quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.5% → 86.0%

Time: 7.6s

Alternatives: 10

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: 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: 86.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.05 \cdot 10^{-108}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.1 \cdot 10^{+133}:\\ \;\;\;\;\frac{b\_2 + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.05e-108)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1.1e+133)
     (/ (+ b_2 (sqrt (- (* b_2 b_2) (* a c)))) (- a))
     (/ (* -2.0 b_2) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.05e-108) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.1e+133) {
		tmp = (b_2 + sqrt(((b_2 * b_2) - (a * c)))) / -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 <= (-3.05d-108)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 1.1d+133) then
        tmp = (b_2 + sqrt(((b_2 * b_2) - (a * c)))) / -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 <= -3.05e-108) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.1e+133) {
		tmp = (b_2 + Math.sqrt(((b_2 * b_2) - (a * c)))) / -a;
	} else {
		tmp = (-2.0 * b_2) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.05e-108:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 1.1e+133:
		tmp = (b_2 + math.sqrt(((b_2 * b_2) - (a * c)))) / -a
	else:
		tmp = (-2.0 * b_2) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.05e-108)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1.1e+133)
		tmp = Float64(Float64(b_2 + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / 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 <= -3.05e-108)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 1.1e+133)
		tmp = (b_2 + sqrt(((b_2 * b_2) - (a * c)))) / -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, -3.05e-108], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.1e+133], 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], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.05000000000000004e-108

   1. Initial program 17.8%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 83.4

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

   5. Applied rewrites 83.4%

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

   7. Applied rewrites 83.4%

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

   if -3.05000000000000004e-108 < b_2 < 1.1e133

   1. Initial program 86.9%

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

   if 1.1e133 < b_2

   1. Initial program 44.0%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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. lower-*.f64 96.3

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

   5. Applied rewrites 96.3%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.05 \cdot 10^{-108}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.1 \cdot 10^{+133}:\\ \;\;\;\;\frac{b\_2 + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.05 \cdot 10^{-108}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{b\_2 + \sqrt{\left(-a\right) \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{b\_2}, c, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.05e-108)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 6.2e-39)
     (/ (+ b_2 (sqrt (* (- a) c))) (- 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 <= -3.05e-108) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 6.2e-39) {
		tmp = (b_2 + sqrt((-a * c))) / -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 <= -3.05e-108)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 6.2e-39)
		tmp = Float64(Float64(b_2 + sqrt(Float64(Float64(-a) * c))) / 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, -3.05e-108], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 6.2e-39], N[(N[(b$95$2 + N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision]), $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 < -3.05000000000000004e-108

   1. Initial program 17.8%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 83.4

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

   5. Applied rewrites 83.4%

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

   7. Applied rewrites 83.4%

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

   if -3.05000000000000004e-108 < b_2 < 6.1999999999999994e-39

   1. Initial program 84.4%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 79.4

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

   5. Applied rewrites 79.4%

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

   if 6.1999999999999994e-39 < b_2

   1. Initial program 63.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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. +-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 89.3

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

   5. Applied rewrites 89.3%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.05 \cdot 10^{-108}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{b\_2 + \sqrt{\left(-a\right) \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{b\_2}, c, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.1 \cdot 10^{-88}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 3 \cdot 10^{-39}:\\ \;\;\;\;\frac{b\_2 - \sqrt{-c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{b\_2}, c, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.1e-88)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 3e-39)
     (/ (- b_2 (sqrt (- (* c a)))) 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 <= -3.1e-88) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 3e-39) {
		tmp = (b_2 - sqrt(-(c * a))) / 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 <= -3.1e-88)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 3e-39)
		tmp = Float64(Float64(b_2 - sqrt(Float64(-Float64(c * a)))) / 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, -3.1e-88], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 3e-39], N[(N[(b$95$2 - N[Sqrt[(-N[(c * a), $MachinePrecision])], $MachinePrecision]), $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 < -3.0999999999999998e-88

   1. Initial program 17.1%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 84.0

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

   5. Applied rewrites 84.0%

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

   7. Applied rewrites 84.0%

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

   if -3.0999999999999998e-88 < b_2 < 3.00000000000000028e-39

   1. Initial program 83.5%

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

   4. Applied rewrites 77.2%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 77.4

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

   7. Applied rewrites 77.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 77.6

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

   9. Applied rewrites 77.6%

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

   if 3.00000000000000028e-39 < b_2

   1. Initial program 63.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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. +-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 89.3

