quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.2% → 82.4%

Time: 18.9s

Alternatives: 11

Speedup: 11.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.2% 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: 82.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.9 \cdot 10^{-80}:\\ \;\;\;\;\frac{\left(0.5 \cdot \left(a \cdot \frac{c}{b\_2}\right) - b\_2\right) - b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 6.6 \cdot 10^{-116}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)}{a} - \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.9e-80)
   (/ (- (- (* 0.5 (* a (/ c b_2))) b_2) b_2) a)
   (if (<= b_2 6.6e-116)
     (- (/ (hypot b_2 (sqrt (* a (- c)))) a) (/ b_2 a))
     (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.9e-80) {
		tmp = (((0.5 * (a * (c / b_2))) - b_2) - b_2) / a;
	} else if (b_2 <= 6.6e-116) {
		tmp = (hypot(b_2, sqrt((a * -c))) / a) - (b_2 / a);
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.9e-80) {
		tmp = (((0.5 * (a * (c / b_2))) - b_2) - b_2) / a;
	} else if (b_2 <= 6.6e-116) {
		tmp = (Math.hypot(b_2, Math.sqrt((a * -c))) / a) - (b_2 / a);
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.9e-80:
		tmp = (((0.5 * (a * (c / b_2))) - b_2) - b_2) / a
	elif b_2 <= 6.6e-116:
		tmp = (math.hypot(b_2, math.sqrt((a * -c))) / a) - (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 <= -2.9e-80)
		tmp = Float64(Float64(Float64(Float64(0.5 * Float64(a * Float64(c / b_2))) - b_2) - b_2) / a);
	elseif (b_2 <= 6.6e-116)
		tmp = Float64(Float64(hypot(b_2, sqrt(Float64(a * Float64(-c)))) / a) - 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 <= -2.9e-80)
		tmp = (((0.5 * (a * (c / b_2))) - b_2) - b_2) / a;
	elseif (b_2 <= 6.6e-116)
		tmp = (hypot(b_2, sqrt((a * -c))) / a) - (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, -2.9e-80], N[(N[(N[(N[(0.5 * N[(a * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b$95$2), $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 6.6e-116], N[(N[(N[Sqrt[b$95$2 ^ 2 + N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision] / a), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.89999999999999998e-80

   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. Step-by-step derivation

      1. pow1/2 71.4%

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

      2. pow-to-exp 69.6%

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

      3. pow2 69.6%

         \[\leadsto \frac{e^{\log \left(\color{blue}{{b\_2}^{2}} - a \cdot c\right) \cdot 0.5} - b\_2}{a} \]

   6. Applied egg-rr 69.6%

      \[\leadsto \frac{\color{blue}{e^{\log \left({b\_2}^{2} - a \cdot c\right) \cdot 0.5}} - b\_2}{a} \]

