quad2m (problem 3.2.1, negative)

Percentage Accurate: 51.2% → 85.4%

Time: 7.9s

Alternatives: 7

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3 \cdot 10^{-93}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \mathbf{elif}\;b\_2 \leq 10^{+126}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - c \cdot a} + b\_2}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3e-93)
   (* (/ c b_2) -0.5)
   (if (<= b_2 1e+126)
     (/ (+ (sqrt (- (* b_2 b_2) (* c a))) b_2) (- a))
     (/ (* -2.0 b_2) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3e-93) {
		tmp = (c / b_2) * -0.5;
	} else if (b_2 <= 1e+126) {
		tmp = (sqrt(((b_2 * b_2) - (c * a))) + b_2) / -a;
	} else {
		tmp = (-2.0 * b_2) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-3d-93)) then
        tmp = (c / b_2) * (-0.5d0)
    else if (b_2 <= 1d+126) then
        tmp = (sqrt(((b_2 * b_2) - (c * a))) + b_2) / -a
    else
        tmp = ((-2.0d0) * b_2) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3e-93) {
		tmp = (c / b_2) * -0.5;
	} else if (b_2 <= 1e+126) {
		tmp = (Math.sqrt(((b_2 * b_2) - (c * a))) + b_2) / -a;
	} else {
		tmp = (-2.0 * b_2) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3e-93:
		tmp = (c / b_2) * -0.5
	elif b_2 <= 1e+126:
		tmp = (math.sqrt(((b_2 * b_2) - (c * a))) + b_2) / -a
	else:
		tmp = (-2.0 * b_2) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3e-93)
		tmp = Float64(Float64(c / b_2) * -0.5);
	elseif (b_2 <= 1e+126)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a))) + b_2) / Float64(-a));
	else
		tmp = Float64(Float64(-2.0 * b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3e-93)
		tmp = (c / b_2) * -0.5;
	elseif (b_2 <= 1e+126)
		tmp = (sqrt(((b_2 * b_2) - (c * a))) + b_2) / -a;
	else
		tmp = (-2.0 * b_2) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3e-93], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[b$95$2, 1e+126], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b$95$2), $MachinePrecision] / (-a)), $MachinePrecision], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.0000000000000001e-93

   1. Initial program 16.2%

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

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

      2. lower-/.f64 87.3

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

   5. Applied rewrites 87.3%

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

   if -3.0000000000000001e-93 < b_2 < 9.99999999999999925e125

   1. Initial program 80.8%

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

   if 9.99999999999999925e125 < b_2

   1. Initial program 54.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 98.2

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

   5. Applied rewrites 98.2%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3 \cdot 10^{-93}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \mathbf{elif}\;b\_2 \leq 10^{+126}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - c \cdot a} + b\_2}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3 \cdot 10^{-93}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \mathbf{elif}\;b\_2 \leq 5.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{\left|\left(-a\right) \cdot c\right|} + b\_2}{-a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{b\_2}, c, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3e-93)
   (* (/ c b_2) -0.5)
   (if (<= b_2 5.2e-48)
     (/ (+ (sqrt (fabs (* (- a) c))) b_2) (- a))
     (fma (/ 0.5 b_2) c (* (/ b_2 a) -2.0)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3e-93) {
		tmp = (c / b_2) * -0.5;
	} else if (b_2 <= 5.2e-48) {
		tmp = (sqrt(fabs((-a * c))) + b_2) / -a;
	} else {
		tmp = fma((0.5 / b_2), c, ((b_2 / a) * -2.0));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3e-93)
		tmp = Float64(Float64(c / b_2) * -0.5);
	elseif (b_2 <= 5.2e-48)
		tmp = Float64(Float64(sqrt(abs(Float64(Float64(-a) * c))) + b_2) / Float64(-a));
	else
		tmp = fma(Float64(0.5 / b_2), c, Float64(Float64(b_2 / a) * -2.0));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3e-93], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[b$95$2, 5.2e-48], N[(N[(N[Sqrt[N[Abs[N[((-a) * c), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + b$95$2), $MachinePrecision] / (-a)), $MachinePrecision], N[(N[(0.5 / b$95$2), $MachinePrecision] * c + N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.0000000000000001e-93

   1. Initial program 16.2%

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

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

      2. lower-/.f64 87.3

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

   5. Applied rewrites 87.3%

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

   if -3.0000000000000001e-93 < b_2 < 5.19999999999999975e-48

   1. Initial program 75.9%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 70.3

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

   5. Applied rewrites 70.3%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. rem-sqrt-square N/A

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

      4. lower-fabs.f64 70.9

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

   7. Applied rewrites 70.9%

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

   if 5.19999999999999975e-48 < b_2

   1. Initial program 68.3%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 90.7

          \[\leadsto \mathsf{fma}\left(\frac{0.5}{b\_2}, c, \color{blue}{\frac{b\_2}{a}} \cdot -2\right) \]

