quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.7% → 86.2%

Time: 11.5s

Alternatives: 8

Speedup: 11.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b\_2 \cdot b\_2 - c \cdot a}\\ \mathbf{if}\;b\_2 \leq -1.75 \cdot 10^{+72}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq -1.25 \cdot 10^{-148}:\\ \;\;\;\;\frac{\frac{c \cdot a}{t\_0 - b\_2}}{a}\\ \mathbf{elif}\;b\_2 \leq 60000:\\ \;\;\;\;\frac{\left(0 - b\_2\right) - t\_0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b_2 b_2) (* c a)))))
   (if (<= b_2 -1.75e+72)
     (/ (* c -0.5) b_2)
     (if (<= b_2 -1.25e-148)
       (/ (/ (* c a) (- t_0 b_2)) a)
       (if (<= b_2 60000.0) (/ (- (- 0.0 b_2) t_0) a) (/ (* b_2 -2.0) a))))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(((b_2 * b_2) - (c * a)));
	double tmp;
	if (b_2 <= -1.75e+72) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= -1.25e-148) {
		tmp = ((c * a) / (t_0 - b_2)) / a;
	} else if (b_2 <= 60000.0) {
		tmp = ((0.0 - b_2) - t_0) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b_2 * b_2) - (c * a)))
    if (b_2 <= (-1.75d+72)) then
        tmp = (c * (-0.5d0)) / b_2
    else if (b_2 <= (-1.25d-148)) then
        tmp = ((c * a) / (t_0 - b_2)) / a
    else if (b_2 <= 60000.0d0) then
        tmp = ((0.0d0 - b_2) - t_0) / a
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt(((b_2 * b_2) - (c * a)));
	double tmp;
	if (b_2 <= -1.75e+72) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= -1.25e-148) {
		tmp = ((c * a) / (t_0 - b_2)) / a;
	} else if (b_2 <= 60000.0) {
		tmp = ((0.0 - b_2) - t_0) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt(((b_2 * b_2) - (c * a)))
	tmp = 0
	if b_2 <= -1.75e+72:
		tmp = (c * -0.5) / b_2
	elif b_2 <= -1.25e-148:
		tmp = ((c * a) / (t_0 - b_2)) / a
	elif b_2 <= 60000.0:
		tmp = ((0.0 - b_2) - t_0) / a
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	t_0 = sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))
	tmp = 0.0
	if (b_2 <= -1.75e+72)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= -1.25e-148)
		tmp = Float64(Float64(Float64(c * a) / Float64(t_0 - b_2)) / a);
	elseif (b_2 <= 60000.0)
		tmp = Float64(Float64(Float64(0.0 - b_2) - t_0) / a);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	t_0 = sqrt(((b_2 * b_2) - (c * a)));
	tmp = 0.0;
	if (b_2 <= -1.75e+72)
		tmp = (c * -0.5) / b_2;
	elseif (b_2 <= -1.25e-148)
		tmp = ((c * a) / (t_0 - b_2)) / a;
	elseif (b_2 <= 60000.0)
		tmp = ((0.0 - b_2) - t_0) / a;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$2, -1.75e+72], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, -1.25e-148], N[(N[(N[(c * a), $MachinePrecision] / N[(t$95$0 - b$95$2), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 60000.0], N[(N[(N[(0.0 - b$95$2), $MachinePrecision] - t$95$0), $MachinePrecision] / a), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.75000000000000005e72

   1. Initial program 3.9%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b\_2}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\_2\right) \]

      4. *-lowering-*.f64 97.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\_2\right) \]

   5. Simplified 97.0%

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

   if -1.75000000000000005e72 < b_2 < -1.25e-148

   1. Initial program 49.2%

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

   4. Applied egg-rr 49.3%

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

   5. Taylor expanded in b_2 around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot c\right)\right)}, \mathsf{\_.f64}\left(b\_2, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \mathsf{*.f64}\left(a, c\right)\right)\right)\right)\right), a\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(a \cdot c\right)\right), \mathsf{\_.f64}\left(b\_2, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \mathsf{*.f64}\left(a, c\right)\right)\right)\right)\right), a\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(0 - a \cdot c\right), \mathsf{\_.f64}\left(b\_2, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \mathsf{*.f64}\left(a, c\right)\right)\right)\right)\right), a\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot c\right)\right), \mathsf{\_.f64}\left(b\_2, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \mathsf{*.f64}\left(a, c\right)\right)\right)\right)\right), a\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(c \cdot a\right)\right), \mathsf{\_.f64}\left(b\_2, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \mathsf{*.f64}\left(a, c\right)\right)\right)\right)\right), a\right) \]