   7. Taylor expanded in b_2 around -inf 93.3%

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

      1. neg-mul-1 93.3%

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

      2. +-commutative 93.3%

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

      3. unsub-neg 93.3%

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

      4. associate-/l* 96.7%

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

   9. Simplified 96.7%

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

   if -2.89999999999999998e-80 < b_2 < 6.60000000000000002e-116

   1. Initial program 69.6%

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

      1. +-commutative 69.6%

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

      2. unsub-neg 69.6%

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

   3. Simplified 69.6%

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

      1. div-sub 69.6%

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

      2. sub-neg 69.6%

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

      3. add-sqr-sqrt 69.6%

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

      4. hypot-define 74.8%

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

      5. *-commutative 74.8%

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

      6. distribute-rgt-neg-in 74.8%

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

   6. Applied egg-rr 74.8%

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

   if 6.60000000000000002e-116 < b_2

   1. Initial program 27.2%

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

      1. +-commutative 27.2%

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

      2. unsub-neg 27.2%

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

   3. Simplified 27.2%

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

   5. Taylor expanded in b_2 around inf 84.1%

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

      1. *-commutative 84.1%

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

      2. associate-*l/ 84.1%

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

   7. Simplified 84.1%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.9 \cdot 10^{-80}:\\ \;\;\;\;\frac{\left(0.5 \cdot \left(a \cdot \frac{c}{b\_2}\right) - b\_2\right) - b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 6.6 \cdot 10^{-116}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right)}{a} - \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\left(0.5 \cdot \left(a \cdot \frac{c}{b\_2}\right) - b\_2\right) - b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.65 \cdot 10^{-115}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right) - b\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.2e-85)
   (/ (- (- (* 0.5 (* a (/ c b_2))) b_2) b_2) a)
   (if (<= b_2 1.65e-115)
     (* (/ 1.0 a) (- (hypot b_2 (sqrt (* a (- c)))) b_2))
     (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.2e-85) {
		tmp = (((0.5 * (a * (c / b_2))) - b_2) - b_2) / a;
	} else if (b_2 <= 1.65e-115) {
		tmp = (1.0 / a) * (hypot(b_2, sqrt((a * -c))) - b_2);
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.2e-85) {
		tmp = (((0.5 * (a * (c / b_2))) - b_2) - b_2) / a;
	} else if (b_2 <= 1.65e-115) {
		tmp = (1.0 / a) * (Math.hypot(b_2, Math.sqrt((a * -c))) - b_2);
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.2e-85:
		tmp = (((0.5 * (a * (c / b_2))) - b_2) - b_2) / a
	elif b_2 <= 1.65e-115:
		tmp = (1.0 / a) * (math.hypot(b_2, math.sqrt((a * -c))) - b_2)
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.2e-85)
		tmp = Float64(Float64(Float64(Float64(0.5 * Float64(a * Float64(c / b_2))) - b_2) - b_2) / a);
	elseif (b_2 <= 1.65e-115)
		tmp = Float64(Float64(1.0 / a) * Float64(hypot(b_2, sqrt(Float64(a * Float64(-c)))) - b_2));
	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 <= -2.2e-85)
		tmp = (((0.5 * (a * (c / b_2))) - b_2) - b_2) / a;
	elseif (b_2 <= 1.65e-115)
		tmp = (1.0 / a) * (hypot(b_2, sqrt((a * -c))) - b_2);
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.2e-85], N[(N[(N[(N[(0.5 * N[(a * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b$95$2), $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.65e-115], N[(N[(1.0 / a), $MachinePrecision] * N[(N[Sqrt[b$95$2 ^ 2 + N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision] - b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.2e-85

   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. Step-by-step derivation

      1. pow1/2 71.4%

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

      2. pow-to-exp 69.6%

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

      3. pow2 69.6%

         \[\leadsto \frac{e^{\log \left(\color{blue}{{b\_2}^{2}} - a \cdot c\right) \cdot 0.5} - b\_2}{a} \]

   6. Applied egg-rr 69.6%

      \[\leadsto \frac{\color{blue}{e^{\log \left({b\_2}^{2} - a \cdot c\right) \cdot 0.5}} - b\_2}{a} \]