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

   5. Applied rewrites 89.3%

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

Alternative 4: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.1 \cdot 10^{-88}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 3 \cdot 10^{-39}:\\ \;\;\;\;\frac{b\_2 - \sqrt{-c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.1e-88)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 3e-39) (/ (- b_2 (sqrt (- (* c a)))) a) (/ (* -2.0 b_2) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.1e-88) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 3e-39) {
		tmp = (b_2 - sqrt(-(c * a))) / 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 <= (-3.1d-88)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 3d-39) then
        tmp = (b_2 - sqrt(-(c * a))) / 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 <= -3.1e-88) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 3e-39) {
		tmp = (b_2 - Math.sqrt(-(c * a))) / a;
	} else {
		tmp = (-2.0 * b_2) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.1e-88:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 3e-39:
		tmp = (b_2 - math.sqrt(-(c * a))) / a
	else:
		tmp = (-2.0 * b_2) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.1e-88)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 3e-39)
		tmp = Float64(Float64(b_2 - sqrt(Float64(-Float64(c * a)))) / 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 <= -3.1e-88)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 3e-39)
		tmp = (b_2 - sqrt(-(c * a))) / 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, -3.1e-88], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 3e-39], N[(N[(b$95$2 - N[Sqrt[(-N[(c * a), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.0999999999999998e-88

   1. Initial program 17.1%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 84.0

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

   5. Applied rewrites 84.0%

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

   7. Applied rewrites 84.0%

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

   if -3.0999999999999998e-88 < b_2 < 3.00000000000000028e-39

   1. Initial program 83.5%

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

   4. Applied rewrites 77.2%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 77.4

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

   7. Applied rewrites 77.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 77.6

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

   9. Applied rewrites 77.6%

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

   if 3.00000000000000028e-39 < b_2

   1. Initial program 63.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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. lower-*.f64 89.3

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

   5. Applied rewrites 89.3%

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

Alternative 5: 67.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-279}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-279) (/ (* -0.5 c) b_2) (/ (* -2.0 b_2) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-279) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (-2.0 * b_2) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1d-279)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = ((-2.0d0) * b_2) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-279) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (-2.0 * b_2) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e-279:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = (-2.0 * b_2) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-279)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	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 <= -1e-279)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = (-2.0 * b_2) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e-279], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.00000000000000006e-279

   1. Initial program 27.0%

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

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

      2. lower-/.f64 73.1

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

   5. Applied rewrites 73.1%

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

   7. Applied rewrites 73.1%

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

   if -1.00000000000000006e-279 < b_2

   1. Initial program 72.2%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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. lower-*.f64 61.4

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

   5. Applied rewrites 61.4%

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

Alternative 6: 67.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-279}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{a} \cdot b\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-279) (/ (* -0.5 c) b_2) (* (/ -2.0 a) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-279) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (-2.0 / a) * 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 <= (-1d-279)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = ((-2.0d0) / a) * 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 <= -1e-279) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (-2.0 / a) * b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e-279:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = (-2.0 / a) * b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-279)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(Float64(-2.0 / a) * b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e-279)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = (-2.0 / a) * b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e-279], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(-2.0 / a), $MachinePrecision] * b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.00000000000000006e-279

   1. Initial program 27.0%

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

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

      2. lower-/.f64 73.1

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

   5. Applied rewrites 73.1%

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

   7. Applied rewrites 73.1%

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

   if -1.00000000000000006e-279 < b_2

   1. Initial program 72.2%

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

   3. Taylor expanded in b_2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. distribute-neg-frac N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 61.0

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

   5. Applied rewrites 61.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 61.2%

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

Alternative 7: 67.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-279}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{a} \cdot b\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-279) (* -0.5 (/ c b_2)) (* (/ -2.0 a) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-279) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = (-2.0 / a) * 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 <= (-1d-279)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = ((-2.0d0) / a) * 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 <= -1e-279) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = (-2.0 / a) * b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e-279:
		tmp = -0.5 * (c / b_2)
	else:
		tmp = (-2.0 / a) * b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-279)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(Float64(-2.0 / a) * b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e-279)
		tmp = -0.5 * (c / b_2);
	else
		tmp = (-2.0 / a) * b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e-279], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 / a), $MachinePrecision] * b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.00000000000000006e-279

   1. Initial program 27.0%

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

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

      2. lower-/.f64 73.1

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

   5. Applied rewrites 73.1%

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

   if -1.00000000000000006e-279 < b_2

   1. Initial program 72.2%

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

   3. Taylor expanded in b_2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. distribute-neg-frac N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 61.0

