quad2m (problem 3.2.1, negative)

Percentage Accurate: 51.9% → 86.7%

Time: 15.6s

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: 51.9% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -3.8 \cdot 10^{+31}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq -1.72 \cdot 10^{-217}:\\ \;\;\;\;\frac{\frac{a \cdot \left(-c\right)}{b_2 - \sqrt{{b_2}^{2} - c \cdot a}}}{a}\\ \mathbf{elif}\;b_2 \leq 4.6 \cdot 10^{+57}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.8e+31)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 -1.72e-217)
     (/ (/ (* a (- c)) (- b_2 (sqrt (- (pow b_2 2.0) (* c a))))) a)
     (if (<= b_2 4.6e+57)
       (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
       (- (- (* 0.5 (/ c b_2)) (/ b_2 a)) (/ b_2 a))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.8e+31) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= -1.72e-217) {
		tmp = ((a * -c) / (b_2 - sqrt((pow(b_2, 2.0) - (c * a))))) / a;
	} else if (b_2 <= 4.6e+57) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (b_2 / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-3.8d+31)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= (-1.72d-217)) then
        tmp = ((a * -c) / (b_2 - sqrt(((b_2 ** 2.0d0) - (c * a))))) / a
    else if (b_2 <= 4.6d+57) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = ((0.5d0 * (c / b_2)) - (b_2 / a)) - (b_2 / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.8e+31) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= -1.72e-217) {
		tmp = ((a * -c) / (b_2 - Math.sqrt((Math.pow(b_2, 2.0) - (c * a))))) / a;
	} else if (b_2 <= 4.6e+57) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.8e+31:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= -1.72e-217:
		tmp = ((a * -c) / (b_2 - math.sqrt((math.pow(b_2, 2.0) - (c * a))))) / a
	elif b_2 <= 4.6e+57:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.8e+31)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= -1.72e-217)
		tmp = Float64(Float64(Float64(a * Float64(-c)) / Float64(b_2 - sqrt(Float64((b_2 ^ 2.0) - Float64(c * a))))) / a);
	elseif (b_2 <= 4.6e+57)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(Float64(0.5 * Float64(c / b_2)) - Float64(b_2 / a)) - Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.8e+31)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= -1.72e-217)
		tmp = ((a * -c) / (b_2 - sqrt(((b_2 ^ 2.0) - (c * a))))) / a;
	elseif (b_2 <= 4.6e+57)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.8e+31], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, -1.72e-217], N[(N[(N[(a * (-c)), $MachinePrecision] / N[(b$95$2 - N[Sqrt[N[(N[Power[b$95$2, 2.0], $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 4.6e+57], 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[(N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -3.8000000000000001e31

   1. Initial program 13.2%

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

      1. div-sub 10.0%

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

      2. neg-sub0 10.0%

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

      3. div-sub 10.0%

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

      4. associate--l- 10.0%

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

      5. +-commutative 10.0%

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

      6. pow2 10.0%

         \[\leadsto \frac{0}{a} - \left(\frac{\sqrt{\color{blue}{{b_2}^{2}} - a \cdot c}}{a} + \frac{b_2}{a}\right) \]

   3. Applied egg-rr 10.0%

      \[\leadsto \color{blue}{\frac{0}{a} - \left(\frac{\sqrt{{b_2}^{2} - a \cdot c}}{a} + \frac{b_2}{a}\right)} \]
   4. Step-by-step derivation

      1. associate--r+ 10.0%

         \[\leadsto \color{blue}{\left(\frac{0}{a} - \frac{\sqrt{{b_2}^{2} - a \cdot c}}{a}\right) - \frac{b_2}{a}} \]

      2. div0 10.0%

         \[\leadsto \left(\color{blue}{0} - \frac{\sqrt{{b_2}^{2} - a \cdot c}}{a}\right) - \frac{b_2}{a} \]

      3. neg-sub0 10.0%

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

      4. distribute-neg-frac 10.0%

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

   5. Simplified 10.0%

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

   6. Taylor expanded in b_2 around -inf 90.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   7. Step-by-step derivation

