 2.9e-49 < b_2

   1. Initial program 70.5%

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

   3. Taylor expanded in c around 0 93.4%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.4 \cdot 10^{-125}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.9 \cdot 10^{-49}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.45 \cdot 10^{-125}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6.5 \cdot 10^{-226}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)}}{-a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.45e-125)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 6.5e-226)
     (/ (sqrt (* 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 <= -4.45e-125) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 6.5e-226) {
		tmp = sqrt((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 <= (-4.45d-125)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 6.5d-226) then
        tmp = sqrt((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 <= -4.45e-125) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 6.5e-226) {
		tmp = Math.sqrt((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 <= -4.45e-125:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 6.5e-226:
		tmp = math.sqrt((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 <= -4.45e-125)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 6.5e-226)
		tmp = Float64(sqrt(Float64(c * Float64(-a))) / Float64(-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 <= -4.45e-125)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 6.5e-226)
		tmp = sqrt((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, -4.45e-125], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 6.5e-226], N[(N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision] / (-a)), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.45000000000000001e-125

   1. Initial program 19.1%

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

   3. Taylor expanded in b_2 around -inf 86.2%

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

      1. associate-*r/ 86.2%

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

   5. Simplified 86.2%

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

   if -4.45000000000000001e-125 < b_2 < 6.50000000000000033e-226

   1. Initial program 74.1%

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

      1. prod-diff 73.8%

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

      2. *-commutative 73.8%

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

      3. fma-define 73.8%

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

      4. associate-+l+ 73.8%

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

      5. pow2 73.8%

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

      6. distribute-rgt-neg-in 73.8%

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

      7. fma-define 73.7%

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

      8. *-commutative 73.7%

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

      9. fma-undefine 73.8%

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

      10. distribute-lft-neg-in 73.8%

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

      11. *-commutative 73.8%

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

      12. distribute-rgt-neg-in 73.8%

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

      13. fma-define 73.7%

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

   4. Applied egg-rr 73.7%

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

      1. unpow2 73.7%

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

      2. fma-undefine 73.7%

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

      3. *-commutative 73.7%

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

   6. Simplified 73.7%

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

   7. Taylor expanded in b_2 around 0 73.8%

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

      1. mul-1-neg 73.8%

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

      2. +-commutative 73.8%

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

      3. associate-*r* 73.8%

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

      4. distribute-rgt-in 74.2%

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

      5. distribute-rgt1-in 74.2%

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

      6. metadata-eval 74.2%

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

      7. mul-1-neg 74.2%

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

   9. Simplified 74.2%

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

   if 6.50000000000000033e-226 < b_2

   1. Initial program 74.4%

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

   3. Taylor expanded in c around 0 83.7%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.45 \cdot 10^{-125}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6.5 \cdot 10^{-226}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)}}{-a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.95 \cdot 10^{-126}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5 \cdot 10^{-203}:\\ \;\;\;\;-\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.95e-126)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 5e-203)
     (- (sqrt (/ c (- a))))
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.95e-126) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 5e-203) {
		tmp = -sqrt((c / -a));
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.95d-126)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 5d-203) then
        tmp = -sqrt((c / -a))
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.95e-126) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 5e-203) {
		tmp = -Math.sqrt((c / -a));
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.95e-126:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 5e-203:
		tmp = -math.sqrt((c / -a))
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.95e-126)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 5e-203)
		tmp = Float64(-sqrt(Float64(c / Float64(-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 <= -1.95e-126)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 5e-203)
		tmp = -sqrt((c / -a));
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.95e-126], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 5e-203], (-N[Sqrt[N[(c / (-a)), $MachinePrecision]], $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 < -1.9499999999999999e-126

