quad2m (problem 3.2.1, negative)

Percentage Accurate: 53.1% → 86.7%

Time: 12.4s

Alternatives: 10

Speedup: 11.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.1% 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: 86.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -9.5 \cdot 10^{-93}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5 \cdot 10^{+74}:\\ \;\;\;\;\frac{b\_2 + \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\_2\right) - b\_2 \cdot \sqrt{1 - a \cdot \left(c \cdot {b\_2}^{-2}\right)}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -9.5e-93)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 5e+74)
     (/ (+ b_2 (sqrt (- (* b_2 b_2) (* c a)))) (- a))
     (/ (- (- b_2) (* b_2 (sqrt (- 1.0 (* a (* c (pow b_2 -2.0))))))) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9.5e-93) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 5e+74) {
		tmp = (b_2 + sqrt(((b_2 * b_2) - (c * a)))) / -a;
	} else {
		tmp = (-b_2 - (b_2 * sqrt((1.0 - (a * (c * pow(b_2, -2.0))))))) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-9.5d-93)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 5d+74) then
        tmp = (b_2 + sqrt(((b_2 * b_2) - (c * a)))) / -a
    else
        tmp = (-b_2 - (b_2 * sqrt((1.0d0 - (a * (c * (b_2 ** (-2.0d0)))))))) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9.5e-93) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 5e+74) {
		tmp = (b_2 + Math.sqrt(((b_2 * b_2) - (c * a)))) / -a;
	} else {
		tmp = (-b_2 - (b_2 * Math.sqrt((1.0 - (a * (c * Math.pow(b_2, -2.0))))))) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -9.5e-93:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 5e+74:
		tmp = (b_2 + math.sqrt(((b_2 * b_2) - (c * a)))) / -a
	else:
		tmp = (-b_2 - (b_2 * math.sqrt((1.0 - (a * (c * math.pow(b_2, -2.0))))))) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -9.5e-93)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 5e+74)
		tmp = Float64(Float64(b_2 + sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / Float64(-a));
	else
		tmp = Float64(Float64(Float64(-b_2) - Float64(b_2 * sqrt(Float64(1.0 - Float64(a * Float64(c * (b_2 ^ -2.0))))))) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -9.5e-93)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 5e+74)
		tmp = (b_2 + sqrt(((b_2 * b_2) - (c * a)))) / -a;
	else
		tmp = (-b_2 - (b_2 * sqrt((1.0 - (a * (c * (b_2 ^ -2.0))))))) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -9.5e-93], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 5e+74], N[(N[(b$95$2 + N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-a)), $MachinePrecision], N[(N[((-b$95$2) - N[(b$95$2 * N[Sqrt[N[(1.0 - N[(a * N[(c * N[Power[b$95$2, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -9.5000000000000001e-93