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

   5. Applied rewrites 61.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 61.2%

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

Alternative 8: 39.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.7 \cdot 10^{-306}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.7e-306) (* -0.5 (/ c b_2)) (/ c 0.0)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.7e-306) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = c / 0.0;
	}
	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 <= (-2.7d-306)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = c / 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.7e-306) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = c / 0.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.7e-306:
		tmp = -0.5 * (c / b_2)
	else:
		tmp = c / 0.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.7e-306)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(c / 0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.7e-306)
		tmp = -0.5 * (c / b_2);
	else
		tmp = c / 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.7e-306], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(c / 0.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.70000000000000009e-306

   1. Initial program 29.7%

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

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

      2. lower-/.f64 69.7

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

   5. Applied rewrites 69.7%

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

   if -2.70000000000000009e-306 < b_2

   1. Initial program 71.7%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 45.6

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

   5. Applied rewrites 45.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 38.2%

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

   8. Taylor expanded in a around 0

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

         \[\leadsto \frac{c}{\color{blue}{0}} \]

      4. lower-/.f64 8.5

         \[\leadsto \color{blue}{\frac{c}{0}} \]

   10. Applied rewrites 8.5%

       \[\leadsto \color{blue}{\frac{c}{0}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 14.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.45 \cdot 10^{-51}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (if (<= b_2 2.45e-51) 0.0 (/ c 0.0)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 2.45e-51) {
		tmp = 0.0;
	} else {
		tmp = c / 0.0;
	}
	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 <= 2.45d-51) then
        tmp = 0.0d0
    else
        tmp = c / 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 2.45e-51) {
		tmp = 0.0;
	} else {
		tmp = c / 0.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 2.45e-51:
		tmp = 0.0
	else:
		tmp = c / 0.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 2.45e-51)
		tmp = 0.0;
	else
		tmp = Float64(c / 0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 2.45e-51)
		tmp = 0.0;
	else
		tmp = c / 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 2.45e-51], 0.0, N[(c / 0.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 2.44999999999999987e-51

   1. Initial program 44.0%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 38.5

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

   5. Applied rewrites 38.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 31.7%

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

   8. Taylor expanded in a around 0

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

         \[\leadsto \frac{c}{\color{blue}{0}} \]

      4. lower-/.f64 2.8

         \[\leadsto \color{blue}{\frac{c}{0}} \]

   10. Applied rewrites 2.8%

       \[\leadsto \color{blue}{\frac{c}{0}} \]
   11. Step-by-step derivation

   12. Applied rewrites 11.6%

       \[\leadsto \color{blue}{0} \]

   if 2.44999999999999987e-51 < b_2

   1. Initial program 65.0%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 30.5

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

   5. Applied rewrites 30.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 25.4%

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

   8. Taylor expanded in a around 0

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

         \[\leadsto \frac{c}{\color{blue}{0}} \]

      4. lower-/.f64 11.1

         \[\leadsto \color{blue}{\frac{c}{0}} \]

   10. Applied rewrites 11.1%

       \[\leadsto \color{blue}{\frac{c}{0}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 11.3% accurate, 40.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 0.0)

C

double code(double a, double b_2, double c) {
	return 0.0;
}

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 = 0.0d0
end function

Java

public static double code(double a, double b_2, double c) {
	return 0.0;
}

Python

def code(a, b_2, c):
	return 0.0

Julia

function code(a, b_2, c)
	return 0.0
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = 0.0;
end

Wolfram

code[a_, b$95$2_, c_] := 0.0

Derivation

1. Initial program 51.4%

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

3. Taylor expanded in a around inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 35.7

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

5. Applied rewrites 35.7%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. associate-/l/ N/A

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

   5. lower-/.f64 N/A

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

7. Applied rewrites 29.4%

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

8. Taylor expanded in a around 0

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

   1. distribute-rgt1-in N/A

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

   2. metadata-eval N/A

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

   3. mul0-lft N/A

      \[\leadsto \frac{c}{\color{blue}{0}} \]

   4. lower-/.f64 5.7

      \[\leadsto \color{blue}{\frac{c}{0}} \]

10. Applied rewrites 5.7%

    \[\leadsto \color{blue}{\frac{c}{0}} \]
11. Step-by-step derivation

12. Applied rewrites 8.3%

    \[\leadsto \color{blue}{0} \]
13. Add Preprocessing

Developer Target 1: 99.6% 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 2024322 
(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))