      1. associate-*r/ 90.9%

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

   8. Simplified 90.9%

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

   if -3.8000000000000001e31 < b_2 < -1.72e-217

   1. Initial program 38.6%

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

      1. add-sqr-sqrt 38.2%

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

      2. pow2 38.2%

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

      3. pow1/2 38.2%

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

      4. sqrt-pow1 38.2%

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

      5. pow2 38.2%

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

      6. metadata-eval 38.2%

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

   3. Applied egg-rr 38.2%

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

      1. flip-- 38.0%

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

      2. frac-2neg 38.0%

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

   5. Applied egg-rr 67.9%

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

      1. neg-sub0 67.9%

         \[\leadsto \frac{\frac{\color{blue}{0 - \left(a \cdot c + \left({b_2}^{2} - {b_2}^{2}\right)\right)}}{b_2 - \sqrt{{b_2}^{2} - a \cdot c}}}{a} \]

      2. +-inverses 67.9%

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

      3. +-commutative 67.9%

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

      4. associate--r+ 67.9%

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

      5. +-inverses 67.9%

         \[\leadsto \frac{\frac{\left(\color{blue}{0} - \left({b_2}^{2} - {b_2}^{2}\right)\right) - a \cdot c}{b_2 - \sqrt{{b_2}^{2} - a \cdot c}}}{a} \]

      6. +-inverses 67.9%

         \[\leadsto \frac{\frac{\left(0 - \color{blue}{0}\right) - a \cdot c}{b_2 - \sqrt{{b_2}^{2} - a \cdot c}}}{a} \]

      7. metadata-eval 67.9%

         \[\leadsto \frac{\frac{\color{blue}{0} - a \cdot c}{b_2 - \sqrt{{b_2}^{2} - a \cdot c}}}{a} \]

      8. neg-sub0 67.9%

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

      9. distribute-rgt-neg-in 67.9%

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

   7. Simplified 67.9%

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

   if -1.72e-217 < b_2 < 4.5999999999999998e57

   1. Initial program 89.0%

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

   if 4.5999999999999998e57 < b_2

   1. Initial program 58.6%

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

      1. div-sub 58.6%

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

      2. neg-sub0 58.6%

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

      3. div-sub 58.6%

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

      4. associate--l- 58.6%

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

      5. +-commutative 58.6%

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

      6. pow2 58.6%

         \[\leadsto \frac{0}{a} - \left(\frac{\sqrt{\color{blue}{{b_2}^{2}} - a \cdot c}}{a} + \frac{b_2}{a}\right) \]

   3. Applied egg-rr 58.6%

      \[\leadsto \color{blue}{\frac{0}{a} - \left(\frac{\sqrt{{b_2}^{2} - a \cdot c}}{a} + \frac{b_2}{a}\right)} \]
   4. Step-by-step derivation

      1. associate--r+ 58.6%

         \[\leadsto \color{blue}{\left(\frac{0}{a} - \frac{\sqrt{{b_2}^{2} - a \cdot c}}{a}\right) - \frac{b_2}{a}} \]

      2. div0 58.6%

         \[\leadsto \left(\color{blue}{0} - \frac{\sqrt{{b_2}^{2} - a \cdot c}}{a}\right) - \frac{b_2}{a} \]

      3. neg-sub0 58.6%

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

      4. distribute-neg-frac 58.6%

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

   5. Simplified 58.6%

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

   6. Taylor expanded in a around 0 92.7%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\right) - \frac{b_2}{a}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -3.8 \cdot 10^{+31}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq -1.72 \cdot 10^{-217}:\\ \;\;\;\;\frac{\frac{a \cdot \left(-c\right)}{b_2 - \sqrt{{b_2}^{2} - c \cdot a}}}{a}\\ \mathbf{elif}\;b_2 \leq 4.6 \cdot 10^{+57}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\ \end{array} \]

