quad2p (problem 3.2.1, positive)

Percentage Accurate: 60.5% → 83.6%

Time: 11.8s

Alternatives: 12

Speedup: 7.5×

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: 60.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{+154}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.5 \cdot 10^{+61}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5 \cdot c}{\frac{b\_2}{a}}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e+154)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 1.5e+61)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (/ (/ (- (* 0.5 c)) (/ b_2 a)) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e+154) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 1.5e+61) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (-(0.5 * c) / (b_2 / a)) / 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+154)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= 1.5d+61) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
    else
        tmp = (-(0.5d0 * c) / (b_2 / a)) / 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+154) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 1.5e+61) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (-(0.5 * c) / (b_2 / a)) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e+154:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 1.5e+61:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
	else:
		tmp = (-(0.5 * c) / (b_2 / a)) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e+154)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 1.5e+61)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(Float64(Float64(-Float64(0.5 * c)) / Float64(b_2 / a)) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e+154)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 1.5e+61)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	else
		tmp = (-(0.5 * c) / (b_2 / a)) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e+154], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.5e+61], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[((-N[(0.5 * c), $MachinePrecision]) / N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.00000000000000004e154

   1. Initial program 45.0%

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

      1. +-commutative 45.0%

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

      2. unsub-neg 45.0%

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

   3. Simplified 45.0%

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

   5. Taylor expanded in b_2 around -inf 96.4%

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

   if -1.00000000000000004e154 < b_2 < 1.5e61

   1. Initial program 84.6%

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

      1. +-commutative 84.6%

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

      2. unsub-neg 84.6%

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

   3. Simplified 84.6%

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

   if 1.5e61 < b_2

   1. Initial program 18.0%

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

      1. +-commutative 18.0%

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

      2. unsub-neg 18.0%

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

   3. Simplified 18.0%

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

   5. Taylor expanded in b_2 around inf 74.9%

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

      1. associate-/l* 79.2%

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

      2. *-un-lft-identity 79.2%

         \[\leadsto \frac{-0.5 \cdot \frac{\color{blue}{1 \cdot a}}{\frac{b\_2}{c}}}{a} \]

      3. div-inv 79.1%

         \[\leadsto \frac{-0.5 \cdot \frac{1 \cdot a}{\color{blue}{b\_2 \cdot \frac{1}{c}}}}{a} \]

      4. times-frac 74.8%

         \[\leadsto \frac{-0.5 \cdot \color{blue}{\left(\frac{1}{b\_2} \cdot \frac{a}{\frac{1}{c}}\right)}}{a} \]

   7. Applied egg-rr 74.8%

      \[\leadsto \frac{-0.5 \cdot \color{blue}{\left(\frac{1}{b\_2} \cdot \frac{a}{\frac{1}{c}}\right)}}{a} \]
   8. Step-by-step derivation

      1. associate-*l/ 74.8%

         \[\leadsto \frac{-0.5 \cdot \color{blue}{\frac{1 \cdot \frac{a}{\frac{1}{c}}}{b\_2}}}{a} \]

      2. *-un-lft-identity 74.8%

         \[\leadsto \frac{-0.5 \cdot \frac{\color{blue}{\frac{a}{\frac{1}{c}}}}{b\_2}}{a} \]

      3. associate-*r/ 74.8%

         \[\leadsto \frac{\color{blue}{\frac{-0.5 \cdot \frac{a}{\frac{1}{c}}}{b\_2}}}{a} \]

      4. metadata-eval 74.8%

         \[\leadsto \frac{\frac{\color{blue}{\left(-0.5\right)} \cdot \frac{a}{\frac{1}{c}}}{b\_2}}{a} \]

      5. associate-/r/ 74.9%

         \[\leadsto \frac{\frac{\left(-0.5\right) \cdot \color{blue}{\left(\frac{a}{1} \cdot c\right)}}{b\_2}}{a} \]