      5. *-lowering-*.f64 87.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(c, a\right)\right), \mathsf{\_.f64}\left(b\_2, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \mathsf{*.f64}\left(a, c\right)\right)\right)\right)\right), a\right) \]

   7. Simplified 87.1%

      \[\leadsto \frac{\frac{\color{blue}{0 - c \cdot a}}{b\_2 - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

   if -1.25e-148 < b_2 < 6e4

   1. Initial program 82.2%

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

   if 6e4 < b_2

   1. Initial program 71.4%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-2 \cdot b\_2\right)}, a\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b\_2 \cdot -2\right), a\right) \]

      2. *-lowering-*.f64 95.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b\_2, -2\right), a\right) \]

   5. Simplified 95.8%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.75 \cdot 10^{+72}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq -1.25 \cdot 10^{-148}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\sqrt{b\_2 \cdot b\_2 - c \cdot a} - b\_2}}{a}\\ \mathbf{elif}\;b\_2 \leq 60000:\\ \;\;\;\;\frac{\left(0 - b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.8 \cdot 10^{-85}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 60000:\\ \;\;\;\;\frac{\left(0 - b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.8e-85)
   (/ (* c -0.5) b_2)
   (if (<= b_2 60000.0)
     (/ (- (- 0.0 b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (/ (* b_2 -2.0) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.8e-85) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 60000.0) {
		tmp = ((0.0 - b_2) - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-4.8d-85)) then
        tmp = (c * (-0.5d0)) / b_2
    else if (b_2 <= 60000.0d0) then
        tmp = ((0.0d0 - b_2) - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.8e-85) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 60000.0) {
		tmp = ((0.0 - b_2) - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4.8e-85:
		tmp = (c * -0.5) / b_2
	elif b_2 <= 60000.0:
		tmp = ((0.0 - b_2) - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4.8e-85)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 60000.0)
		tmp = Float64(Float64(Float64(0.0 - b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4.8e-85)
		tmp = (c * -0.5) / b_2;
	elseif (b_2 <= 60000.0)
		tmp = ((0.0 - b_2) - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4.8e-85], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 60000.0], N[(N[(N[(0.0 - b$95$2), $MachinePrecision] - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.8000000000000001e-85

   1. Initial program 17.4%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b\_2}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\_2\right) \]

      4. *-lowering-*.f64 86.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\_2\right) \]

   5. Simplified 86.7%

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

   if -4.8000000000000001e-85 < b_2 < 6e4

   1. Initial program 82.3%

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

   if 6e4 < b_2

   1. Initial program 71.4%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-2 \cdot b\_2\right)}, a\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b\_2 \cdot -2\right), a\right) \]

      2. *-lowering-*.f64 95.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b\_2, -2\right), a\right) \]

   5. Simplified 95.8%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.8 \cdot 10^{-85}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 60000:\\ \;\;\;\;\frac{\left(0 - b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.2 \cdot 10^{-81}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.4 \cdot 10^{-36}:\\ \;\;\;\;\frac{\left(0 - b\_2\right) - \sqrt{c \cdot \left(0 - a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.2e-81)
   (/ (* c -0.5) b_2)
   (if (<= b_2 1.4e-36)
     (/ (- (- 0.0 b_2) (sqrt (* c (- 0.0 a)))) a)
     (/ (* b_2 -2.0) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.2e-81) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 1.4e-36) {
		tmp = ((0.0 - b_2) - sqrt((c * (0.0 - a)))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-3.2d-81)) then
        tmp = (c * (-0.5d0)) / b_2
    else if (b_2 <= 1.4d-36) then
        tmp = ((0.0d0 - b_2) - sqrt((c * (0.0d0 - a)))) / a
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.2e-81) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 1.4e-36) {
		tmp = ((0.0 - b_2) - Math.sqrt((c * (0.0 - a)))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.2e-81:
		tmp = (c * -0.5) / b_2
	elif b_2 <= 1.4e-36:
		tmp = ((0.0 - b_2) - math.sqrt((c * (0.0 - a)))) / a
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.2e-81)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 1.4e-36)
		tmp = Float64(Float64(Float64(0.0 - b_2) - sqrt(Float64(c * Float64(0.0 - a)))) / a);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.2e-81)
		tmp = (c * -0.5) / b_2;
	elseif (b_2 <= 1.4e-36)
		tmp = ((0.0 - b_2) - sqrt((c * (0.0 - a)))) / a;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.2e-81], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.4e-36], N[(N[(N[(0.0 - b$95$2), $MachinePrecision] - N[Sqrt[N[(c * N[(0.0 - a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.2e-81

