quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.5% → 86.2%

Time: 11.4s

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: 52.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.2 \cdot 10^{-53}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 7.2 \cdot 10^{+104}:\\ \;\;\;\;\frac{\left(0 - b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.2e-53)
   (/ (* c -0.5) b_2)
   (if (<= b_2 7.2e+104)
     (/ (- (- 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 <= -3.2e-53) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 7.2e+104) {
		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 <= (-3.2d-53)) then
        tmp = (c * (-0.5d0)) / b_2
    else if (b_2 <= 7.2d+104) 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 <= -3.2e-53) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 7.2e+104) {
		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 <= -3.2e-53:
		tmp = (c * -0.5) / b_2
	elif b_2 <= 7.2e+104:
		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 <= -3.2e-53)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 7.2e+104)
		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 <= -3.2e-53)
		tmp = (c * -0.5) / b_2;
	elseif (b_2 <= 7.2e+104)
		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, -3.2e-53], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 7.2e+104], 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 < -3.2000000000000001e-53

   1. Initial program 19.0%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   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 82.9%

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

   5. Simplified 82.9%

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

   if -3.2000000000000001e-53 < b_2 < 7.20000000000000001e104

   1. Initial program 82.4%

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

   if 7.20000000000000001e104 < b_2

   1. Initial program 51.0%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 98.4%

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

   5. Simplified 98.4%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.2 \cdot 10^{-53}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 7.2 \cdot 10^{+104}:\\ \;\;\;\;\frac{\left(0 - b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3 \cdot 10^{-54}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 9 \cdot 10^{-87}:\\ \;\;\;\;\frac{\left(0 - b\_2\right) - \sqrt{0 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c \cdot 0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3e-54)
   (/ (* c -0.5) b_2)
   (if (<= b_2 9e-87)
     (/ (- (- 0.0 b_2) (sqrt (- 0.0 (* c a)))) a)
     (+ (* -2.0 (/ b_2 a)) (/ (* c 0.5) b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3e-54) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 9e-87) {
		tmp = ((0.0 - b_2) - sqrt((0.0 - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-3d-54)) then
        tmp = (c * (-0.5d0)) / b_2
    else if (b_2 <= 9d-87) then
        tmp = ((0.0d0 - b_2) - sqrt((0.0d0 - (c * a)))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((c * 0.5d0) / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3e-54) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 9e-87) {
		tmp = ((0.0 - b_2) - Math.sqrt((0.0 - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3e-54:
		tmp = (c * -0.5) / b_2
	elif b_2 <= 9e-87:
		tmp = ((0.0 - b_2) - math.sqrt((0.0 - (c * a)))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3e-54)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 9e-87)
		tmp = Float64(Float64(Float64(0.0 - b_2) - sqrt(Float64(0.0 - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c * 0.5) / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3e-54)
		tmp = (c * -0.5) / b_2;
	elseif (b_2 <= 9e-87)
		tmp = ((0.0 - b_2) - sqrt((0.0 - (c * a)))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + ((c * 0.5) / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3e-54], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 9e-87], N[(N[(N[(0.0 - b$95$2), $MachinePrecision] - N[Sqrt[N[(0.0 - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c * 0.5), $MachinePrecision] / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.00000000000000009e-54

   1. Initial program 19.0%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   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 82.9%

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

   5. Simplified 82.9%

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

   if -3.00000000000000009e-54 < b_2 < 8.99999999999999915e-87

   1. Initial program 76.8%

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

   3. Taylor expanded in b_2 around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot c\right)\right)}\right)\right), a\right) \]
   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. *-lowering-*.f64 76.7%

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

   5. Simplified 76.7%

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

      1. sub0-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. +-lft-identity N/A

