quad2m (problem 3.2.1, negative)

Percentage Accurate: 51.1% → 87.5%

Time: 13.6s

Alternatives: 8

Speedup: 15.9×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.1% 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: 87.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - c \cdot a}\\ \mathbf{if}\;b_2 \leq -2.3 \cdot 10^{+48}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -2 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{c \cdot \left(-a\right)}{b_2 - t_0}}{a}\\ \mathbf{elif}\;b_2 \leq 8.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{\left(-b_2\right) - t_0}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b_2 b_2) (* c a)))))
   (if (<= b_2 -2.3e+48)
     (* -0.5 (/ c b_2))
     (if (<= b_2 -2e-110)
       (/ (/ (* c (- a)) (- b_2 t_0)) a)
       (if (<= b_2 8.5e+72)
         (/ (- (- b_2) t_0) a)
         (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5)))))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(((b_2 * b_2) - (c * a)));
	double tmp;
	if (b_2 <= -2.3e+48) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= -2e-110) {
		tmp = ((c * -a) / (b_2 - t_0)) / a;
	} else if (b_2 <= 8.5e+72) {
		tmp = (-b_2 - t_0) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b_2 * b_2) - (c * a)))
    if (b_2 <= (-2.3d+48)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= (-2d-110)) then
        tmp = ((c * -a) / (b_2 - t_0)) / a
    else if (b_2 <= 8.5d+72) then
        tmp = (-b_2 - t_0) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((c / b_2) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt(((b_2 * b_2) - (c * a)));
	double tmp;
	if (b_2 <= -2.3e+48) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= -2e-110) {
		tmp = ((c * -a) / (b_2 - t_0)) / a;
	} else if (b_2 <= 8.5e+72) {
		tmp = (-b_2 - t_0) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt(((b_2 * b_2) - (c * a)))
	tmp = 0
	if b_2 <= -2.3e+48:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= -2e-110:
		tmp = ((c * -a) / (b_2 - t_0)) / a
	elif b_2 <= 8.5e+72:
		tmp = (-b_2 - t_0) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5)
	return tmp

Julia

function code(a, b_2, c)
	t_0 = sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))
	tmp = 0.0
	if (b_2 <= -2.3e+48)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= -2e-110)
		tmp = Float64(Float64(Float64(c * Float64(-a)) / Float64(b_2 - t_0)) / a);
	elseif (b_2 <= 8.5e+72)
		tmp = Float64(Float64(Float64(-b_2) - t_0) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c / b_2) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	t_0 = sqrt(((b_2 * b_2) - (c * a)));
	tmp = 0.0;
	if (b_2 <= -2.3e+48)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= -2e-110)
		tmp = ((c * -a) / (b_2 - t_0)) / a;
	elseif (b_2 <= 8.5e+72)
		tmp = (-b_2 - t_0) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$2, -2.3e+48], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, -2e-110], N[(N[(N[(c * (-a)), $MachinePrecision] / N[(b$95$2 - t$95$0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 8.5e+72], N[(N[((-b$95$2) - t$95$0), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -2.3e48

   1. Initial program 12.2%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 96.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]

   if -2.3e48 < b_2 < -2.0000000000000001e-110

   1. Initial program 36.6%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 36.4%

         \[\leadsto \frac{\left(-b_2\right) - \color{blue}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a} \]

      2. pow2 36.4%

         \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right)}^{2}}}{a} \]

      3. pow1/2 36.4%

         \[\leadsto \frac{\left(-b_2\right) - {\left(\sqrt{\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5}}}\right)}^{2}}{a} \]

      4. sqrt-pow1 36.5%

         \[\leadsto \frac{\left(-b_2\right) - {\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}}{a} \]

      5. metadata-eval 36.5%

         \[\leadsto \frac{\left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

   3. Applied egg-rr 36.5%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]
   4. Step-by-step derivation

      1. flip-- 36.4%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2} \cdot {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}{\left(-b_2\right) + {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}}{a} \]