   1. Initial program 17.4%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b\_2}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\_2\right) \]

      4. *-lowering-*.f64 86.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\_2\right) \]

   5. Simplified 86.7%

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

   if -3.2e-81 < b_2 < 1.4000000000000001e-36

   1. Initial program 81.1%

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

   3. Taylor expanded in b_2 around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot c\right)\right)}\right)\right), a\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(a \cdot c\right)\right)\right)\right), a\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\left(0 - a \cdot c\right)\right)\right), a\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot c\right)\right)\right)\right), a\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \left(c \cdot a\right)\right)\right)\right), a\right) \]

      5. *-lowering-*.f64 72.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(c, a\right)\right)\right)\right), a\right) \]

   5. Simplified 72.9%

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

   if 1.4000000000000001e-36 < b_2

   1. Initial program 74.2%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-2 \cdot b\_2\right)}, a\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b\_2 \cdot -2\right), a\right) \]

      2. *-lowering-*.f64 91.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b\_2, -2\right), a\right) \]

   5. Simplified 91.8%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.2 \cdot 10^{-81}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.4 \cdot 10^{-36}:\\ \;\;\;\;\frac{\left(0 - b\_2\right) - \sqrt{c \cdot \left(0 - a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.5% accurate, 11.2× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310) (/ (* 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 <= -5e-310) {
		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 <= (-5d-310)) 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 <= -5e-310) {
		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 <= -5e-310:
		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 <= -5e-310)
		tmp = Float64(Float64(c * -0.5) / 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 <= -5e-310)
		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, -5e-310], N[(N[(c * -0.5), $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 < -4.999999999999985e-310

   1. Initial program 31.8%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b\_2}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\_2\right) \]

      4. *-lowering-*.f64 70.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\_2\right) \]

   5. Simplified 70.5%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 77.0%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-2 \cdot b\_2\right)}, a\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b\_2 \cdot -2\right), a\right) \]

      2. *-lowering-*.f64 64.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b\_2, -2\right), a\right) \]

   5. Simplified 64.1%

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

Alternative 5: 42.6% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4 \cdot 10^{-5}:\\ \;\;\;\;\frac{c \cdot 0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e-5) (/ (* 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 <= -4e-5) {
		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 <= (-4d-5)) 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 <= -4e-5) {
		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 <= -4e-5:
		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 <= -4e-5)
		tmp = Float64(Float64(c * 0.5) / 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 <= -4e-5)
		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, -4e-5], N[(N[(c * 0.5), $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 < -4.00000000000000033e-5

   1. Initial program 14.6%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + \frac{1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. fma-define N/A

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

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-2, \frac{b\_2}{a}, \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \frac{c}{b\_2}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(-2, \frac{b\_2}{a}, \mathsf{neg}\left(\frac{-1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      4. fmm-undef N/A