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

      3. flip3-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{{0}^{3} + {\left(a \cdot c\right)}^{3}}{0 \cdot 0 + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) - 0 \cdot \left(a \cdot c\right)\right)}\right)\right)\right)\right), a\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{0 + {\left(a \cdot c\right)}^{3}}{0 \cdot 0 + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) - 0 \cdot \left(a \cdot c\right)\right)}\right)\right)\right)\right), a\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{0 + {\left(a \cdot c\right)}^{\left(\frac{3}{2} + \frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) - 0 \cdot \left(a \cdot c\right)\right)}\right)\right)\right)\right), a\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{0 + {\left(a \cdot c\right)}^{\left(\frac{1}{2} \cdot 3 + \frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) - 0 \cdot \left(a \cdot c\right)\right)}\right)\right)\right)\right), a\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{0 + {\left(a \cdot c\right)}^{\left(\frac{1}{2} \cdot 3 + \frac{1}{2} \cdot 3\right)}}{0 \cdot 0 + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) - 0 \cdot \left(a \cdot c\right)\right)}\right)\right)\right)\right), a\right) \]

      8. pow-prod-up N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{0 + {\left(a \cdot c\right)}^{\left(\frac{1}{2} \cdot 3\right)} \cdot {\left(a \cdot c\right)}^{\left(\frac{1}{2} \cdot 3\right)}}{0 \cdot 0 + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) - 0 \cdot \left(a \cdot c\right)\right)}\right)\right)\right)\right), a\right) \]

      9. pow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{0 + {\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right)}^{\left(\frac{1}{2} \cdot 3\right)}}{0 \cdot 0 + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) - 0 \cdot \left(a \cdot c\right)\right)}\right)\right)\right)\right), a\right) \]

      10. sqr-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{0 + {\left(\left(\mathsf{neg}\left(a \cdot c\right)\right) \cdot \left(\mathsf{neg}\left(a \cdot c\right)\right)\right)}^{\left(\frac{1}{2} \cdot 3\right)}}{0 \cdot 0 + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) - 0 \cdot \left(a \cdot c\right)\right)}\right)\right)\right)\right), a\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{0 + {\left(\left(\mathsf{neg}\left(a \cdot c\right)\right) \cdot \left(\mathsf{neg}\left(a \cdot c\right)\right)\right)}^{\frac{3}{2}}}{0 \cdot 0 + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) - 0 \cdot \left(a \cdot c\right)\right)}\right)\right)\right)\right), a\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{0 + {\left(\left(\mathsf{neg}\left(a \cdot c\right)\right) \cdot \left(\mathsf{neg}\left(a \cdot c\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) - 0 \cdot \left(a \cdot c\right)\right)}\right)\right)\right)\right), a\right) \]

      13. sqrt-pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{0 + {\left(\sqrt{\left(\mathsf{neg}\left(a \cdot c\right)\right) \cdot \left(\mathsf{neg}\left(a \cdot c\right)\right)}\right)}^{3}}{0 \cdot 0 + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) - 0 \cdot \left(a \cdot c\right)\right)}\right)\right)\right)\right), a\right) \]

      14. sqrt-unprod N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{0 + {\left(\sqrt{\mathsf{neg}\left(a \cdot c\right)} \cdot \sqrt{\mathsf{neg}\left(a \cdot c\right)}\right)}^{3}}{0 \cdot 0 + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) - 0 \cdot \left(a \cdot c\right)\right)}\right)\right)\right)\right), a\right) \]

      15. rem-square-sqrt N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{0 + {\left(\mathsf{neg}\left(a \cdot c\right)\right)}^{3}}{0 \cdot 0 + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) - 0 \cdot \left(a \cdot c\right)\right)}\right)\right)\right)\right), a\right) \]

      16. cube-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{0 + \left(\mathsf{neg}\left({\left(a \cdot c\right)}^{3}\right)\right)}{0 \cdot 0 + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) - 0 \cdot \left(a \cdot c\right)\right)}\right)\right)\right)\right), a\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{0 - {\left(a \cdot c\right)}^{3}}{0 \cdot 0 + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) - 0 \cdot \left(a \cdot c\right)\right)}\right)\right)\right)\right), a\right) \]

      18. mul0-lft N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{0 - {\left(a \cdot c\right)}^{3}}{0 \cdot 0 + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) - 0\right)}\right)\right)\right)\right), a\right) \]