Alternative 2: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -8.2 \cdot 10^{-118}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 4.6 \cdot 10^{+57}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -8.2e-118)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 4.6e+57)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (- (- (* 0.5 (/ c b_2)) (/ b_2 a)) (/ b_2 a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -8.2e-118) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 4.6e+57) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (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 <= (-8.2d-118)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 4.6d+57) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = ((0.5d0 * (c / b_2)) - (b_2 / a)) - (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 <= -8.2e-118) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 4.6e+57) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -8.2e-118:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 4.6e+57:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -8.2e-118)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 4.6e+57)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(Float64(0.5 * Float64(c / b_2)) - Float64(b_2 / a)) - Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -8.2e-118)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 4.6e+57)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -8.2e-118], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 4.6e+57], 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[(N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -8.2000000000000006e-118

   1. Initial program 17.4%

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

      1. div-sub 15.1%

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

      2. neg-sub0 15.1%

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

      3. div-sub 15.1%

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

      4. associate--l- 15.1%

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

      5. +-commutative 15.1%

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

      6. pow2 15.1%

         \[\leadsto \frac{0}{a} - \left(\frac{\sqrt{\color{blue}{{b_2}^{2}} - a \cdot c}}{a} + \frac{b_2}{a}\right) \]

   3. Applied egg-rr 15.1%

      \[\leadsto \color{blue}{\frac{0}{a} - \left(\frac{\sqrt{{b_2}^{2} - a \cdot c}}{a} + \frac{b_2}{a}\right)} \]
   4. Step-by-step derivation

      1. associate--r+ 15.1%

         \[\leadsto \color{blue}{\left(\frac{0}{a} - \frac{\sqrt{{b_2}^{2} - a \cdot c}}{a}\right) - \frac{b_2}{a}} \]

      2. div0 15.1%

         \[\leadsto \left(\color{blue}{0} - \frac{\sqrt{{b_2}^{2} - a \cdot c}}{a}\right) - \frac{b_2}{a} \]

      3. neg-sub0 15.1%

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

      4. distribute-neg-frac 15.1%

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

   5. Simplified 15.1%

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

   6. Taylor expanded in b_2 around -inf 82.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   7. Step-by-step derivation

      1. associate-*r/ 82.4%

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

   8. Simplified 82.4%

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

   if -8.2000000000000006e-118 < b_2 < 4.5999999999999998e57

   1. Initial program 81.9%

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

   if 4.5999999999999998e57 < b_2

   1. Initial program 58.6%

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

      1. div-sub 58.6%

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

      2. neg-sub0 58.6%

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

      3. div-sub 58.6%

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

      4. associate--l- 58.6%

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

      5. +-commutative 58.6%

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

      6. pow2 58.6%

         \[\leadsto \frac{0}{a} - \left(\frac{\sqrt{\color{blue}{{b_2}^{2}} - a \cdot c}}{a} + \frac{b_2}{a}\right) \]

   3. Applied egg-rr 58.6%

      \[\leadsto \color{blue}{\frac{0}{a} - \left(\frac{\sqrt{{b_2}^{2} - a \cdot c}}{a} + \frac{b_2}{a}\right)} \]
   4. Step-by-step derivation

      1. associate--r+ 58.6%

         \[\leadsto \color{blue}{\left(\frac{0}{a} - \frac{\sqrt{{b_2}^{2} - a \cdot c}}{a}\right) - \frac{b_2}{a}} \]

      2. div0 58.6%

         \[\leadsto \left(\color{blue}{0} - \frac{\sqrt{{b_2}^{2} - a \cdot c}}{a}\right) - \frac{b_2}{a} \]

      3. neg-sub0 58.6%

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

      4. distribute-neg-frac 58.6%

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

   5. Simplified 58.6%

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

   6. Taylor expanded in a around 0 92.7%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\right) - \frac{b_2}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -8.2 \cdot 10^{-118}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 4.6 \cdot 10^{+57}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\ \end{array} \]