      6. /-rgt-identity 74.9%

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

      7. *-commutative 74.9%

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

      8. distribute-lft-neg-in 74.9%

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

      9. associate-*l* 74.9%

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

      10. distribute-neg-frac 74.9%

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

      11. associate-/l* 86.2%

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

      12. distribute-neg-frac 86.2%

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

   9. Applied egg-rr 86.2%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{+154}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.5 \cdot 10^{+61}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5 \cdot c}{\frac{b\_2}{a}}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.5 \cdot 10^{-46}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 4 \cdot 10^{-16}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5 \cdot c}{\frac{b\_2}{a}}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5.5e-46)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 4e-16)
     (/ (- (sqrt (* c (- a))) b_2) a)
     (/ (/ (- (* 0.5 c)) (/ b_2 a)) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.5e-46) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 4e-16) {
		tmp = (sqrt((c * -a)) - b_2) / a;
	} else {
		tmp = (-(0.5 * c) / (b_2 / a)) / 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 <= (-5.5d-46)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= 4d-16) then
        tmp = (sqrt((c * -a)) - b_2) / a
    else
        tmp = (-(0.5d0 * c) / (b_2 / a)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.5e-46) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 4e-16) {
		tmp = (Math.sqrt((c * -a)) - b_2) / a;
	} else {
		tmp = (-(0.5 * c) / (b_2 / a)) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5.5e-46:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 4e-16:
		tmp = (math.sqrt((c * -a)) - b_2) / a
	else:
		tmp = (-(0.5 * c) / (b_2 / a)) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5.5e-46)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 4e-16)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(-a))) - b_2) / a);
	else
		tmp = Float64(Float64(Float64(-Float64(0.5 * c)) / Float64(b_2 / a)) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5.5e-46)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 4e-16)
		tmp = (sqrt((c * -a)) - b_2) / a;
	else
		tmp = (-(0.5 * c) / (b_2 / a)) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5.5e-46], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 4e-16], N[(N[(N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[((-N[(0.5 * c), $MachinePrecision]) / N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -5.49999999999999983e-46

   1. Initial program 69.4%

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

      1. +-commutative 69.4%

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

      2. unsub-neg 69.4%

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

   3. Simplified 69.4%

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

   5. Taylor expanded in b_2 around -inf 90.5%

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

   if -5.49999999999999983e-46 < b_2 < 3.9999999999999999e-16

   1. Initial program 78.9%

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

      1. +-commutative 78.9%

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

      2. unsub-neg 78.9%

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

   3. Simplified 78.9%

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

   5. Taylor expanded in b_2 around 0 70.9%

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

      1. associate-*r* 70.9%

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

      2. neg-mul-1 70.9%

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

   7. Simplified 70.9%

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

   if 3.9999999999999999e-16 < b_2

   1. Initial program 22.0%

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

      1. +-commutative 22.0%

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

      2. unsub-neg 22.0%

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

   3. Simplified 22.0%

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

   5. Taylor expanded in b_2 around inf 73.1%

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

      1. associate-/l* 77.1%

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

      2. *-un-lft-identity 77.1%

         \[\leadsto \frac{-0.5 \cdot \frac{\color{blue}{1 \cdot a}}{\frac{b\_2}{c}}}{a} \]

      3. div-inv 76.9%

         \[\leadsto \frac{-0.5 \cdot \frac{1 \cdot a}{\color{blue}{b\_2 \cdot \frac{1}{c}}}}{a} \]

      4. times-frac 73.0%

         \[\leadsto \frac{-0.5 \cdot \color{blue}{\left(\frac{1}{b\_2} \cdot \frac{a}{\frac{1}{c}}\right)}}{a} \]

   7. Applied egg-rr 73.0%

      \[\leadsto \frac{-0.5 \cdot \color{blue}{\left(\frac{1}{b\_2} \cdot \frac{a}{\frac{1}{c}}\right)}}{a} \]
   8. Step-by-step derivation

      1. associate-*l/ 73.0%

         \[\leadsto \frac{-0.5 \cdot \color{blue}{\frac{1 \cdot \frac{a}{\frac{1}{c}}}{b\_2}}}{a} \]

      2. *-un-lft-identity 73.0%

         \[\leadsto \frac{-0.5 \cdot \frac{\color{blue}{\frac{a}{\frac{1}{c}}}}{b\_2}}{a} \]

      3. associate-*r/ 73.0%

         \[\leadsto \frac{\color{blue}{\frac{-0.5 \cdot \frac{a}{\frac{1}{c}}}{b\_2}}}{a} \]

      4. metadata-eval 73.0%

         \[\leadsto \frac{\frac{\color{blue}{\left(-0.5\right)} \cdot \frac{a}{\frac{1}{c}}}{b\_2}}{a} \]

      5. associate-/r/ 73.1%

         \[\leadsto \frac{\frac{\left(-0.5\right) \cdot \color{blue}{\left(\frac{a}{1} \cdot c\right)}}{b\_2}}{a} \]