      19. fmm-def N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{0 - {\left(a \cdot c\right)}^{3}}{0 \cdot 0 + \mathsf{fma}\left(a \cdot c, a \cdot c, \mathsf{neg}\left(0\right)\right)}\right)\right)\right)\right), a\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{0 - {\left(a \cdot c\right)}^{3}}{0 \cdot 0 + \mathsf{fma}\left(a \cdot c, a \cdot c, 0\right)}\right)\right)\right)\right), a\right) \]

      21. mul0-lft N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{neg.f64}\left(b\_2\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{0 - {\left(a \cdot c\right)}^{3}}{0 \cdot 0 + \mathsf{fma}\left(a \cdot c, a \cdot c, 0 \cdot \left(a \cdot c\right)\right)}\right)\right)\right)\right), a\right) \]

   7. Applied egg-rr 76.7%

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

   if 8.99999999999999915e-87 < b_2

   1. Initial program 68.6%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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. +-lowering-+.f64 N/A

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

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

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

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

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

      4. associate-*r/ N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 88.7%

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

   5. Simplified 88.7%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3 \cdot 10^{-54}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 9 \cdot 10^{-87}:\\ \;\;\;\;\frac{\left(0 - b\_2\right) - \sqrt{0 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + \frac{c \cdot 0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.7% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 37.2%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   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 63.2%

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

   5. Simplified 63.2%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 69.8%

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

   3. Taylor expanded in c around 0

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

      1. +-lowering-+.f64 N/A

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

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

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

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

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

      4. associate-*r/ N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 73.1%

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

   5. Simplified 73.1%

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

Alternative 4: 68.4% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.9 \cdot 10^{-305}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.9e-305) (/ (* 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 <= -1.9e-305) {
		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 <= (-1.9d-305)) 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 <= -1.9e-305) {
		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 <= -1.9e-305:
		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 <= -1.9e-305)
		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 <= -1.9e-305)
		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, -1.9e-305], 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 < -1.9e-305

   1. Initial program 36.7%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   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 63.6%

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

   5. Simplified 63.6%

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

   if -1.9e-305 < b_2

   1. Initial program 70.0%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 72.3%

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

   5. Simplified 72.3%

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

Alternative 5: 68.3% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.9 \cdot 10^{-305}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.9e-305) (* 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 <= -1.9e-305) {
		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 <= (-1.9d-305)) 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 <= -1.9e-305) {
		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 <= -1.9e-305:
		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 <= -1.9e-305)
		tmp = Float64(c * Float64(-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 <= -1.9e-305)
		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, -1.9e-305], N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.9e-305

   1. Initial program 36.7%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   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 63.6%

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

   5. Simplified 63.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 63.4%

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

   7. Applied egg-rr 63.4%

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

   if -1.9e-305 < b_2

   1. Initial program 70.0%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 72.3%

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

   5. Simplified 72.3%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.9 \cdot 10^{-305}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.2% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.9 \cdot 10^{-305}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.9e-305) (* 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 <= -1.9e-305) {
		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 <= (-1.9d-305)) 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 <= -1.9e-305) {
		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 <= -1.9e-305:
		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 <= -1.9e-305)
		tmp = Float64(c * Float64(-0.5 / b_2));
	else
		tmp = Float64(b_2 * Float64(-2.0 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.9e-305)
		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, -1.9e-305], N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.9e-305

   1. Initial program 36.7%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   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 63.6%

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

   5. Simplified 63.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 63.4%

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

   7. Applied egg-rr 63.4%

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

   if -1.9e-305 < b_2

   1. Initial program 70.0%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 72.3%

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

   5. Simplified 72.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 72.2%

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

   7. Applied egg-rr 72.2%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.9 \cdot 10^{-305}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 36.2% 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 53.7%

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

3. Taylor expanded in b_2 around inf

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 38.3%

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

5. Simplified 38.3%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

   4. /-lowering-/.f64 38.2%

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

7. Applied egg-rr 38.2%

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

8. Final simplification 38.2%

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

Alternative 8: 2.5% 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 53.7%

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

4. Applied egg-rr 35.4%

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

5. Taylor expanded in b_2 around -inf

   \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\color{blue}{\left(\frac{\frac{1}{2}}{b\_2}\right)}, \mathsf{\_.f64}\left(0, a\right)\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f64 2.7%