      2. frac-2neg 36.4%

         \[\leadsto \frac{\color{blue}{\frac{-\left(\left(-b_2\right) \cdot \left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2} \cdot {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}\right)}{-\left(\left(-b_2\right) + {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}\right)}}}{a} \]

   5. Applied egg-rr 36.6%

      \[\leadsto \frac{\color{blue}{\frac{-\left(b_2 \cdot b_2 - \left(b_2 \cdot b_2 - a \cdot c\right)\right)}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a} \]
   6. Step-by-step derivation

      1. unpow2 36.6%

         \[\leadsto \frac{\frac{-\left(\color{blue}{{b_2}^{2}} - \left(b_2 \cdot b_2 - a \cdot c\right)\right)}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]

      2. unpow2 36.6%

         \[\leadsto \frac{\frac{-\left({b_2}^{2} - \left(\color{blue}{{b_2}^{2}} - a \cdot c\right)\right)}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]

      3. associate--r- 87.4%

         \[\leadsto \frac{\frac{-\color{blue}{\left(\left({b_2}^{2} - {b_2}^{2}\right) + a \cdot c\right)}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]

      4. +-inverses 87.4%

         \[\leadsto \frac{\frac{-\left(\color{blue}{0} + a \cdot c\right)}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]

      5. distribute-neg-in 87.4%

         \[\leadsto \frac{\frac{\color{blue}{\left(-0\right) + \left(-a \cdot c\right)}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]

      6. metadata-eval 87.4%

         \[\leadsto \frac{\frac{\color{blue}{0} + \left(-a \cdot c\right)}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]

      7. distribute-rgt-neg-in 87.4%

         \[\leadsto \frac{\frac{0 + \color{blue}{a \cdot \left(-c\right)}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]

   7. Simplified 87.4%

      \[\leadsto \frac{\color{blue}{\frac{0 + a \cdot \left(-c\right)}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a} \]

   if -2.0000000000000001e-110 < b_2 < 8.5000000000000004e72

   1. Initial program 85.4%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   if 8.5000000000000004e72 < b_2

   1. Initial program 64.4%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 100.0%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.3 \cdot 10^{+48}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -2 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{c \cdot \left(-a\right)}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a}\\ \mathbf{elif}\;b_2 \leq 8.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 2: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -5.1 \cdot 10^{-70}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 8.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5.1e-70)
   (* -0.5 (/ c b_2))
   (if (<= b_2 8.5e+72)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.1e-70) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 8.5e+72) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	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 <= (-5.1d-70)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 8.5d+72) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((c / b_2) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.1e-70) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 8.5e+72) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5.1e-70:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 8.5e+72:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5.1e-70)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 8.5e+72)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c / b_2) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5.1e-70)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 8.5e+72)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5.1e-70], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 8.5e+72], 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[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -5.10000000000000025e-70

   1. Initial program 16.5%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 90.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]

   if -5.10000000000000025e-70 < b_2 < 8.5000000000000004e72

   1. Initial program 85.1%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   if 8.5000000000000004e72 < b_2

   1. Initial program 64.4%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 100.0%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -5.1 \cdot 10^{-70}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 8.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 3: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -8.8 \cdot 10^{-67}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 8.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -8.8e-67)
   (* -0.5 (/ c b_2))
   (if (<= b_2 8.5e-30)
     (/ (- (- b_2) (sqrt (* c (- a)))) a)
     (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -8.8e-67) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 8.5e-30) {
		tmp = (-b_2 - sqrt((c * -a))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	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 <= (-8.8d-67)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 8.5d-30) then
        tmp = (-b_2 - sqrt((c * -a))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((c / b_2) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -8.8e-67) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 8.5e-30) {
		tmp = (-b_2 - Math.sqrt((c * -a))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -8.8e-67:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 8.5e-30:
		tmp = (-b_2 - math.sqrt((c * -a))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -8.8e-67)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 8.5e-30)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(c * Float64(-a)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c / b_2) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -8.8e-67)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 8.5e-30)
		tmp = (-b_2 - sqrt((c * -a))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -8.8e-67], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 8.5e-30], 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[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -8.8000000000000004e-67