   1. Initial program 19.1%

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

   3. Taylor expanded in b_2 around -inf 86.2%

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

      1. associate-*r/ 86.2%

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

   5. Simplified 86.2%

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

   if -1.9499999999999999e-126 < b_2 < 5.0000000000000002e-203

   1. Initial program 71.6%

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

      1. prod-diff 71.2%

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

      2. *-commutative 71.2%

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

      3. fma-define 71.2%

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

      4. associate-+l+ 71.2%

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

      5. pow2 71.2%

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

      6. distribute-rgt-neg-in 71.2%

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

      7. fma-define 71.1%

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

      8. *-commutative 71.1%

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

      9. fma-undefine 71.2%

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

      10. distribute-lft-neg-in 71.2%

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

      11. *-commutative 71.2%

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

      12. distribute-rgt-neg-in 71.2%

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

      13. fma-define 71.1%

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

   4. Applied egg-rr 71.1%

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

      1. unpow2 71.1%

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

      2. fma-undefine 71.1%

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

      3. *-commutative 71.1%

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

   6. Simplified 71.1%

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

   7. Taylor expanded in a around inf 40.4%

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

      1. mul-1-neg 40.4%

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

      2. distribute-rgt1-in 40.4%

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

      3. metadata-eval 40.4%

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

      4. mul-1-neg 40.4%

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

   9. Simplified 40.4%

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

   if 5.0000000000000002e-203 < b_2

   1. Initial program 75.2%

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

   3. Taylor expanded in c around 0 84.9%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.95 \cdot 10^{-126}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5 \cdot 10^{-203}:\\ \;\;\;\;-\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.6% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 31.4%

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

   3. Taylor expanded in b_2 around -inf 67.2%

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

      1. associate-*r/ 67.2%

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

   5. Simplified 67.2%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 74.9%

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

   3. Taylor expanded in c around 0 79.5%

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

Alternative 6: 68.4% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -8 \cdot 10^{-283}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -8e-283) (/ (* -0.5 c) b_2) (* -2.0 (/ b_2 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -8e-283) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = -2.0 * (b_2 / 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 <= (-8d-283)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = (-2.0d0) * (b_2 / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -8e-283) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -8e-283:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -8e-283)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -8e-283)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = -2.0 * (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -8e-283], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -7.99999999999999957e-283

   1. Initial program 30.0%

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

   3. Taylor expanded in b_2 around -inf 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 -7.99999999999999957e-283 < b_2

   1. Initial program 74.9%

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

   3. Taylor expanded in b_2 around inf 76.6%

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

      1. *-commutative 76.6%

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

   5. Simplified 76.6%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -8 \cdot 10^{-283}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.8% accurate, 22.4× speedup

Math

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

FPCore

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

C

double code(double a, double b_2, double c) {
	return -2.0 * (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 = (-2.0d0) * (b_2 / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.8%

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

3. Taylor expanded in b_2 around inf 42.0%

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

   1. *-commutative 42.0%

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

5. Simplified 42.0%

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

6. Final simplification 42.0%

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

Alternative 8: 35.7% 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 53.8%

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

3. Taylor expanded in b_2 around inf 42.0%

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

   1. *-commutative 42.0%

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

5. Simplified 42.0%

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

   1. *-commutative 42.0%

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

   2. clear-num 41.9%

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

   3. un-div-inv 41.9%

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

7. Applied egg-rr 41.9%

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

   1. associate-/r/ 42.0%

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

9. Simplified 42.0%

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

10. Final simplification 42.0%

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

Alternative 9: 15.1% 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 53.8%

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

3. Taylor expanded in b_2 around 0 31.9%

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

   1. associate-*r* 31.9%

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

   2. neg-mul-1 31.9%

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

5. Simplified 31.9%

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

6. Taylor expanded in b_2 around inf 19.2%

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

   1. neg-mul-1 19.2%

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

   2. distribute-neg-frac2 19.2%

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

8. Simplified 19.2%

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

Alternative 10: 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 53.8%

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

3. Taylor expanded in b_2 around 0 31.9%

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

   1. associate-*r* 31.9%

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

   2. neg-mul-1 31.9%

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

5. Simplified 31.9%

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

6. Taylor expanded in b_2 around inf 19.2%

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

   1. neg-mul-1 19.2%

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

   2. distribute-neg-frac2 19.2%

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

8. Simplified 19.2%

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

   1. *-un-lft-identity 19.2%

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

   2. add-sqr-sqrt 9.3%

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

   3. sqrt-unprod 12.2%

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

   4. sqr-neg 12.2%

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

   5. sqrt-unprod 1.2%

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

   6. add-sqr-sqrt 2.4%

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

10. Applied egg-rr 2.4%

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

    1. *-lft-identity 2.4%

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

12. Simplified 2.4%

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

Developer Target 1: 99.7% accurate, 0.1× speedup

Math

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

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 2024144 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  :herbie-expected 10

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

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