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \mathsf{\_.f64}\left(\color{blue}{0}, a\right)\right)\right) \]

7. Simplified 2.7%

   \[\leadsto \frac{-1}{\color{blue}{\frac{0.5}{b\_2}} \cdot \left(0 - a\right)} \]
8. Step-by-step derivation

   1. flip-- N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \left(\frac{0 \cdot 0 - a \cdot a}{\color{blue}{0 + a}}\right)\right)\right) \]

   2. div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \left(\left(0 \cdot 0 - a \cdot a\right) \cdot \color{blue}{\frac{1}{0 + a}}\right)\right)\right) \]

   3. +-lft-identity N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \left(\left(0 \cdot 0 - a \cdot a\right) \cdot \frac{1}{a}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \mathsf{*.f64}\left(\left(0 \cdot 0 - a \cdot a\right), \color{blue}{\left(\frac{1}{a}\right)}\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \mathsf{*.f64}\left(\left(0 - a \cdot a\right), \left(\frac{1}{a}\right)\right)\right)\right) \]

   6. +-lft-identity N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \mathsf{*.f64}\left(\left(0 - a \cdot \left(0 + a\right)\right), \left(\frac{1}{a}\right)\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \mathsf{*.f64}\left(\left(0 - a \cdot \left(a + 0\right)\right), \left(\frac{1}{a}\right)\right)\right)\right) \]

   8. distribute-rgt-out N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \mathsf{*.f64}\left(\left(0 - \left(a \cdot a + 0 \cdot a\right)\right), \left(\frac{1}{a}\right)\right)\right)\right) \]

   9. +-lft-identity N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \mathsf{*.f64}\left(\left(0 - \left(0 + \left(a \cdot a + 0 \cdot a\right)\right)\right), \left(\frac{1}{a}\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \mathsf{*.f64}\left(\left(0 - \left(0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)\right)\right), \left(\frac{1}{a}\right)\right)\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \left(0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)\right)\right), \left(\frac{\color{blue}{1}}{a}\right)\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \left(0 + \left(a \cdot a + 0 \cdot a\right)\right)\right), \left(\frac{1}{a}\right)\right)\right)\right) \]

   13. +-lft-identity N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot a + 0 \cdot a\right)\right), \left(\frac{1}{a}\right)\right)\right)\right) \]

   14. distribute-rgt-out N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot \left(a + 0\right)\right)\right), \left(\frac{1}{a}\right)\right)\right)\right) \]

   15. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot \left(0 + a\right)\right)\right), \left(\frac{1}{a}\right)\right)\right)\right) \]

   16. +-lft-identity N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot a\right)\right), \left(\frac{1}{a}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{1}{a}\right)\right)\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, a\right)\right), \left(\frac{\mathsf{neg}\left(-1\right)}{a}\right)\right)\right)\right) \]

   19. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(\left(\mathsf{neg}\left(-1\right)\right), \color{blue}{a}\right)\right)\right)\right) \]

   20. metadata-eval 11.9%

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{/.f64}\left(1, a\right)\right)\right)\right) \]

9. Applied egg-rr 11.9%

   \[\leadsto \frac{-1}{\frac{0.5}{b\_2} \cdot \color{blue}{\left(\left(0 - a \cdot a\right) \cdot \frac{1}{a}\right)}} \]

10. Taylor expanded in b_2 around 0

    \[\leadsto \color{blue}{2 \cdot \frac{b\_2}{a}} \]
11. 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

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

    5. associate-*r/ N/A

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

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

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

    7. associate-*r/ N/A

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

    8. metadata-eval N/A

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

    9. /-lowering-/.f64 2.7%

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

12. Simplified 2.7%

    \[\leadsto \color{blue}{b\_2 \cdot \frac{2}{a}} \]
13. 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 2024139 
(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))