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

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(-2 \cdot \frac{b\_2}{a}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{b\_2}\right)}\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{-2 \cdot b\_2}{a}\right), \left(\color{blue}{\frac{-1}{2}} \cdot \frac{c}{b\_2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{b\_2 \cdot -2}{a}\right), \left(\frac{-1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(b\_2 \cdot \frac{-2}{a}\right), \left(\color{blue}{\frac{-1}{2}} \cdot \frac{c}{b\_2}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(b\_2 \cdot \frac{\mathsf{neg}\left(2\right)}{a}\right), \left(\frac{-1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(b\_2 \cdot \left(\mathsf{neg}\left(\frac{2}{a}\right)\right)\right), \left(\frac{-1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(b\_2 \cdot \left(\mathsf{neg}\left(\frac{2 \cdot 1}{a}\right)\right)\right), \left(\frac{-1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(b\_2 \cdot \left(\mathsf{neg}\left(2 \cdot \frac{1}{a}\right)\right)\right), \left(\frac{-1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, \left(\mathsf{neg}\left(2 \cdot \frac{1}{a}\right)\right)\right), \left(\color{blue}{\frac{-1}{2}} \cdot \frac{c}{b\_2}\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, \left(\mathsf{neg}\left(\frac{2 \cdot 1}{a}\right)\right)\right), \left(\frac{-1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, \left(\mathsf{neg}\left(\frac{2}{a}\right)\right)\right), \left(\frac{-1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      16. distribute-neg-frac N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, \left(\frac{\mathsf{neg}\left(2\right)}{a}\right)\right), \left(\frac{-1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, \left(\frac{-2}{a}\right)\right), \left(\frac{-1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, \mathsf{/.f64}\left(-2, a\right)\right), \left(\frac{-1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      19. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, \mathsf{/.f64}\left(-2, a\right)\right), \left(\frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}}\right)\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, \mathsf{/.f64}\left(-2, a\right)\right), \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b\_2}\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, \mathsf{/.f64}\left(-2, a\right)\right), \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\_2\right)\right) \]

      22. *-lowering-*.f64 2.7%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, \mathsf{/.f64}\left(-2, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\_2\right)\right) \]

   5. Simplified 2.7%

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

   6. Taylor expanded in b_2 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b\_2}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot c\right), \color{blue}{b\_2}\right) \]

      3. *-lowering-*.f64 28.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, c\right), b\_2\right) \]

   8. Simplified 28.6%

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

   if -4.00000000000000033e-5 < 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 b_2 around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-2 \cdot b\_2\right)}, a\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b\_2 \cdot -2\right), a\right) \]

      2. *-lowering-*.f64 50.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b\_2, -2\right), a\right) \]

   5. Simplified 50.2%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -4 \cdot 10^{-5}:\\ \;\;\;\;\frac{c \cdot 0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.5% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{c \cdot 0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.6e-5) (/ (* 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 <= -1.6e-5) {
		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 <= (-1.6d-5)) 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 <= -1.6e-5) {
		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 <= -1.6e-5:
		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 <= -1.6e-5)
		tmp = Float64(Float64(c * 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 <= -1.6e-5)
		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, -1.6e-5], N[(N[(c * 0.5), $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 < -1.59999999999999993e-5

   1. Initial program 14.6%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + \frac{1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. fma-define N/A

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

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-2, \frac{b\_2}{a}, \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \frac{c}{b\_2}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(-2, \frac{b\_2}{a}, \mathsf{neg}\left(\frac{-1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      4. fmm-undef N/A

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

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(-2 \cdot \frac{b\_2}{a}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{b\_2}\right)}\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{-2 \cdot b\_2}{a}\right), \left(\color{blue}{\frac{-1}{2}} \cdot \frac{c}{b\_2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{b\_2 \cdot -2}{a}\right), \left(\frac{-1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(b\_2 \cdot \frac{-2}{a}\right), \left(\color{blue}{\frac{-1}{2}} \cdot \frac{c}{b\_2}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(b\_2 \cdot \frac{\mathsf{neg}\left(2\right)}{a}\right), \left(\frac{-1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(b\_2 \cdot \left(\mathsf{neg}\left(\frac{2}{a}\right)\right)\right), \left(\frac{-1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(b\_2 \cdot \left(\mathsf{neg}\left(\frac{2 \cdot 1}{a}\right)\right)\right), \left(\frac{-1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(b\_2 \cdot \left(\mathsf{neg}\left(2 \cdot \frac{1}{a}\right)\right)\right), \left(\frac{-1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, \left(\mathsf{neg}\left(2 \cdot \frac{1}{a}\right)\right)\right), \left(\color{blue}{\frac{-1}{2}} \cdot \frac{c}{b\_2}\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, \left(\mathsf{neg}\left(\frac{2 \cdot 1}{a}\right)\right)\right), \left(\frac{-1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, \left(\mathsf{neg}\left(\frac{2}{a}\right)\right)\right), \left(\frac{-1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      16. distribute-neg-frac N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, \left(\frac{\mathsf{neg}\left(2\right)}{a}\right)\right), \left(\frac{-1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, \left(\frac{-2}{a}\right)\right), \left(\frac{-1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, \mathsf{/.f64}\left(-2, a\right)\right), \left(\frac{-1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      19. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, \mathsf{/.f64}\left(-2, a\right)\right), \left(\frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}}\right)\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, \mathsf{/.f64}\left(-2, a\right)\right), \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b\_2}\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, \mathsf{/.f64}\left(-2, a\right)\right), \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\_2\right)\right) \]