      6. /-rgt-identity 73.1%

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

      7. *-commutative 73.1%

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

      8. distribute-lft-neg-in 73.1%

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

      9. associate-*l* 73.1%

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

      10. distribute-neg-frac 73.1%

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

      11. associate-/l* 83.5%

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

      12. distribute-neg-frac 83.5%

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

   9. Applied egg-rr 83.5%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.5 \cdot 10^{-46}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 4 \cdot 10^{-16}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5 \cdot c}{\frac{b\_2}{a}}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.9% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-310)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (/ (/ (- (* 0.5 c)) (/ b_2 a)) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-310) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else {
		tmp = (-(0.5 * c) / (b_2 / a)) / 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 <= (-2d-310)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else
        tmp = (-(0.5d0 * c) / (b_2 / a)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-310) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else {
		tmp = (-(0.5 * c) / (b_2 / a)) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e-310:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	else:
		tmp = (-(0.5 * c) / (b_2 / a)) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e-310)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	else
		tmp = Float64(Float64(Float64(-Float64(0.5 * c)) / Float64(b_2 / a)) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2e-310)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	else
		tmp = (-(0.5 * c) / (b_2 / a)) / a;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.999999999999994e-310

   1. Initial program 72.8%

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

      1. +-commutative 72.8%

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

      2. unsub-neg 72.8%

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

   3. Simplified 72.8%

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

   5. Taylor expanded in b_2 around -inf 65.8%

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

   if -1.999999999999994e-310 < b_2

   1. Initial program 41.2%

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

      1. +-commutative 41.2%

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

      2. unsub-neg 41.2%

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

   3. Simplified 41.2%

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

   5. Taylor expanded in b_2 around inf 52.1%

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

      1. associate-/l* 54.0%

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

      2. *-un-lft-identity 54.0%

         \[\leadsto \frac{-0.5 \cdot \frac{\color{blue}{1 \cdot a}}{\frac{b\_2}{c}}}{a} \]

      3. div-inv 53.9%

         \[\leadsto \frac{-0.5 \cdot \frac{1 \cdot a}{\color{blue}{b\_2 \cdot \frac{1}{c}}}}{a} \]

      4. times-frac 52.0%

         \[\leadsto \frac{-0.5 \cdot \color{blue}{\left(\frac{1}{b\_2} \cdot \frac{a}{\frac{1}{c}}\right)}}{a} \]

   7. Applied egg-rr 52.0%

      \[\leadsto \frac{-0.5 \cdot \color{blue}{\left(\frac{1}{b\_2} \cdot \frac{a}{\frac{1}{c}}\right)}}{a} \]
   8. Step-by-step derivation

      1. associate-*l/ 52.0%

         \[\leadsto \frac{-0.5 \cdot \color{blue}{\frac{1 \cdot \frac{a}{\frac{1}{c}}}{b\_2}}}{a} \]

      2. *-un-lft-identity 52.0%

         \[\leadsto \frac{-0.5 \cdot \frac{\color{blue}{\frac{a}{\frac{1}{c}}}}{b\_2}}{a} \]

      3. associate-*r/ 52.0%

         \[\leadsto \frac{\color{blue}{\frac{-0.5 \cdot \frac{a}{\frac{1}{c}}}{b\_2}}}{a} \]

      4. metadata-eval 52.0%

         \[\leadsto \frac{\frac{\color{blue}{\left(-0.5\right)} \cdot \frac{a}{\frac{1}{c}}}{b\_2}}{a} \]

      5. associate-/r/ 52.1%

         \[\leadsto \frac{\frac{\left(-0.5\right) \cdot \color{blue}{\left(\frac{a}{1} \cdot c\right)}}{b\_2}}{a} \]

      6. /-rgt-identity 52.1%

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

      7. *-commutative 52.1%

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

      8. distribute-lft-neg-in 52.1%

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

      9. associate-*l* 52.1%

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

      10. distribute-neg-frac 52.1%

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

      11. associate-/l* 58.1%

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

      12. distribute-neg-frac 58.1%

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

   9. Applied egg-rr 58.1%

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

4. Final simplification 62.3%

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

Alternative 4: 60.6% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 9 \cdot 10^{-263}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5 \cdot c}{\frac{b\_2}{a}}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 9e-263) (/ (* b_2 -2.0) a) (/ (/ (- (* 0.5 c)) (/ b_2 a)) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 9e-263) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (-(0.5 * c) / (b_2 / a)) / 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 <= 9d-263) then
        tmp = (b_2 * (-2.0d0)) / a
    else
        tmp = (-(0.5d0 * c) / (b_2 / a)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 9e-263) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (-(0.5 * c) / (b_2 / a)) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 9e-263:
		tmp = (b_2 * -2.0) / a
	else:
		tmp = (-(0.5 * c) / (b_2 / a)) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 9e-263)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = Float64(Float64(Float64(-Float64(0.5 * c)) / Float64(b_2 / a)) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 9e-263)
		tmp = (b_2 * -2.0) / a;
	else
		tmp = (-(0.5 * c) / (b_2 / a)) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 9e-263], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[((-N[(0.5 * c), $MachinePrecision]) / N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 8.9999999999999994e-263