   5. Applied rewrites 90.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{b\_2}, c, \frac{b\_2}{a} \cdot -2\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3 \cdot 10^{-93}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \mathbf{elif}\;b\_2 \leq 5.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{\left|\left(-a\right) \cdot c\right|} + b\_2}{-a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{b\_2}, c, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3 \cdot 10^{-93}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \mathbf{elif}\;b\_2 \leq 5.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c} + b\_2}{-a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{b\_2}, c, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3e-93)
   (* (/ c b_2) -0.5)
   (if (<= b_2 5.2e-48)
     (/ (+ (sqrt (* (- a) c)) b_2) (- a))
     (fma (/ 0.5 b_2) c (* (/ b_2 a) -2.0)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3e-93) {
		tmp = (c / b_2) * -0.5;
	} else if (b_2 <= 5.2e-48) {
		tmp = (sqrt((-a * c)) + b_2) / -a;
	} else {
		tmp = fma((0.5 / b_2), c, ((b_2 / a) * -2.0));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3e-93)
		tmp = Float64(Float64(c / b_2) * -0.5);
	elseif (b_2 <= 5.2e-48)
		tmp = Float64(Float64(sqrt(Float64(Float64(-a) * c)) + b_2) / Float64(-a));
	else
		tmp = fma(Float64(0.5 / b_2), c, Float64(Float64(b_2 / a) * -2.0));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3e-93], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[b$95$2, 5.2e-48], N[(N[(N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision] + b$95$2), $MachinePrecision] / (-a)), $MachinePrecision], N[(N[(0.5 / b$95$2), $MachinePrecision] * c + N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.0000000000000001e-93

   1. Initial program 16.2%

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

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

      2. lower-/.f64 87.3

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

   5. Applied rewrites 87.3%

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

   if -3.0000000000000001e-93 < b_2 < 5.19999999999999975e-48

   1. Initial program 75.9%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 70.3

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

   5. Applied rewrites 70.3%

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

   if 5.19999999999999975e-48 < b_2

   1. Initial program 68.3%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 90.7

          \[\leadsto \mathsf{fma}\left(\frac{0.5}{b\_2}, c, \color{blue}{\frac{b\_2}{a}} \cdot -2\right) \]

   5. Applied rewrites 90.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{b\_2}, c, \frac{b\_2}{a} \cdot -2\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3 \cdot 10^{-93}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \mathbf{elif}\;b\_2 \leq 5.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c} + b\_2}{-a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{b\_2}, c, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{b\_2}, c, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-311)
   (* (/ c b_2) -0.5)
   (fma (/ 0.5 b_2) c (* (/ b_2 a) -2.0))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-311) {
		tmp = (c / b_2) * -0.5;
	} else {
		tmp = fma((0.5 / b_2), c, ((b_2 / a) * -2.0));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-311)
		tmp = Float64(Float64(c / b_2) * -0.5);
	else
		tmp = fma(Float64(0.5 / b_2), c, Float64(Float64(b_2 / a) * -2.0));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-311], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(0.5 / b$95$2), $MachinePrecision] * c + N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -5.00000000000023e-311

   1. Initial program 29.5%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 69.9

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

   5. Applied rewrites 69.9%

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

   if -5.00000000000023e-311 < b_2

   1. Initial program 72.7%

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

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 65.7

          \[\leadsto \mathsf{fma}\left(\frac{0.5}{b\_2}, c, \color{blue}{\frac{b\_2}{a}} \cdot -2\right) \]

   5. Applied rewrites 65.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{b\_2}, c, \frac{b\_2}{a} \cdot -2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{b\_2}, c, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-311) (* (/ c b_2) -0.5) (/ (* -2.0 b_2) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-311) {
		tmp = (c / b_2) * -0.5;
	} else {
		tmp = (-2.0 * b_2) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d-311)) then
        tmp = (c / b_2) * (-0.5d0)
    else
        tmp = ((-2.0d0) * b_2) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-311) {
		tmp = (c / b_2) * -0.5;
	} else {
		tmp = (-2.0 * b_2) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-311:
		tmp = (c / b_2) * -0.5
	else:
		tmp = (-2.0 * b_2) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-311)
		tmp = Float64(Float64(c / b_2) * -0.5);
	else
		tmp = Float64(Float64(-2.0 * b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-311)
		tmp = (c / b_2) * -0.5;
	else
		tmp = (-2.0 * b_2) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-311], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -5.00000000000023e-311

   1. Initial program 29.5%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 69.9

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

   5. Applied rewrites 69.9%

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

   if -5.00000000000023e-311 < b_2

   1. Initial program 72.7%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 65.0

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

   5. Applied rewrites 65.0%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 47.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-311) (* (/ c b_2) -0.5) (/ (- b_2) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-311) {
		tmp = (c / b_2) * -0.5;
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d-311)) then
        tmp = (c / b_2) * (-0.5d0)
    else
        tmp = -b_2 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-311) {
		tmp = (c / b_2) * -0.5;
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-311:
		tmp = (c / b_2) * -0.5
	else:
		tmp = -b_2 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-311)
		tmp = Float64(Float64(c / b_2) * -0.5);
	else
		tmp = Float64(Float64(-b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-311)
		tmp = (c / b_2) * -0.5;
	else
		tmp = -b_2 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-311], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision], N[((-b$95$2) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -5.00000000000023e-311

   1. Initial program 29.5%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 69.9

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

   5. Applied rewrites 69.9%

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

   if -5.00000000000023e-311 < b_2

   1. Initial program 72.7%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 46.4

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

   5. Applied rewrites 46.4%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. rem-sqrt-square N/A

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

      4. lower-fabs.f64 56.6

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

   7. Applied rewrites 56.6%

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

   8. Taylor expanded in b_2 around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 29.9

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

   10. Applied rewrites 29.9%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 15.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.4%

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

3. Taylor expanded in a around inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 34.6

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

5. Applied rewrites 34.6%

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

   1. rem-square-sqrt N/A

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

   2. sqrt-unprod N/A

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

   3. rem-sqrt-square N/A

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

   4. lower-fabs.f64 40.0

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

7. Applied rewrites 40.0%

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

8. Taylor expanded in b_2 around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 15.4

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

10. Applied rewrites 15.4%

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

Developer Target 1: 99.7% accurate, 0.2× 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 2024235 
(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))