   1. Initial program 16.5%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 90.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]

   if -8.8000000000000004e-67 < b_2 < 8.49999999999999931e-30

   1. Initial program 84.5%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around 0 72.4%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]
   3. Step-by-step derivation

      1. associate-*r* 72.4%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}}}{a} \]

      2. neg-mul-1 72.4%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(-a\right)} \cdot c}}{a} \]

      3. *-commutative 72.4%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{c \cdot \left(-a\right)}}}{a} \]

   4. Simplified 72.4%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{c \cdot \left(-a\right)}}}{a} \]

   if 8.49999999999999931e-30 < b_2

   1. Initial program 70.2%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 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 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -8.8 \cdot 10^{-67}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 8.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 4: 68.3% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-309)
   (* -0.5 (/ c b_2))
   (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-309) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	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-309)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((c / b_2) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-309) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e-309:
		tmp = -0.5 * (c / b_2)
	else:
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-309)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c / b_2) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e-309)
		tmp = -0.5 * (c / b_2);
	else
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e-309], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.000000000000002e-309

   1. Initial program 30.5%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 73.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]

   if -1.000000000000002e-309 < b_2

   1. Initial program 78.5%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 64.7%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 5: 43.6% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-309) (/ 0.0 a) (* -2.0 (/ b_2 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-309) {
		tmp = 0.0 / 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 <= (-1d-309)) then
        tmp = 0.0d0 / 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 <= -1e-309) {
		tmp = 0.0 / a;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e-309:
		tmp = 0.0 / a
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-309)
		tmp = Float64(0.0 / a);
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e-309)
		tmp = 0.0 / 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, -1e-309], N[(0.0 / a), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.000000000000002e-309

   1. Initial program 30.5%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 27.9%

         \[\leadsto \frac{\left(-b_2\right) - \color{blue}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a} \]

      2. pow2 27.9%

         \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right)}^{2}}}{a} \]

      3. pow1/2 27.9%

         \[\leadsto \frac{\left(-b_2\right) - {\left(\sqrt{\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5}}}\right)}^{2}}{a} \]

      4. sqrt-pow1 27.9%

         \[\leadsto \frac{\left(-b_2\right) - {\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}}{a} \]

      5. metadata-eval 27.9%

         \[\leadsto \frac{\left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

   3. Applied egg-rr 27.9%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

   4. Taylor expanded in b_2 around -inf 19.7%

      \[\leadsto \frac{\color{blue}{b_2 + -1 \cdot b_2}}{a} \]
   5. Step-by-step derivation

      1. distribute-rgt1-in 19.7%

         \[\leadsto \frac{\color{blue}{\left(-1 + 1\right) \cdot b_2}}{a} \]

      2. metadata-eval 19.7%

         \[\leadsto \frac{\color{blue}{0} \cdot b_2}{a} \]

      3. mul0-lft 19.7%

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

   6. Simplified 19.7%

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

   if -1.000000000000002e-309 < b_2

   1. Initial program 78.5%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 64.6%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 6: 68.1% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-309) (* -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-309) {
		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-309)) 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-309) {
		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-309:
		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-309)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e-309)
		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-309], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.000000000000002e-309

   1. Initial program 30.5%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 73.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]

   if -1.000000000000002e-309 < b_2

   1. Initial program 78.5%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 64.6%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 7: 23.3% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-309) (/ 0.0 a) (/ (- b_2) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-309) {
		tmp = 0.0 / a;
	} 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 <= (-1d-309)) then
        tmp = 0.0d0 / a
    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 <= -1e-309) {
		tmp = 0.0 / a;
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e-309:
		tmp = 0.0 / a
	else:
		tmp = -b_2 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-309)
		tmp = Float64(0.0 / a);
	else
		tmp = Float64(Float64(-b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e-309)
		tmp = 0.0 / a;
	else
		tmp = -b_2 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e-309], N[(0.0 / a), $MachinePrecision], N[((-b$95$2) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.000000000000002e-309