   1. Initial program 72.9%

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

      1. +-commutative 72.9%

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

      2. unsub-neg 72.9%

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

   3. Simplified 72.9%

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

   5. Taylor expanded in b_2 around -inf 63.8%

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

      1. *-commutative 63.8%

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

   7. Simplified 63.8%

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

   if 8.9999999999999994e-263 < b_2

   1. Initial program 40.0%

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

      1. +-commutative 40.0%

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

      2. unsub-neg 40.0%

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

   3. Simplified 40.0%

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

   5. Taylor expanded in b_2 around inf 53.8%

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

      1. associate-/l* 55.8%

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

      2. *-un-lft-identity 55.8%

         \[\leadsto \frac{-0.5 \cdot \frac{\color{blue}{1 \cdot a}}{\frac{b\_2}{c}}}{a} \]

      3. div-inv 55.7%

         \[\leadsto \frac{-0.5 \cdot \frac{1 \cdot a}{\color{blue}{b\_2 \cdot \frac{1}{c}}}}{a} \]

      4. times-frac 53.7%

         \[\leadsto \frac{-0.5 \cdot \color{blue}{\left(\frac{1}{b\_2} \cdot \frac{a}{\frac{1}{c}}\right)}}{a} \]

   7. Applied egg-rr 53.7%

      \[\leadsto \frac{-0.5 \cdot \color{blue}{\left(\frac{1}{b\_2} \cdot \frac{a}{\frac{1}{c}}\right)}}{a} \]
   8. Step-by-step derivation

      1. associate-*l/ 53.7%

         \[\leadsto \frac{-0.5 \cdot \color{blue}{\frac{1 \cdot \frac{a}{\frac{1}{c}}}{b\_2}}}{a} \]

      2. *-un-lft-identity 53.7%

         \[\leadsto \frac{-0.5 \cdot \frac{\color{blue}{\frac{a}{\frac{1}{c}}}}{b\_2}}{a} \]

      3. associate-*r/ 53.7%

         \[\leadsto \frac{\color{blue}{\frac{-0.5 \cdot \frac{a}{\frac{1}{c}}}{b\_2}}}{a} \]

      4. metadata-eval 53.7%

         \[\leadsto \frac{\frac{\color{blue}{\left(-0.5\right)} \cdot \frac{a}{\frac{1}{c}}}{b\_2}}{a} \]

      5. associate-/r/ 53.8%

         \[\leadsto \frac{\frac{\left(-0.5\right) \cdot \color{blue}{\left(\frac{a}{1} \cdot c\right)}}{b\_2}}{a} \]

      6. /-rgt-identity 53.8%

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

      7. *-commutative 53.8%

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

      8. distribute-lft-neg-in 53.8%

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

      9. associate-*l* 53.8%

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

      10. distribute-neg-frac 53.8%

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

      11. associate-/l* 60.0%

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

      12. distribute-neg-frac 60.0%

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

   9. Applied egg-rr 60.0%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 9 \cdot 10^{-263}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5 \cdot c}{\frac{b\_2}{a}}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.9% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 9 \cdot 10^{-263}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(a \cdot \frac{c}{b\_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 9e-263) (/ (* b_2 -2.0) a) (* (/ -0.5 a) (* a (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 9e-263) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (-0.5 / a) * (a * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= 9d-263) then
        tmp = (b_2 * (-2.0d0)) / a
    else
        tmp = ((-0.5d0) / a) * (a * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 9e-263) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (-0.5 / a) * (a * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 9e-263:
		tmp = (b_2 * -2.0) / a
	else:
		tmp = (-0.5 / a) * (a * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 9e-263)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = Float64(Float64(-0.5 / a) * Float64(a * Float64(c / b_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 9e-263)
		tmp = (b_2 * -2.0) / a;
	else
		tmp = (-0.5 / a) * (a * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 9e-263], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 / a), $MachinePrecision] * N[(a * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 8.9999999999999994e-263

   1. Initial program 72.9%

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

      1. +-commutative 72.9%

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

      2. unsub-neg 72.9%

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

   3. Simplified 72.9%

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

   5. Taylor expanded in b_2 around -inf 63.8%

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

      1. *-commutative 63.8%

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

   7. Simplified 63.8%

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

   if 8.9999999999999994e-263 < b_2

   1. Initial program 40.0%

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

      1. +-commutative 40.0%

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

      2. unsub-neg 40.0%

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

   3. Simplified 40.0%

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

   5. Taylor expanded in b_2 around inf 53.8%

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

      1. associate-/l* 55.8%

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

      2. *-un-lft-identity 55.8%

         \[\leadsto \frac{-0.5 \cdot \frac{\color{blue}{1 \cdot a}}{\frac{b\_2}{c}}}{a} \]