   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 85.6%

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

      1. associate-*r/ 85.6%

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

   5. Simplified 85.6%

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

   if -9.5000000000000001e-93 < b_2 < 4.99999999999999963e74

   1. Initial program 79.4%

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

   if 4.99999999999999963e74 < b_2

   1. Initial program 50.7%

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

   3. Taylor expanded in b_2 around inf 50.4%

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

      1. mul-1-neg 50.4%

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

      2. unsub-neg 50.4%

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

      3. associate-/l* 50.7%

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

   5. Simplified 50.7%

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

      1. *-commutative 50.7%

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

      2. sqrt-prod 55.5%

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

      3. div-inv 55.5%

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

      4. pow-flip 55.5%

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

      5. metadata-eval 55.5%

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

      6. sqrt-pow1 97.9%

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

      7. metadata-eval 97.9%

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

      8. pow1 97.9%

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

   7. Applied egg-rr 97.9%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\sqrt{1 - a \cdot \left(c \cdot {b\_2}^{-2}\right)} \cdot b\_2}}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -9.5 \cdot 10^{-93}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5 \cdot 10^{+74}:\\ \;\;\;\;\frac{b\_2 + \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\_2\right) - b\_2 \cdot \sqrt{1 - a \cdot \left(c \cdot {b\_2}^{-2}\right)}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.9 \cdot 10^{-93}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 4 \cdot 10^{+81}:\\ \;\;\;\;\frac{b\_2 + \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{-a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.9e-93)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 4e+81)
     (/ (+ b_2 (sqrt (- (* b_2 b_2) (* c a)))) (- a))
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.9e-93) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 4e+81) {
		tmp = (b_2 + sqrt(((b_2 * b_2) - (c * a)))) / -a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.9d-93)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 4d+81) then
        tmp = (b_2 + sqrt(((b_2 * b_2) - (c * a)))) / -a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.9e-93) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 4e+81) {
		tmp = (b_2 + Math.sqrt(((b_2 * b_2) - (c * a)))) / -a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.9e-93:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 4e+81:
		tmp = (b_2 + math.sqrt(((b_2 * b_2) - (c * a)))) / -a
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.9e-93)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 4e+81)
		tmp = Float64(Float64(b_2 + sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / Float64(-a));
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.9e-93)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 4e+81)
		tmp = (b_2 + sqrt(((b_2 * b_2) - (c * a)))) / -a;
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.9e-93], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 4e+81], N[(N[(b$95$2 + N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-a)), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.8999999999999999e-93

   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 85.6%

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

      1. associate-*r/ 85.6%

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

   5. Simplified 85.6%

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

   if -1.8999999999999999e-93 < b_2 < 3.99999999999999969e81

   1. Initial program 79.4%

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

   if 3.99999999999999969e81 < b_2

   1. Initial program 50.7%

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

   3. Taylor expanded in c around 0 93.9%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.9 \cdot 10^{-93}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 4 \cdot 10^{+81}:\\ \;\;\;\;\frac{b\_2 + \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{-a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.8e-93)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 6.2e-5)
     (/ (- (- b_2) (sqrt (* c (- a)))) a)
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.8e-93) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 6.2e-5) {
		tmp = (-b_2 - sqrt((c * -a))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.8d-93)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 6.2d-5) then
        tmp = (-b_2 - sqrt((c * -a))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.8e-93) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 6.2e-5) {
		tmp = (-b_2 - Math.sqrt((c * -a))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.8e-93:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 6.2e-5:
		tmp = (-b_2 - math.sqrt((c * -a))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.8e-93)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 6.2e-5)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(c * Float64(-a)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.8e-93)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 6.2e-5)
		tmp = (-b_2 - sqrt((c * -a))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.8e-93], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 6.2e-5], N[(N[((-b$95$2) - N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.79999999999999998e-93

   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 85.6%

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

      1. associate-*r/ 85.6%

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

   5. Simplified 85.6%

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

   if -2.79999999999999998e-93 < b_2 < 6.20000000000000027e-5

   1. Initial program 76.8%

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

   3. Taylor expanded in b_2 around 0 68.4%

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

      1. mul-1-neg 68.4%

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

      2. distribute-rgt-neg-out 68.4%

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

   5. Simplified 68.4%

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

   if 6.20000000000000027e-5 < b_2

   1. Initial program 60.6%

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

   3. Taylor expanded in c around 0 90.7%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-92}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)}}{-a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-92)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 6.2e-5)
     (/ (sqrt (* c (- a))) (- a))
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-92) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 6.2e-5) {
		tmp = sqrt((c * -a)) / -a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1d-92)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 6.2d-5) then
        tmp = sqrt((c * -a)) / -a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-92) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 6.2e-5) {
		tmp = Math.sqrt((c * -a)) / -a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e-92:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 6.2e-5:
		tmp = math.sqrt((c * -a)) / -a
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-92)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 6.2e-5)
		tmp = Float64(sqrt(Float64(c * Float64(-a))) / Float64(-a));
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e-92)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 6.2e-5)
		tmp = sqrt((c * -a)) / -a;
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e-92], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 6.2e-5], N[(N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision] / (-a)), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -9.99999999999999988e-93