      22. *-lowering-*.f64 2.7%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, \mathsf{/.f64}\left(-2, a\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\_2\right)\right) \]

   5. Simplified 2.7%

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

   6. Taylor expanded in b_2 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b\_2}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot c\right), \color{blue}{b\_2}\right) \]

      3. *-lowering-*.f64 28.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, c\right), b\_2\right) \]

   8. Simplified 28.6%

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

   if -1.59999999999999993e-5 < 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 b_2 around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

         \[\leadsto b\_2 \cdot \frac{\mathsf{neg}\left(2\right)}{a} \]

      5. distribute-neg-frac N/A

         \[\leadsto b\_2 \cdot \left(\mathsf{neg}\left(\frac{2}{a}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto b\_2 \cdot \left(\mathsf{neg}\left(\frac{2 \cdot 1}{a}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto b\_2 \cdot \left(\mathsf{neg}\left(2 \cdot \frac{1}{a}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(b\_2, \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{1}{a}\right)\right)}\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(b\_2, \left(\mathsf{neg}\left(\frac{2 \cdot 1}{a}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(b\_2, \left(\mathsf{neg}\left(\frac{2}{a}\right)\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f64}\left(b\_2, \left(\frac{\mathsf{neg}\left(2\right)}{\color{blue}{a}}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(b\_2, \left(\frac{-2}{a}\right)\right) \]

      13. /-lowering-/.f64 50.1%

          \[\leadsto \mathsf{*.f64}\left(b\_2, \mathsf{/.f64}\left(-2, \color{blue}{a}\right)\right) \]

   5. Simplified 50.1%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{c \cdot 0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.0% 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 55.1%

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

3. Taylor expanded in b_2 around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. metadata-eval N/A

      \[\leadsto b\_2 \cdot \frac{\mathsf{neg}\left(2\right)}{a} \]

   5. distribute-neg-frac N/A

      \[\leadsto b\_2 \cdot \left(\mathsf{neg}\left(\frac{2}{a}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto b\_2 \cdot \left(\mathsf{neg}\left(\frac{2 \cdot 1}{a}\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto b\_2 \cdot \left(\mathsf{neg}\left(2 \cdot \frac{1}{a}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(b\_2, \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{1}{a}\right)\right)}\right) \]

   9. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(b\_2, \left(\mathsf{neg}\left(\frac{2 \cdot 1}{a}\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(b\_2, \left(\mathsf{neg}\left(\frac{2}{a}\right)\right)\right) \]

   11. distribute-neg-frac N/A

       \[\leadsto \mathsf{*.f64}\left(b\_2, \left(\frac{\mathsf{neg}\left(2\right)}{\color{blue}{a}}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(b\_2, \left(\frac{-2}{a}\right)\right) \]

   13. /-lowering-/.f64 34.3%

       \[\leadsto \mathsf{*.f64}\left(b\_2, \mathsf{/.f64}\left(-2, \color{blue}{a}\right)\right) \]

5. Simplified 34.3%

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

Alternative 8: 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 55.1%

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

4. Applied egg-rr 32.7%

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

5. Taylor expanded in b_2 around 0

   \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(b\_2, \mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot c\right)\right)}\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(b\_2, \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(a \cdot c\right)\right)\right)\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(b\_2, \mathsf{sqrt.f64}\left(\left(0 - a \cdot c\right)\right)\right)\right)\right) \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(b\_2, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot c\right)\right)\right)\right)\right)\right) \]

   4. *-lowering-*.f64 27.6%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(a, \mathsf{\_.f64}\left(b\_2, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, c\right)\right)\right)\right)\right)\right) \]

7. Simplified 27.6%

   \[\leadsto \frac{1}{\frac{a}{b\_2 - \sqrt{\color{blue}{0 - a \cdot c}}}} \]

8. Taylor expanded in b_2 around inf

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

   1. /-lowering-/.f64 2.4%

      \[\leadsto \mathsf{/.f64}\left(b\_2, \color{blue}{a}\right) \]

10. Simplified 2.4%

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

Developer Target 1: 99.6% accurate, 0.1× speedup

Math

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

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 2024161 
(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))