      3. div-inv 55.7%

         \[\leadsto \frac{-0.5 \cdot \frac{1 \cdot a}{\color{blue}{b\_2 \cdot \frac{1}{c}}}}{a} \]

      4. times-frac 53.7%

         \[\leadsto \frac{-0.5 \cdot \color{blue}{\left(\frac{1}{b\_2} \cdot \frac{a}{\frac{1}{c}}\right)}}{a} \]

   7. Applied egg-rr 53.7%

      \[\leadsto \frac{-0.5 \cdot \color{blue}{\left(\frac{1}{b\_2} \cdot \frac{a}{\frac{1}{c}}\right)}}{a} \]
   8. Step-by-step derivation

      1. associate-/l* 53.7%

         \[\leadsto \color{blue}{\frac{-0.5}{\frac{a}{\frac{1}{b\_2} \cdot \frac{a}{\frac{1}{c}}}}} \]

      2. associate-/r/ 53.7%

         \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(\frac{1}{b\_2} \cdot \frac{a}{\frac{1}{c}}\right)} \]

      3. frac-times 55.7%

         \[\leadsto \frac{-0.5}{a} \cdot \color{blue}{\frac{1 \cdot a}{b\_2 \cdot \frac{1}{c}}} \]

      4. div-inv 55.8%

         \[\leadsto \frac{-0.5}{a} \cdot \frac{1 \cdot a}{\color{blue}{\frac{b\_2}{c}}} \]

      5. *-un-lft-identity 55.8%

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

      6. div-inv 55.7%

         \[\leadsto \frac{-0.5}{a} \cdot \color{blue}{\left(a \cdot \frac{1}{\frac{b\_2}{c}}\right)} \]

      7. clear-num 55.8%

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

   9. Applied egg-rr 55.8%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 9 \cdot 10^{-263}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(a \cdot \frac{c}{b\_2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.5% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 1.16 \cdot 10^{-262}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot \left(c \cdot \frac{a}{b\_2}\right)}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 1.16e-262) (/ (* b_2 -2.0) a) (/ (* -0.5 (* c (/ a b_2))) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1.16e-262) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (-0.5 * (c * (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 <= 1.16d-262) then
        tmp = (b_2 * (-2.0d0)) / a
    else
        tmp = ((-0.5d0) * (c * (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 <= 1.16e-262) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (-0.5 * (c * (a / b_2))) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 1.16e-262:
		tmp = (b_2 * -2.0) / a
	else:
		tmp = (-0.5 * (c * (a / b_2))) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 1.16e-262)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = Float64(Float64(-0.5 * Float64(c * Float64(a / b_2))) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 1.16e-262)
		tmp = (b_2 * -2.0) / a;
	else
		tmp = (-0.5 * (c * (a / b_2))) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 1.16e-262], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * N[(c * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 1.16000000000000001e-262

   1. Initial program 72.9%

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

      1. +-commutative 72.9%

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

      2. unsub-neg 72.9%

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

   3. Simplified 72.9%

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

   5. Taylor expanded in b_2 around -inf 63.8%

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

      1. *-commutative 63.8%

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

   7. Simplified 63.8%

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

   if 1.16000000000000001e-262 < b_2

   1. Initial program 40.0%

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

      1. +-commutative 40.0%

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

      2. unsub-neg 40.0%

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

   3. Simplified 40.0%

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

   5. Taylor expanded in b_2 around inf 53.8%

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

      1. associate-/l* 55.8%

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

      2. *-un-lft-identity 55.8%

         \[\leadsto \frac{-0.5 \cdot \frac{\color{blue}{1 \cdot a}}{\frac{b\_2}{c}}}{a} \]

      3. div-inv 55.7%

         \[\leadsto \frac{-0.5 \cdot \frac{1 \cdot a}{\color{blue}{b\_2 \cdot \frac{1}{c}}}}{a} \]

      4. times-frac 53.7%

         \[\leadsto \frac{-0.5 \cdot \color{blue}{\left(\frac{1}{b\_2} \cdot \frac{a}{\frac{1}{c}}\right)}}{a} \]

   7. Applied egg-rr 53.7%

      \[\leadsto \frac{-0.5 \cdot \color{blue}{\left(\frac{1}{b\_2} \cdot \frac{a}{\frac{1}{c}}\right)}}{a} \]