Alternative 3: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -7 \cdot 10^{-121}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.35 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7e-121)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 2.35e-35)
     (/ (- (- b_2) (sqrt (* a (- c)))) a)
     (- (- (* 0.5 (/ c b_2)) (/ b_2 a)) (/ b_2 a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7e-121) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 2.35e-35) {
		tmp = (-b_2 - sqrt((a * -c))) / a;
	} else {
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (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 <= (-7d-121)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 2.35d-35) then
        tmp = (-b_2 - sqrt((a * -c))) / a
    else
        tmp = ((0.5d0 * (c / b_2)) - (b_2 / a)) - (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 <= -7e-121) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 2.35e-35) {
		tmp = (-b_2 - Math.sqrt((a * -c))) / a;
	} else {
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -7e-121:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 2.35e-35:
		tmp = (-b_2 - math.sqrt((a * -c))) / a
	else:
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -7e-121)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 2.35e-35)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(a * Float64(-c)))) / a);
	else
		tmp = Float64(Float64(Float64(0.5 * Float64(c / b_2)) - Float64(b_2 / a)) - Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -7e-121)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 2.35e-35)
		tmp = (-b_2 - sqrt((a * -c))) / a;
	else
		tmp = ((0.5 * (c / b_2)) - (b_2 / a)) - (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7e-121], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 2.35e-35], N[(N[((-b$95$2) - N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -6.99999999999999985e-121

   1. Initial program 17.4%

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

      1. div-sub 15.1%

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

      2. neg-sub0 15.1%

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

      3. div-sub 15.1%

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

      4. associate--l- 15.1%

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

      5. +-commutative 15.1%

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

      6. pow2 15.1%

         \[\leadsto \frac{0}{a} - \left(\frac{\sqrt{\color{blue}{{b_2}^{2}} - a \cdot c}}{a} + \frac{b_2}{a}\right) \]

   3. Applied egg-rr 15.1%

      \[\leadsto \color{blue}{\frac{0}{a} - \left(\frac{\sqrt{{b_2}^{2} - a \cdot c}}{a} + \frac{b_2}{a}\right)} \]
   4. Step-by-step derivation

      1. associate--r+ 15.1%

         \[\leadsto \color{blue}{\left(\frac{0}{a} - \frac{\sqrt{{b_2}^{2} - a \cdot c}}{a}\right) - \frac{b_2}{a}} \]

      2. div0 15.1%

         \[\leadsto \left(\color{blue}{0} - \frac{\sqrt{{b_2}^{2} - a \cdot c}}{a}\right) - \frac{b_2}{a} \]

      3. neg-sub0 15.1%

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

      4. distribute-neg-frac 15.1%

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

   5. Simplified 15.1%

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

   6. Taylor expanded in b_2 around -inf 82.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   7. Step-by-step derivation

      1. associate-*r/ 82.4%

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

   8. Simplified 82.4%

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

   if -6.99999999999999985e-121 < b_2 < 2.35e-35

   1. Initial program 76.7%

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

   2. Taylor expanded in b_2 around 0 68.2%

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

      1. mul-1-neg 68.2%

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

      2. distribute-rgt-neg-out 68.2%

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

   4. Simplified 68.2%

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

   if 2.35e-35 < b_2

   1. Initial program 68.6%

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

      1. div-sub 68.6%

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

      2. neg-sub0 68.6%

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

      3. div-sub 68.6%

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

      4. associate--l- 68.6%

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

      5. +-commutative 68.6%

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

      6. pow2 68.6%

         \[\leadsto \frac{0}{a} - \left(\frac{\sqrt{\color{blue}{{b_2}^{2}} - a \cdot c}}{a} + \frac{b_2}{a}\right) \]

   3. Applied egg-rr 68.6%

      \[\leadsto \color{blue}{\frac{0}{a} - \left(\frac{\sqrt{{b_2}^{2} - a \cdot c}}{a} + \frac{b_2}{a}\right)} \]
   4. Step-by-step derivation

      1. associate--r+ 68.6%

         \[\leadsto \color{blue}{\left(\frac{0}{a} - \frac{\sqrt{{b_2}^{2} - a \cdot c}}{a}\right) - \frac{b_2}{a}} \]

      2. div0 68.6%

         \[\leadsto \left(\color{blue}{0} - \frac{\sqrt{{b_2}^{2} - a \cdot c}}{a}\right) - \frac{b_2}{a} \]