   1. Initial program 30.5%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 27.9%

         \[\leadsto \frac{\left(-b_2\right) - \color{blue}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a} \]

      2. pow2 27.9%

         \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right)}^{2}}}{a} \]

      3. pow1/2 27.9%

         \[\leadsto \frac{\left(-b_2\right) - {\left(\sqrt{\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5}}}\right)}^{2}}{a} \]

      4. sqrt-pow1 27.9%

         \[\leadsto \frac{\left(-b_2\right) - {\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}}{a} \]

      5. metadata-eval 27.9%

         \[\leadsto \frac{\left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

   3. Applied egg-rr 27.9%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

   4. Taylor expanded in b_2 around -inf 19.7%

      \[\leadsto \frac{\color{blue}{b_2 + -1 \cdot b_2}}{a} \]
   5. Step-by-step derivation

      1. distribute-rgt1-in 19.7%

         \[\leadsto \frac{\color{blue}{\left(-1 + 1\right) \cdot b_2}}{a} \]

      2. metadata-eval 19.7%

         \[\leadsto \frac{\color{blue}{0} \cdot b_2}{a} \]

      3. mul0-lft 19.7%

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

   6. Simplified 19.7%

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

   if -1.000000000000002e-309 < b_2

   1. Initial program 78.5%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around 0 50.5%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]
   3. Step-by-step derivation

      1. associate-*r* 50.5%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}}}{a} \]

      2. neg-mul-1 50.5%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(-a\right)} \cdot c}}{a} \]

      3. *-commutative 50.5%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{c \cdot \left(-a\right)}}}{a} \]

   4. Simplified 50.5%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{c \cdot \left(-a\right)}}}{a} \]

   5. Taylor expanded in b_2 around inf 26.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{b_2}{a}} \]
   6. Step-by-step derivation

      1. associate-*r/ 26.4%

         \[\leadsto \color{blue}{\frac{-1 \cdot b_2}{a}} \]

      2. mul-1-neg 26.4%

         \[\leadsto \frac{\color{blue}{-b_2}}{a} \]

   7. Simplified 26.4%

      \[\leadsto \color{blue}{\frac{-b_2}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 8: 11.7% accurate, 37.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = 0.0d0 / a
end function

Java

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

Python

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

Julia

function code(a, b_2, c)
	return Float64(0.0 / a)
end

MATLAB

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

Wolfram

code[a_, b$95$2_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 53.2%

   \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
2. Step-by-step derivation

   1. add-sqr-sqrt 51.6%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a} \]

   2. pow2 51.6%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right)}^{2}}}{a} \]

   3. pow1/2 51.6%

      \[\leadsto \frac{\left(-b_2\right) - {\left(\sqrt{\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5}}}\right)}^{2}}{a} \]

   4. sqrt-pow1 51.6%

      \[\leadsto \frac{\left(-b_2\right) - {\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}}{a} \]

   5. metadata-eval 51.6%

      \[\leadsto \frac{\left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

3. Applied egg-rr 51.6%

   \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

4. Taylor expanded in b_2 around -inf 11.6%

   \[\leadsto \frac{\color{blue}{b_2 + -1 \cdot b_2}}{a} \]
5. Step-by-step derivation

   1. distribute-rgt1-in 11.6%

      \[\leadsto \frac{\color{blue}{\left(-1 + 1\right) \cdot b_2}}{a} \]

   2. metadata-eval 11.6%

      \[\leadsto \frac{\color{blue}{0} \cdot b_2}{a} \]

   3. mul0-lft 11.6%

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

6. Simplified 11.6%

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

7. Final simplification 11.6%

   \[\leadsto \frac{0}{a} \]

Reproduce

herbie shell --seed 2023280 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))