   8. Taylor expanded in b_2 around 0 53.8%

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

      1. *-commutative 53.8%

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

      2. associate-*r/ 59.2%

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

   10. Simplified 59.2%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 1.16 \cdot 10^{-262}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot \left(c \cdot \frac{a}{b\_2}\right)}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 22.9% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 7.2 \cdot 10^{+87}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 7.2e+87) (/ (- b_2) a) (* 0.5 (/ c b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 7.2e+87) {
		tmp = -b_2 / a;
	} else {
		tmp = 0.5 * (c / b_2);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= 7.2d+87) then
        tmp = -b_2 / a
    else
        tmp = 0.5d0 * (c / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 7.2e+87) {
		tmp = -b_2 / a;
	} else {
		tmp = 0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 7.2e+87:
		tmp = -b_2 / a
	else:
		tmp = 0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 7.2e+87)
		tmp = Float64(Float64(-b_2) / a);
	else
		tmp = Float64(0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 7.2e+87)
		tmp = -b_2 / a;
	else
		tmp = 0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 7.2e+87], N[((-b$95$2) / a), $MachinePrecision], N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 7.19999999999999988e87

   1. Initial program 71.4%

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

      1. +-commutative 71.4%

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

      2. unsub-neg 71.4%

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

   3. Simplified 71.4%

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

   5. Taylor expanded in b_2 around 0 47.4%

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

      1. associate-*r* 47.4%

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

      2. neg-mul-1 47.4%

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

   7. Simplified 47.4%

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

   8. Taylor expanded in a around 0 23.3%

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

      1. associate-*r/ 23.3%

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

      2. neg-mul-1 23.3%

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

   10. Simplified 23.3%

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

   if 7.19999999999999988e87 < b_2

   1. Initial program 17.0%

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

      1. +-commutative 17.0%

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

      2. unsub-neg 17.0%

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

   3. Simplified 17.0%

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

   5. Taylor expanded in b_2 around -inf 2.2%

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

   6. Taylor expanded in b_2 around 0 42.4%

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

4. Final simplification 27.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 7.2 \cdot 10^{+87}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.5% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 3 \cdot 10^{-262}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 3e-262) (/ (- b_2) a) (* c (/ -0.5 b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 3e-262) {
		tmp = -b_2 / a;
	} else {
		tmp = c * (-0.5 / b_2);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= 3d-262) then
        tmp = -b_2 / a
    else
        tmp = c * ((-0.5d0) / b_2)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 3e-262) {
		tmp = -b_2 / a;
	} else {
		tmp = c * (-0.5 / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 3e-262:
		tmp = -b_2 / a
	else:
		tmp = c * (-0.5 / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 3e-262)
		tmp = Float64(Float64(-b_2) / a);
	else
		tmp = Float64(c * Float64(-0.5 / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 3e-262)
		tmp = -b_2 / a;
	else
		tmp = c * (-0.5 / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 3e-262], N[((-b$95$2) / a), $MachinePrecision], N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 3.00000000000000018e-262

   1. Initial program 72.9%

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

      1. +-commutative 72.9%

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

      2. unsub-neg 72.9%

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

   3. Simplified 72.9%

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

   5. Taylor expanded in b_2 around 0 46.0%

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

      1. associate-*r* 46.0%

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

      2. neg-mul-1 46.0%

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

   7. Simplified 46.0%

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

   8. Taylor expanded in a around 0 30.5%

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

      1. associate-*r/ 30.5%

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

      2. neg-mul-1 30.5%

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

   10. Simplified 30.5%

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

   if 3.00000000000000018e-262 < b_2

   1. Initial program 40.0%

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

      1. +-commutative 40.0%

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

      2. unsub-neg 40.0%

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

   3. Simplified 40.0%

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

   5. Taylor expanded in b_2 around inf 51.8%

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

      1. associate-*r/ 51.8%

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

      2. associate-/l* 51.6%

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

   7. Simplified 51.6%

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

      1. associate-/r/ 51.7%

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

   9. Applied egg-rr 51.7%

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

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 3 \cdot 10^{-262}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.6% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 9 \cdot 10^{-263}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 9e-263) (/ (- b_2) a) (/ (* c -0.5) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 9e-263) {
		tmp = -b_2 / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= 9d-263) then
        tmp = -b_2 / a
    else
        tmp = (c * (-0.5d0)) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 9e-263) {
		tmp = -b_2 / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 9e-263:
		tmp = -b_2 / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 9e-263)
		tmp = Float64(Float64(-b_2) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 9e-263)
		tmp = -b_2 / a;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 9e-263], N[((-b$95$2) / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 8.9999999999999994e-263