      3. neg-sub0 68.6%

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

      4. distribute-neg-frac 68.6%

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

   5. Simplified 68.6%

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

   6. Taylor expanded in a around 0 84.9%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\right) - \frac{b_2}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -7 \cdot 10^{-121}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.35 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\ \end{array} \]

Alternative 4: 43.1% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -1.95 \cdot 10^{+24}:\\ \;\;\;\;\frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.95e+24) (/ c b_2) (* (/ b_2 a) -2.0)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.95e+24) {
		tmp = c / b_2;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.95d+24)) then
        tmp = c / b_2
    else
        tmp = (b_2 / a) * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.95e+24) {
		tmp = c / b_2;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.95e+24:
		tmp = c / b_2
	else:
		tmp = (b_2 / a) * -2.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.95e+24)
		tmp = Float64(c / b_2);
	else
		tmp = Float64(Float64(b_2 / a) * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.95e+24)
		tmp = c / b_2;
	else
		tmp = (b_2 / a) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.95e+24], N[(c / b$95$2), $MachinePrecision], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.9499999999999999e24

   1. Initial program 13.8%

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

      1. div-inv 13.8%

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

      2. *-commutative 13.8%

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

      3. add-sqr-sqrt 11.8%

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

      4. sqrt-unprod 12.9%

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

      5. sqr-neg 12.9%

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

      6. sqrt-prod 0.0%

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

      7. add-sqr-sqrt 4.4%

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

      8. pow2 4.4%

         \[\leadsto \frac{1}{a} \cdot \left(b_2 - \sqrt{\color{blue}{{b_2}^{2}} - a \cdot c}\right) \]

   3. Applied egg-rr 4.4%

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(b_2 - \sqrt{{b_2}^{2} - a \cdot c}\right)} \]

   5. Applied egg-rr 1.7%

      \[\leadsto \color{blue}{\frac{\left(a \cdot c + \left({b_2}^{2} - {b_2}^{2}\right)\right) \cdot -1}{\left(b_2 + \sqrt{{b_2}^{2} - a \cdot c}\right) \cdot \left(-a\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 1.7%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(a \cdot c + \left({b_2}^{2} - {b_2}^{2}\right)\right)}}{\left(b_2 + \sqrt{{b_2}^{2} - a \cdot c}\right) \cdot \left(-a\right)} \]

      2. *-commutative 1.7%

         \[\leadsto \frac{-1 \cdot \left(a \cdot c + \left({b_2}^{2} - {b_2}^{2}\right)\right)}{\color{blue}{\left(-a\right) \cdot \left(b_2 + \sqrt{{b_2}^{2} - a \cdot c}\right)}} \]

      3. mul-1-neg 1.7%

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

      4. neg-sub0 1.7%

         \[\leadsto \frac{\color{blue}{0 - \left(a \cdot c + \left({b_2}^{2} - {b_2}^{2}\right)\right)}}{\left(-a\right) \cdot \left(b_2 + \sqrt{{b_2}^{2} - a \cdot c}\right)} \]

      5. +-inverses 1.7%

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

      6. +-commutative 1.7%

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

      7. associate--r+ 1.7%

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

      8. +-inverses 1.7%

         \[\leadsto \frac{\left(\color{blue}{0} - \left({b_2}^{2} - {b_2}^{2}\right)\right) - a \cdot c}{\left(-a\right) \cdot \left(b_2 + \sqrt{{b_2}^{2} - a \cdot c}\right)} \]

      9. +-inverses 21.3%

         \[\leadsto \frac{\left(0 - \color{blue}{0}\right) - a \cdot c}{\left(-a\right) \cdot \left(b_2 + \sqrt{{b_2}^{2} - a \cdot c}\right)} \]

      10. metadata-eval 21.3%

          \[\leadsto \frac{\color{blue}{0} - a \cdot c}{\left(-a\right) \cdot \left(b_2 + \sqrt{{b_2}^{2} - a \cdot c}\right)} \]