   7. Taylor expanded in b_2 around -inf 93.3%

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

      1. neg-mul-1 93.3%

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

      2. +-commutative 93.3%

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

      3. unsub-neg 93.3%

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

      4. associate-/l* 96.7%

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

   9. Simplified 96.7%

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

   if -2.2e-85 < b_2 < 1.64999999999999995e-115

   1. Initial program 69.6%

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

      1. +-commutative 69.6%

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

      2. unsub-neg 69.6%

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

   3. Simplified 69.6%

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

      1. clear-num 69.5%

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

      2. associate-/r/ 69.3%

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

      3. sub-neg 69.3%

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

      4. add-sqr-sqrt 69.3%

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

      5. hypot-define 74.6%

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

      6. *-commutative 74.6%

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

      7. distribute-rgt-neg-in 74.6%

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

   6. Applied egg-rr 74.6%

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

   if 1.64999999999999995e-115 < b_2

   1. Initial program 27.2%

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

      1. +-commutative 27.2%

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

      2. unsub-neg 27.2%

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

   3. Simplified 27.2%

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

   5. Taylor expanded in b_2 around inf 84.1%

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

      1. *-commutative 84.1%

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

      2. associate-*l/ 84.1%

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

   7. Simplified 84.1%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\left(0.5 \cdot \left(a \cdot \frac{c}{b\_2}\right) - b\_2\right) - b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.65 \cdot 10^{-115}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\mathsf{hypot}\left(b\_2, \sqrt{a \cdot \left(-c\right)}\right) - b\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.1 \cdot 10^{+164}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{elif}\;b\_2 \leq 1.35 \cdot 10^{-115}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.1e+164)
   (* (/ b_2 a) -2.0)
   (if (<= b_2 1.35e-115)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.1e+164) {
		tmp = (b_2 / a) * -2.0;
	} else if (b_2 <= 1.35e-115) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - 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 <= (-2.1d+164)) then
        tmp = (b_2 / a) * (-2.0d0)
    else if (b_2 <= 1.35d-115) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) - 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 <= -2.1e+164) {
		tmp = (b_2 / a) * -2.0;
	} else if (b_2 <= 1.35e-115) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.1e+164:
		tmp = (b_2 / a) * -2.0
	elif b_2 <= 1.35e-115:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - 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 <= -2.1e+164)
		tmp = Float64(Float64(b_2 / a) * -2.0);
	elseif (b_2 <= 1.35e-115)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - 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 <= -2.1e+164)
		tmp = (b_2 / a) * -2.0;
	elseif (b_2 <= 1.35e-115)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - 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, -2.1e+164], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[b$95$2, 1.35e-115], 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[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.0999999999999999e164

   1. Initial program 48.1%

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

      1. +-commutative 48.1%

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

      2. unsub-neg 48.1%

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

   3. Simplified 48.1%

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

   5. Taylor expanded in b_2 around -inf 100.0%

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

   if -2.0999999999999999e164 < b_2 < 1.35e-115

   1. Initial program 80.7%

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

      1. +-commutative 80.7%

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

      2. unsub-neg 80.7%

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

   3. Simplified 80.7%

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

   if 1.35e-115 < b_2

   1. Initial program 27.2%

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

      1. +-commutative 27.2%

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

      2. unsub-neg 27.2%

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

   3. Simplified 27.2%

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

   5. Taylor expanded in b_2 around inf 84.1%

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

      1. *-commutative 84.1%

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

      2. associate-*l/ 84.1%

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

   7. Simplified 84.1%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.1 \cdot 10^{+164}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{elif}\;b\_2 \leq 1.35 \cdot 10^{-115}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.15 \cdot 10^{-143}:\\ \;\;\;\;\frac{\left(0.5 \cdot \left(a \cdot \frac{c}{b\_2}\right) - b\_2\right) - b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.9 \cdot 10^{-115}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.15e-143)
   (/ (- (- (* 0.5 (* a (/ c b_2))) b_2) b_2) a)
   (if (<= b_2 1.9e-115) (/ (- (sqrt (* a (- c))) b_2) a) (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.15e-143) {
		tmp = (((0.5 * (a * (c / b_2))) - b_2) - b_2) / a;
	} else if (b_2 <= 1.9e-115) {
		tmp = (sqrt((a * -c)) - 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 <= (-1.15d-143)) then
        tmp = (((0.5d0 * (a * (c / b_2))) - b_2) - b_2) / a
    else if (b_2 <= 1.9d-115) then
        tmp = (sqrt((a * -c)) - 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 <= -1.15e-143) {
		tmp = (((0.5 * (a * (c / b_2))) - b_2) - b_2) / a;
	} else if (b_2 <= 1.9e-115) {
		tmp = (Math.sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.15e-143:
		tmp = (((0.5 * (a * (c / b_2))) - b_2) - b_2) / a
	elif b_2 <= 1.9e-115:
		tmp = (math.sqrt((a * -c)) - 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 <= -1.15e-143)
		tmp = Float64(Float64(Float64(Float64(0.5 * Float64(a * Float64(c / b_2))) - b_2) - b_2) / a);
	elseif (b_2 <= 1.9e-115)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-c))) - 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 <= -1.15e-143)
		tmp = (((0.5 * (a * (c / b_2))) - b_2) - b_2) / a;
	elseif (b_2 <= 1.9e-115)
		tmp = (sqrt((a * -c)) - 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, -1.15e-143], N[(N[(N[(N[(0.5 * N[(a * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b$95$2), $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.9e-115], N[(N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.15000000000000006e-143