   1. Initial program 72.9%

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

      1. +-commutative 72.9%

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

      2. unsub-neg 72.9%

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

   3. Simplified 72.9%

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

   5. Taylor expanded in b_2 around 0 46.0%

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

      1. associate-*r* 46.0%

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

      2. neg-mul-1 46.0%

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

   7. Simplified 46.0%

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

   8. Taylor expanded in a around 0 30.5%

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

      1. associate-*r/ 30.5%

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

      2. neg-mul-1 30.5%

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

   10. Simplified 30.5%

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

   if 8.9999999999999994e-263 < b_2

   1. Initial program 40.0%

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

      1. +-commutative 40.0%

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

      2. unsub-neg 40.0%

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

   3. Simplified 40.0%

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

   5. Taylor expanded in b_2 around inf 51.8%

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

      1. associate-*r/ 51.8%

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

   7. Simplified 51.8%

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

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 9 \cdot 10^{-263}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.2% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 9 \cdot 10^{-263}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 9e-263) (/ (* b_2 -2.0) a) (/ (* c -0.5) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 9e-263) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= 9d-263) then
        tmp = (b_2 * (-2.0d0)) / a
    else
        tmp = (c * (-0.5d0)) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 9e-263) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 9e-263:
		tmp = (b_2 * -2.0) / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 9e-263)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 9e-263)
		tmp = (b_2 * -2.0) / a;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 9e-263], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 8.9999999999999994e-263

   1. Initial program 72.9%

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

      1. +-commutative 72.9%

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

      2. unsub-neg 72.9%

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

   3. Simplified 72.9%

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

   5. Taylor expanded in b_2 around -inf 63.8%

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

      1. *-commutative 63.8%

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

   7. Simplified 63.8%

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

   if 8.9999999999999994e-263 < b_2

   1. Initial program 40.0%

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

      1. +-commutative 40.0%

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

      2. unsub-neg 40.0%

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

   3. Simplified 40.0%

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

   5. Taylor expanded in b_2 around inf 51.8%

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

      1. associate-*r/ 51.8%

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

   7. Simplified 51.8%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 9 \cdot 10^{-263}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 15.0% accurate, 28.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.6%

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

   1. +-commutative 58.6%

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

   2. unsub-neg 58.6%

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

3. Simplified 58.6%

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

5. Taylor expanded in b_2 around 0 37.0%

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

   1. associate-*r* 37.0%

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

   2. neg-mul-1 37.0%

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

7. Simplified 37.0%

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

8. Taylor expanded in a around 0 18.4%

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

   1. associate-*r/ 18.4%

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

   2. neg-mul-1 18.4%

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

10. Simplified 18.4%

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

11. Final simplification 18.4%

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

Alternative 12: 2.5% accurate, 37.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.6%

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

   1. +-commutative 58.6%

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

   2. unsub-neg 58.6%

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

3. Simplified 58.6%

   \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 55.3%

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

   2. add-cube-cbrt 53.7%

      \[\leadsto \frac{\sqrt{\sqrt{b\_2 \cdot b\_2 - a \cdot c}} \cdot \sqrt{\sqrt{b\_2 \cdot b\_2 - a \cdot c}} - \color{blue}{\left(\sqrt[3]{b\_2} \cdot \sqrt[3]{b\_2}\right) \cdot \sqrt[3]{b\_2}}}{a} \]

   3. prod-diff 52.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}, \sqrt{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}, -\sqrt[3]{b\_2} \cdot \left(\sqrt[3]{b\_2} \cdot \sqrt[3]{b\_2}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{b\_2}, \sqrt[3]{b\_2} \cdot \sqrt[3]{b\_2}, \sqrt[3]{b\_2} \cdot \left(\sqrt[3]{b\_2} \cdot \sqrt[3]{b\_2}\right)\right)}}{a} \]

6. Applied egg-rr 25.0%

   \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c, {b\_2}^{2}\right)} - b\_2\right) + \mathsf{fma}\left(-\sqrt[3]{b\_2}, {\left(\sqrt[3]{b\_2}\right)}^{2}, b\_2\right)}}{a} \]
7. Step-by-step derivation

   1. fma-def 24.9%

      \[\leadsto \frac{\left(\sqrt{\color{blue}{a \cdot c + {b\_2}^{2}}} - b\_2\right) + \mathsf{fma}\left(-\sqrt[3]{b\_2}, {\left(\sqrt[3]{b\_2}\right)}^{2}, b\_2\right)}{a} \]

   2. unpow1 24.9%

      \[\leadsto \frac{\left(\sqrt{\color{blue}{{\left(a \cdot c\right)}^{1}} + {b\_2}^{2}} - b\_2\right) + \mathsf{fma}\left(-\sqrt[3]{b\_2}, {\left(\sqrt[3]{b\_2}\right)}^{2}, b\_2\right)}{a} \]

   3. sqr-pow 15.9%

      \[\leadsto \frac{\left(\sqrt{\color{blue}{{\left(a \cdot c\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(a \cdot c\right)}^{\left(\frac{1}{2}\right)}} + {b\_2}^{2}} - b\_2\right) + \mathsf{fma}\left(-\sqrt[3]{b\_2}, {\left(\sqrt[3]{b\_2}\right)}^{2}, b\_2\right)}{a} \]