      11. neg-sub0 21.3%

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

      12. distribute-rgt-neg-in 21.3%

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

   7. Simplified 21.3%

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

   8. Taylor expanded in b_2 around 0 22.7%

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

      1. mul-1-neg 22.7%

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

      2. distribute-lft-neg-in 22.7%

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

      3. *-commutative 22.7%

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

   10. Simplified 22.7%

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

   11. Taylor expanded in a around 0 27.9%

       \[\leadsto \color{blue}{\frac{c}{b_2}} \]

   if -1.9499999999999999e24 < b_2

   1. Initial program 65.1%

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

   2. Taylor expanded in b_2 around inf 47.1%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.95 \cdot 10^{+24}:\\ \;\;\;\;\frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array} \]

Alternative 5: 67.2% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -1.72 \cdot 10^{-245}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.72e-245) (* -0.5 (/ c b_2)) (* (/ b_2 a) -2.0)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.72e-245) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.72d-245)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = (b_2 / a) * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.72e-245) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.72e-245:
		tmp = -0.5 * (c / b_2)
	else:
		tmp = (b_2 / a) * -2.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.72e-245)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(Float64(b_2 / a) * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.72e-245)
		tmp = -0.5 * (c / b_2);
	else
		tmp = (b_2 / a) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.72e-245], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.71999999999999997e-245

   1. Initial program 24.0%

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

   2. Taylor expanded in b_2 around -inf 73.5%

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

   if -1.71999999999999997e-245 < b_2

   1. Initial program 73.8%

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

   2. Taylor expanded in b_2 around inf 63.8%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.72 \cdot 10^{-245}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array} \]

Alternative 6: 67.2% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -1.72 \cdot 10^{-245}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.72e-245) (/ (* -0.5 c) b_2) (* (/ b_2 a) -2.0)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.72e-245) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.72d-245)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = (b_2 / a) * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.72e-245) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.72e-245:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = (b_2 / a) * -2.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.72e-245)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(Float64(b_2 / a) * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.72e-245)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = (b_2 / a) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.72e-245], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.71999999999999997e-245

   1. Initial program 24.0%

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

      1. div-sub 22.1%

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

      2. neg-sub0 22.1%

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

      3. div-sub 22.1%

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

      4. associate--l- 22.1%

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

      5. +-commutative 22.1%

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

      6. pow2 22.1%

         \[\leadsto \frac{0}{a} - \left(\frac{\sqrt{\color{blue}{{b_2}^{2}} - a \cdot c}}{a} + \frac{b_2}{a}\right) \]

   3. Applied egg-rr 22.1%

      \[\leadsto \color{blue}{\frac{0}{a} - \left(\frac{\sqrt{{b_2}^{2} - a \cdot c}}{a} + \frac{b_2}{a}\right)} \]
   4. Step-by-step derivation

      1. associate--r+ 22.1%

         \[\leadsto \color{blue}{\left(\frac{0}{a} - \frac{\sqrt{{b_2}^{2} - a \cdot c}}{a}\right) - \frac{b_2}{a}} \]

      2. div0 22.1%

         \[\leadsto \left(\color{blue}{0} - \frac{\sqrt{{b_2}^{2} - a \cdot c}}{a}\right) - \frac{b_2}{a} \]

      3. neg-sub0 22.1%

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

      4. distribute-neg-frac 22.1%

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

   5. Simplified 22.1%

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

   6. Taylor expanded in b_2 around -inf 73.5%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   7. Step-by-step derivation

      1. associate-*r/ 73.5%

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

   8. Simplified 73.5%

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

   if -1.71999999999999997e-245 < b_2

   1. Initial program 73.8%

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

   2. Taylor expanded in b_2 around inf 63.8%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.72 \cdot 10^{-245}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array} \]