   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 85.6%

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

      1. associate-*r/ 85.6%

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

   5. Simplified 85.6%

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

   if -9.99999999999999988e-93 < b_2 < 6.20000000000000027e-5

   1. Initial program 76.8%

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

      1. prod-diff 76.3%

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

      2. *-commutative 76.3%

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

      3. fmm-def 76.3%

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

      4. prod-diff 76.3%

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

      5. *-commutative 76.3%

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

      6. fmm-def 76.3%

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

      7. associate-+l+ 76.2%

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

      8. pow2 76.2%

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

      9. *-commutative 76.2%

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

      10. fma-undefine 76.3%

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

      11. distribute-lft-neg-in 76.3%

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

      12. *-commutative 76.3%

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

      13. distribute-rgt-neg-in 76.3%

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

      14. fma-define 76.2%

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

      15. *-commutative 76.2%

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

      16. fma-undefine 76.3%

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

      17. distribute-lft-neg-in 76.3%

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

      18. *-commutative 76.3%

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

      19. distribute-rgt-neg-in 76.3%

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

   4. Applied egg-rr 76.2%

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

      1. count-2 76.2%

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

   6. Simplified 76.2%

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

   7. Taylor expanded in b_2 around 0 66.0%

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

      1. mul-1-neg 66.0%

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

      2. distribute-lft1-in 66.0%

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

      3. metadata-eval 66.0%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(\color{blue}{0} \cdot \left(a \cdot c\right)\right) - a \cdot c}}{a} \]

      4. mul0-lft 66.6%

         \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{0} - a \cdot c}}{a} \]

      5. metadata-eval 66.6%

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

      6. neg-sub0 66.6%

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

      7. *-commutative 66.6%

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

      8. distribute-rgt-neg-in 66.6%

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

   9. Simplified 66.6%

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

   if 6.20000000000000027e-5 < b_2

   1. Initial program 60.6%

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

   3. Taylor expanded in c around 0 90.7%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-92}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)}}{-a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.2 \cdot 10^{-178}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.9 \cdot 10^{-74}:\\ \;\;\;\;-\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.2e-178)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 2.9e-74)
     (- (sqrt (/ c (- a))))
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.2e-178) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 2.9e-74) {
		tmp = -sqrt((c / -a));
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.2d-178)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 2.9d-74) then
        tmp = -sqrt((c / -a))
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.2e-178) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 2.9e-74) {
		tmp = -Math.sqrt((c / -a));
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.2e-178:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 2.9e-74:
		tmp = -math.sqrt((c / -a))
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.2e-178)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 2.9e-74)
		tmp = Float64(-sqrt(Float64(c / Float64(-a))));
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.2e-178)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 2.9e-74)
		tmp = -sqrt((c / -a));
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.2e-178], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 2.9e-74], (-N[Sqrt[N[(c / (-a)), $MachinePrecision]], $MachinePrecision]), N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.20000000000000002e-178

   1. Initial program 26.3%

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

   3. Taylor expanded in b_2 around -inf 75.8%

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

      1. associate-*r/ 75.8%

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

   5. Simplified 75.8%

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

   if -1.20000000000000002e-178 < b_2 < 2.9e-74

   1. Initial program 70.2%

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

      1. prod-diff 69.4%

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

      2. *-commutative 69.4%

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

      3. fmm-def 69.4%

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

      4. prod-diff 69.4%

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

      5. *-commutative 69.4%

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

      6. fmm-def 69.4%

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

      7. associate-+l+ 69.3%

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

      8. pow2 69.3%

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

      9. *-commutative 69.3%

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

      10. fma-undefine 69.4%

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

      11. distribute-lft-neg-in 69.4%

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

      12. *-commutative 69.4%

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

      13. distribute-rgt-neg-in 69.4%

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

      14. fma-define 69.3%

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

      15. *-commutative 69.3%

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

      16. fma-undefine 69.4%

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

      17. distribute-lft-neg-in 69.4%

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

      18. *-commutative 69.4%

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

      19. distribute-rgt-neg-in 69.4%

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

   4. Applied egg-rr 69.3%

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

      1. count-2 69.3%

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

   6. Simplified 69.3%

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

   7. Taylor expanded in a around inf 40.3%

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

      1. mul-1-neg 40.3%

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

      2. distribute-rgt1-in 40.3%

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

      3. metadata-eval 40.3%

         \[\leadsto -\sqrt{\frac{2 \cdot \left(\color{blue}{0} \cdot c\right) - c}{a}} \]

      4. mul0-lft 40.3%

         \[\leadsto -\sqrt{\frac{2 \cdot \color{blue}{0} - c}{a}} \]

      5. metadata-eval 40.3%

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

      6. neg-sub0 40.3%

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

   9. Simplified 40.3%

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

   if 2.9e-74 < b_2

   1. Initial program 66.8%

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

   3. Taylor expanded in c around 0 81.8%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.2 \cdot 10^{-178}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.9 \cdot 10^{-74}:\\ \;\;\;\;-\sqrt{\frac{c}{-a}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.5% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 33.1%