   4. unpow2 15.9%

      \[\leadsto \frac{\left(\sqrt{{\left(a \cdot c\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(a \cdot c\right)}^{\left(\frac{1}{2}\right)} + \color{blue}{b\_2 \cdot b\_2}} - b\_2\right) + \mathsf{fma}\left(-\sqrt[3]{b\_2}, {\left(\sqrt[3]{b\_2}\right)}^{2}, b\_2\right)}{a} \]

   5. hypot-def 19.2%

      \[\leadsto \frac{\left(\color{blue}{\mathsf{hypot}\left({\left(a \cdot c\right)}^{\left(\frac{1}{2}\right)}, b\_2\right)} - b\_2\right) + \mathsf{fma}\left(-\sqrt[3]{b\_2}, {\left(\sqrt[3]{b\_2}\right)}^{2}, b\_2\right)}{a} \]

   6. metadata-eval 19.2%

      \[\leadsto \frac{\left(\mathsf{hypot}\left({\left(a \cdot c\right)}^{\color{blue}{0.5}}, b\_2\right) - b\_2\right) + \mathsf{fma}\left(-\sqrt[3]{b\_2}, {\left(\sqrt[3]{b\_2}\right)}^{2}, b\_2\right)}{a} \]

   7. unpow1/2 18.9%

      \[\leadsto \frac{\left(\mathsf{hypot}\left(\color{blue}{\sqrt{a \cdot c}}, b\_2\right) - b\_2\right) + \mathsf{fma}\left(-\sqrt[3]{b\_2}, {\left(\sqrt[3]{b\_2}\right)}^{2}, b\_2\right)}{a} \]

   8. fma-udef 21.2%

      \[\leadsto \frac{\left(\mathsf{hypot}\left(\sqrt{a \cdot c}, b\_2\right) - b\_2\right) + \color{blue}{\left(\left(-\sqrt[3]{b\_2}\right) \cdot {\left(\sqrt[3]{b\_2}\right)}^{2} + b\_2\right)}}{a} \]

   9. +-commutative 21.2%

      \[\leadsto \frac{\left(\mathsf{hypot}\left(\sqrt{a \cdot c}, b\_2\right) - b\_2\right) + \color{blue}{\left(b\_2 + \left(-\sqrt[3]{b\_2}\right) \cdot {\left(\sqrt[3]{b\_2}\right)}^{2}\right)}}{a} \]

   10. distribute-lft-neg-out 21.2%

       \[\leadsto \frac{\left(\mathsf{hypot}\left(\sqrt{a \cdot c}, b\_2\right) - b\_2\right) + \left(b\_2 + \color{blue}{\left(-\sqrt[3]{b\_2} \cdot {\left(\sqrt[3]{b\_2}\right)}^{2}\right)}\right)}{a} \]

   11. unpow2 21.2%

       \[\leadsto \frac{\left(\mathsf{hypot}\left(\sqrt{a \cdot c}, b\_2\right) - b\_2\right) + \left(b\_2 + \left(-\sqrt[3]{b\_2} \cdot \color{blue}{\left(\sqrt[3]{b\_2} \cdot \sqrt[3]{b\_2}\right)}\right)\right)}{a} \]

   12. cube-unmult 21.7%

       \[\leadsto \frac{\left(\mathsf{hypot}\left(\sqrt{a \cdot c}, b\_2\right) - b\_2\right) + \left(b\_2 + \left(-\color{blue}{{\left(\sqrt[3]{b\_2}\right)}^{3}}\right)\right)}{a} \]

8. Simplified 21.7%

   \[\leadsto \frac{\color{blue}{\left(\mathsf{hypot}\left(\sqrt{a \cdot c}, b\_2\right) - b\_2\right) + \left(b\_2 + \left(-{\left(\sqrt[3]{b\_2}\right)}^{3}\right)\right)}}{a} \]

9. Taylor expanded in b_2 around inf 2.4%

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

10. Final simplification 2.4%

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

Developer target: 86.8% accurate, 0.1× speedup

Math

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

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) (/ (- t_1 b_2) a) (/ (- c) (+ b_2 t_1)))))

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 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	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 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	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 = (t_1 - b_2) / a
	else:
		tmp_1 = -c / (b_2 + t_1)
	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(Float64(t_1 - b_2) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(b_2 + t_1));
	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 = (t_1 - b_2) / a;
	else
		tmp_2 = -c / (b_2 + t_1);
	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[(N[(t$95$1 - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(b$95$2 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2024031 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  :herbie-expected 10

  :herbie-target
  (if (< b_2 0.0) (/ (- (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) a) (/ (- c) (+ 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))))))))

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