Alternative 7: 23.0% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2.1 \cdot 10^{+24}:\\ \;\;\;\;\frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.1e+24) (/ c b_2) (/ (- b_2) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.1e+24) {
		tmp = 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 <= (-2.1d+24)) then
        tmp = 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 <= -2.1e+24) {
		tmp = c / b_2;
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.1e+24:
		tmp = c / b_2
	else:
		tmp = -b_2 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.1e+24)
		tmp = Float64(c / b_2);
	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 <= -2.1e+24)
		tmp = 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, -2.1e+24], N[(c / b$95$2), $MachinePrecision], N[((-b$95$2) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.1000000000000001e24

   1. Initial program 13.8%

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

      1. div-inv 13.8%

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

      2. *-commutative 13.8%

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

      3. add-sqr-sqrt 11.8%

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

      4. sqrt-unprod 12.9%

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

      5. sqr-neg 12.9%

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

      6. sqrt-prod 0.0%

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

      7. add-sqr-sqrt 4.4%

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

      8. pow2 4.4%

         \[\leadsto \frac{1}{a} \cdot \left(b_2 - \sqrt{\color{blue}{{b_2}^{2}} - a \cdot c}\right) \]

   3. Applied egg-rr 4.4%

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(b_2 - \sqrt{{b_2}^{2} - a \cdot c}\right)} \]

   5. Applied egg-rr 1.7%

      \[\leadsto \color{blue}{\frac{\left(a \cdot c + \left({b_2}^{2} - {b_2}^{2}\right)\right) \cdot -1}{\left(b_2 + \sqrt{{b_2}^{2} - a \cdot c}\right) \cdot \left(-a\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 1.7%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(a \cdot c + \left({b_2}^{2} - {b_2}^{2}\right)\right)}}{\left(b_2 + \sqrt{{b_2}^{2} - a \cdot c}\right) \cdot \left(-a\right)} \]

      2. *-commutative 1.7%

         \[\leadsto \frac{-1 \cdot \left(a \cdot c + \left({b_2}^{2} - {b_2}^{2}\right)\right)}{\color{blue}{\left(-a\right) \cdot \left(b_2 + \sqrt{{b_2}^{2} - a \cdot c}\right)}} \]

      3. mul-1-neg 1.7%

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

      4. neg-sub0 1.7%

         \[\leadsto \frac{\color{blue}{0 - \left(a \cdot c + \left({b_2}^{2} - {b_2}^{2}\right)\right)}}{\left(-a\right) \cdot \left(b_2 + \sqrt{{b_2}^{2} - a \cdot c}\right)} \]

      5. +-inverses 1.7%

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

      6. +-commutative 1.7%

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

      7. associate--r+ 1.7%

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

      8. +-inverses 1.7%

         \[\leadsto \frac{\left(\color{blue}{0} - \left({b_2}^{2} - {b_2}^{2}\right)\right) - a \cdot c}{\left(-a\right) \cdot \left(b_2 + \sqrt{{b_2}^{2} - a \cdot c}\right)} \]

      9. +-inverses 21.3%

         \[\leadsto \frac{\left(0 - \color{blue}{0}\right) - a \cdot c}{\left(-a\right) \cdot \left(b_2 + \sqrt{{b_2}^{2} - a \cdot c}\right)} \]

      10. metadata-eval 21.3%

          \[\leadsto \frac{\color{blue}{0} - a \cdot c}{\left(-a\right) \cdot \left(b_2 + \sqrt{{b_2}^{2} - a \cdot c}\right)} \]

      11. neg-sub0 21.3%

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

      12. distribute-rgt-neg-in 21.3%

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

   7. Simplified 21.3%

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

   8. Taylor expanded in b_2 around 0 22.7%

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

      1. mul-1-neg 22.7%

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

      2. distribute-lft-neg-in 22.7%

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

      3. *-commutative 22.7%

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

   10. Simplified 22.7%

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

   11. Taylor expanded in a around 0 27.9%

       \[\leadsto \color{blue}{\frac{c}{b_2}} \]

   if -2.1000000000000001e24 < b_2

   1. Initial program 65.1%

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

      1. add-sqr-sqrt 64.9%

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

      2. pow2 64.9%

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

      3. pow1/2 64.9%

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

      4. sqrt-pow1 64.9%

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

      5. pow2 64.9%

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

      6. metadata-eval 64.9%

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

   3. Applied egg-rr 64.9%

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

   4. Taylor expanded in b_2 around inf 20.6%

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

      1. mul-1-neg 20.6%

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

   6. Simplified 20.6%

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

4. Final simplification 23.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.1 \cdot 10^{+24}:\\ \;\;\;\;\frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 8: 11.2% accurate, 37.3× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b_2} \end{array} \]