   1. Initial program 73.3%

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

      1. +-commutative 73.3%

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

      2. unsub-neg 73.3%

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

   3. Simplified 73.3%

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

      1. pow1/2 73.3%

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

      2. pow-to-exp 71.2%

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

      3. pow2 71.2%

         \[\leadsto \frac{e^{\log \left(\color{blue}{{b\_2}^{2}} - a \cdot c\right) \cdot 0.5} - b\_2}{a} \]

   6. Applied egg-rr 71.2%

      \[\leadsto \frac{\color{blue}{e^{\log \left({b\_2}^{2} - a \cdot c\right) \cdot 0.5}} - b\_2}{a} \]

   7. Taylor expanded in b_2 around -inf 89.1%

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

      1. neg-mul-1 89.1%

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

      2. +-commutative 89.1%

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

      3. unsub-neg 89.1%

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

      4. associate-/l* 92.1%

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

   9. Simplified 92.1%

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

   if -1.15000000000000006e-143 < b_2 < 1.89999999999999996e-115

   1. Initial program 65.6%

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

      1. +-commutative 65.6%

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

      2. unsub-neg 65.6%

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

   3. Simplified 65.6%

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

   5. Taylor expanded in b_2 around 0 64.1%

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

      1. associate-*r* 64.1%

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

      2. neg-mul-1 64.1%

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

      3. *-commutative 64.1%

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

   7. Simplified 64.1%

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

   if 1.89999999999999996e-115 < b_2

   1. Initial program 27.2%

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

      1. +-commutative 27.2%

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

      2. unsub-neg 27.2%

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

   3. Simplified 27.2%

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

   5. Taylor expanded in b_2 around inf 84.1%

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

      1. *-commutative 84.1%

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

      2. associate-*l/ 84.1%

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

   7. Simplified 84.1%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.15 \cdot 10^{-143}:\\ \;\;\;\;\frac{\left(0.5 \cdot \left(a \cdot \frac{c}{b\_2}\right) - b\_2\right) - b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.9 \cdot 10^{-115}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.5% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2 + 0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b\_2}{c} + 0.5 \cdot \frac{a}{b\_2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-311)
   (+ (* (/ b_2 a) -2.0) (* 0.5 (/ c b_2)))
   (/ 1.0 (+ (* -2.0 (/ b_2 c)) (* 0.5 (/ a b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-311) {
		tmp = ((b_2 / a) * -2.0) + (0.5 * (c / b_2));
	} else {
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / b_2)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d-311)) then
        tmp = ((b_2 / a) * (-2.0d0)) + (0.5d0 * (c / b_2))
    else
        tmp = 1.0d0 / (((-2.0d0) * (b_2 / c)) + (0.5d0 * (a / b_2)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-311) {
		tmp = ((b_2 / a) * -2.0) + (0.5 * (c / b_2));
	} else {
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / b_2)));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-311:
		tmp = ((b_2 / a) * -2.0) + (0.5 * (c / b_2))
	else:
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / b_2)))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-311)
		tmp = Float64(Float64(Float64(b_2 / a) * -2.0) + Float64(0.5 * Float64(c / b_2)));
	else
		tmp = Float64(1.0 / Float64(Float64(-2.0 * Float64(b_2 / c)) + Float64(0.5 * Float64(a / b_2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-311)
		tmp = ((b_2 / a) * -2.0) + (0.5 * (c / b_2));
	else
		tmp = 1.0 / ((-2.0 * (b_2 / c)) + (0.5 * (a / b_2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-311], N[(N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -5.00000000000023e-311