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

   3. Taylor expanded in b_2 around -inf 64.8%

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

      1. associate-*r/ 64.8%

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

   5. Simplified 64.8%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 67.5%

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

   3. Taylor expanded in c around 0 66.1%

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

4. Final simplification 65.4%

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

Alternative 7: 23.5% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -8 \cdot 10^{-6}:\\ \;\;\;\;0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -8e-6) (* 0.5 (/ c b_2)) (/ b_2 (- a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -8e-6) {
		tmp = 0.5 * (c / b_2);
	} 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 <= (-8d-6)) then
        tmp = 0.5d0 * (c / b_2)
    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 <= -8e-6) {
		tmp = 0.5 * (c / b_2);
	} else {
		tmp = b_2 / -a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -8e-6:
		tmp = 0.5 * (c / b_2)
	else:
		tmp = b_2 / -a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -8e-6)
		tmp = Float64(0.5 * Float64(c / b_2));
	else
		tmp = Float64(b_2 / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -8e-6)
		tmp = 0.5 * (c / b_2);
	else
		tmp = b_2 / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -8e-6], N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(b$95$2 / (-a)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -7.99999999999999964e-6

   1. Initial program 13.6%

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

   3. Taylor expanded in a around 0 2.3%

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

   4. Taylor expanded in a around inf 35.5%

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

   if -7.99999999999999964e-6 < b_2

   1. Initial program 65.7%

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

   3. Taylor expanded in b_2 around inf 43.3%

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

      1. mul-1-neg 43.3%

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

      2. unsub-neg 43.3%

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

      3. associate-/l* 40.8%

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

   5. Simplified 40.8%

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

      1. add-sqr-sqrt 40.8%

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

      2. rem-sqrt-square 40.8%

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

      3. sqrt-prod 42.5%

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

      4. sqrt-pow1 57.6%

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

      5. metadata-eval 57.6%

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

      6. pow1 57.6%

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

      7. div-inv 56.7%

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

      8. pow-flip 56.7%

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

      9. metadata-eval 56.7%

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

   7. Applied egg-rr 56.7%

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

   8. Taylor expanded in b_2 around inf 22.0%

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

      1. mul-1-neg 22.0%

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

   10. Simplified 22.0%

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

4. Final simplification 26.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -8 \cdot 10^{-6}:\\ \;\;\;\;0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.4% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.15 \cdot 10^{-275}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.15e-275) (/ (* -0.5 c) b_2) (/ b_2 (- a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.15e-275) {
		tmp = (-0.5 * c) / b_2;
	} 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 <= (-2.15d-275)) then
        tmp = ((-0.5d0) * c) / b_2
    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 <= -2.15e-275) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = b_2 / -a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.15e-275:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = b_2 / -a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.15e-275)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(b_2 / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.15e-275)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = b_2 / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.15e-275], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(b$95$2 / (-a)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.14999999999999988e-275

   1. Initial program 31.3%

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

   3. Taylor expanded in b_2 around -inf 67.3%

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

      1. associate-*r/ 67.3%

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

   5. Simplified 67.3%

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

   if -2.14999999999999988e-275 < b_2

   1. Initial program 67.9%

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

   3. Taylor expanded in b_2 around inf 51.8%

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

      1. mul-1-neg 51.8%

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

      2. unsub-neg 51.8%

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

      3. associate-/l* 51.2%

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

   5. Simplified 51.2%

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

      1. add-sqr-sqrt 51.2%

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

      2. rem-sqrt-square 51.2%

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

      3. sqrt-prod 53.4%

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

      4. sqrt-pow1 73.9%

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

      5. metadata-eval 73.9%

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

      6. pow1 73.9%

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

      7. div-inv 73.1%

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

      8. pow-flip 73.1%

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

      9. metadata-eval 73.1%

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

   7. Applied egg-rr 73.1%

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

   8. Taylor expanded in b_2 around inf 29.0%

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

      1. mul-1-neg 29.0%

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

   10. Simplified 29.0%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.15 \cdot 10^{-275}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.3% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.9 \cdot 10^{-275}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.9e-275) (/ (* -0.5 c) b_2) (/ (* b_2 -2.0) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.9e-275) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.9d-275)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.9e-275) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.9e-275:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.9e-275)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.9e-275)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.9e-275], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.9e-275