FPCore

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

C

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

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 = c / b_2
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.3%

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

   1. div-inv 48.2%

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

   2. *-commutative 48.2%

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

   3. add-sqr-sqrt 13.6%

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

   4. sqrt-unprod 26.0%

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

   5. sqr-neg 26.0%

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

   6. sqrt-prod 20.5%

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

   7. add-sqr-sqrt 30.8%

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

   8. pow2 30.8%

      \[\leadsto \frac{1}{a} \cdot \left(b_2 - \sqrt{\color{blue}{{b_2}^{2}} - a \cdot c}\right) \]

3. Applied egg-rr 30.8%

   \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(b_2 - \sqrt{{b_2}^{2} - a \cdot c}\right)} \]

5. Applied egg-rr 18.3%

   \[\leadsto \color{blue}{\frac{\left(a \cdot c + \left({b_2}^{2} - {b_2}^{2}\right)\right) \cdot -1}{\left(b_2 + \sqrt{{b_2}^{2} - a \cdot c}\right) \cdot \left(-a\right)}} \]
6. Step-by-step derivation

   1. *-commutative 18.3%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(a \cdot c + \left({b_2}^{2} - {b_2}^{2}\right)\right)}}{\left(b_2 + \sqrt{{b_2}^{2} - a \cdot c}\right) \cdot \left(-a\right)} \]

   2. *-commutative 18.3%

      \[\leadsto \frac{-1 \cdot \left(a \cdot c + \left({b_2}^{2} - {b_2}^{2}\right)\right)}{\color{blue}{\left(-a\right) \cdot \left(b_2 + \sqrt{{b_2}^{2} - a \cdot c}\right)}} \]

   3. mul-1-neg 18.3%

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

   4. neg-sub0 18.3%

      \[\leadsto \frac{\color{blue}{0 - \left(a \cdot c + \left({b_2}^{2} - {b_2}^{2}\right)\right)}}{\left(-a\right) \cdot \left(b_2 + \sqrt{{b_2}^{2} - a \cdot c}\right)} \]

   5. +-inverses 18.3%

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

   6. +-commutative 18.3%

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

   7. associate--r+ 18.3%

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

   8. +-inverses 18.3%

      \[\leadsto \frac{\left(\color{blue}{0} - \left({b_2}^{2} - {b_2}^{2}\right)\right) - a \cdot c}{\left(-a\right) \cdot \left(b_2 + \sqrt{{b_2}^{2} - a \cdot c}\right)} \]

   9. +-inverses 25.1%

      \[\leadsto \frac{\left(0 - \color{blue}{0}\right) - a \cdot c}{\left(-a\right) \cdot \left(b_2 + \sqrt{{b_2}^{2} - a \cdot c}\right)} \]

   10. metadata-eval 25.1%

       \[\leadsto \frac{\color{blue}{0} - a \cdot c}{\left(-a\right) \cdot \left(b_2 + \sqrt{{b_2}^{2} - a \cdot c}\right)} \]

   11. neg-sub0 25.1%

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

   12. distribute-rgt-neg-in 25.1%

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

7. Simplified 25.1%

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

8. Taylor expanded in b_2 around 0 25.3%

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

   1. mul-1-neg 25.3%

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

   2. distribute-lft-neg-in 25.3%

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

   3. *-commutative 25.3%

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

10. Simplified 25.3%

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

11. Taylor expanded in a around 0 11.2%

    \[\leadsto \color{blue}{\frac{c}{b_2}} \]

12. Final simplification 11.2%

    \[\leadsto \frac{c}{b_2} \]

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 2023305 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  :herbie-expected 10

  :herbie-target
  (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))