   1. Initial program 31.3%

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

   3. Taylor expanded in b_2 around -inf 67.3%

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

      1. associate-*r/ 67.3%

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

   5. Simplified 67.3%

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

   if -2.9e-275 < b_2

   1. Initial program 67.9%

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

   3. Taylor expanded in b_2 around inf 63.3%

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

      1. *-commutative 63.3%

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

   5. Simplified 63.3%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.9 \cdot 10^{-275}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 15.5% accurate, 28.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.0%

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

3. Taylor expanded in b_2 around inf 34.3%

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

   1. mul-1-neg 34.3%

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

   2. unsub-neg 34.3%

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

   3. associate-/l* 32.6%

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

5. Simplified 32.6%

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

   1. add-sqr-sqrt 32.6%

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

   2. rem-sqrt-square 32.6%

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

   3. sqrt-prod 34.2%

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

   4. sqrt-pow1 53.3%

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

   5. metadata-eval 53.3%

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

   6. pow1 53.3%

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

   7. div-inv 52.7%

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

   8. pow-flip 52.7%

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

   9. metadata-eval 52.7%

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

7. Applied egg-rr 52.7%

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

8. Taylor expanded in b_2 around inf 16.2%

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

   1. mul-1-neg 16.2%

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

10. Simplified 16.2%

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

11. Final simplification 16.2%

    \[\leadsto \frac{b\_2}{-a} \]
12. Add Preprocessing

Developer target: 99.6% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_1
         (if (== (copysign a c) a)
           (* (sqrt (- (fabs b_2) t_0)) (sqrt (+ (fabs b_2) t_0)))
           (hypot b_2 t_0))))
   (if (< b_2 0.0) (/ c (- t_1 b_2)) (/ (+ b_2 t_1) (- a)))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((fabs(b_2) - t_0)) * sqrt((fabs(b_2) + t_0));
	} else {
		tmp = hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = c / (t_1 - b_2);
	} else {
		tmp_1 = (b_2 + t_1) / -a;
	}
	return tmp_1;
}

Java

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((Math.abs(b_2) - t_0)) * Math.sqrt((Math.abs(b_2) + t_0));
	} else {
		tmp = Math.hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = c / (t_1 - b_2);
	} else {
		tmp_1 = (b_2 + t_1) / -a;
	}
	return tmp_1;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((math.fabs(b_2) - t_0)) * math.sqrt((math.fabs(b_2) + t_0))
	else:
		tmp = math.hypot(b_2, t_0)
	t_1 = tmp
	tmp_1 = 0
	if b_2 < 0.0:
		tmp_1 = c / (t_1 - b_2)
	else:
		tmp_1 = (b_2 + t_1) / -a
	return tmp_1

Julia

function code(a, b_2, c)
	t_0 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(abs(b_2) - t_0)) * sqrt(Float64(abs(b_2) + t_0)));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp
	tmp_1 = 0.0
	if (b_2 < 0.0)
		tmp_1 = Float64(c / Float64(t_1 - b_2));
	else
		tmp_1 = Float64(Float64(b_2 + t_1) / Float64(-a));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b_2, c)
	t_0 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((abs(b_2) - t_0)) * sqrt((abs(b_2) + t_0));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp;
	tmp_2 = 0.0;
	if (b_2 < 0.0)
		tmp_2 = c / (t_1 - b_2);
	else
		tmp_2 = (b_2 + t_1) / -a;
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(N[Abs[b$95$2], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[Abs[b$95$2], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[b$95$2 ^ 2 + t$95$0 ^ 2], $MachinePrecision]]}, If[Less[b$95$2, 0.0], N[(c / N[(t$95$1 - b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$2 + t$95$1), $MachinePrecision] / (-a)), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2024112 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  :herbie-expected 10

  :alt
  (if (< b_2 0.0) (/ c (- (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot b_2 (* (sqrt (fabs a)) (sqrt (fabs c))))) b_2)) (/ (+ b_2 (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot b_2 (* (sqrt (fabs a)) (sqrt (fabs c)))))) (- a)))

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