   1. Initial program 72.0%

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

      1. +-commutative 72.0%

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

      2. unsub-neg 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in b_2 around -inf 76.4%

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

   if -5.00000000000023e-311 < b_2

   1. Initial program 33.8%

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

      1. +-commutative 33.8%

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

      2. unsub-neg 33.8%

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

   3. Simplified 33.8%

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

      1. add-sqr-sqrt 30.4%

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

      2. pow2 30.4%

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

      3. pow1/2 30.4%

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

      4. sqrt-pow1 30.4%

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

      5. pow2 30.4%

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

      6. metadata-eval 30.4%

         \[\leadsto \frac{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2} - b\_2}{a} \]

   6. Applied egg-rr 30.4%

      \[\leadsto \frac{\color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}} - b\_2}{a} \]
   7. Step-by-step derivation

      1. clear-num 30.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2} - b\_2}}} \]

      2. pow-pow 33.8%

         \[\leadsto \frac{1}{\frac{a}{\color{blue}{{\left({b\_2}^{2} - a \cdot c\right)}^{\left(0.25 \cdot 2\right)}} - b\_2}} \]

      3. metadata-eval 33.8%

         \[\leadsto \frac{1}{\frac{a}{{\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.5}} - b\_2}} \]

      4. exp-to-pow 25.0%

         \[\leadsto \frac{1}{\frac{a}{\color{blue}{e^{\log \left({b\_2}^{2} - a \cdot c\right) \cdot 0.5}} - b\_2}} \]

      5. inv-pow 25.0%

         \[\leadsto \color{blue}{{\left(\frac{a}{e^{\log \left({b\_2}^{2} - a \cdot c\right) \cdot 0.5} - b\_2}\right)}^{-1}} \]

      6. exp-to-pow 33.8%

         \[\leadsto {\left(\frac{a}{\color{blue}{{\left({b\_2}^{2} - a \cdot c\right)}^{0.5}} - b\_2}\right)}^{-1} \]

      7. metadata-eval 33.8%

         \[\leadsto {\left(\frac{a}{{\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{\left(0.25 \cdot 2\right)}} - b\_2}\right)}^{-1} \]

      8. pow-pow 30.4%

         \[\leadsto {\left(\frac{a}{\color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}} - b\_2}\right)}^{-1} \]

      9. pow-pow 33.8%

         \[\leadsto {\left(\frac{a}{\color{blue}{{\left({b\_2}^{2} - a \cdot c\right)}^{\left(0.25 \cdot 2\right)}} - b\_2}\right)}^{-1} \]

      10. metadata-eval 33.8%

          \[\leadsto {\left(\frac{a}{{\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.5}} - b\_2}\right)}^{-1} \]

      11. pow1/2 33.8%

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

   8. Applied egg-rr 33.8%

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

      1. unpow-1 33.8%

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

   10. Simplified 33.8%

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

       1. add-cube-cbrt 33.4%

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

       2. pow3 33.4%

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

   12. Applied egg-rr 33.4%

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

   13. Taylor expanded in a around 0 72.4%

       \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b\_2}{c} + 0.5 \cdot \frac{a}{b\_2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2 + 0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b\_2}{c} + 0.5 \cdot \frac{a}{b\_2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.7% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2 + 0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-311)
   (+ (* (/ b_2 a) -2.0) (* 0.5 (/ c b_2)))
   (/ (* c -0.5) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-311) {
		tmp = ((b_2 / a) * -2.0) + (0.5 * (c / b_2));
	} 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 <= (-5d-311)) then
        tmp = ((b_2 / a) * (-2.0d0)) + (0.5d0 * (c / b_2))
    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 <= -5e-311) {
		tmp = ((b_2 / a) * -2.0) + (0.5 * (c / b_2));
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-311:
		tmp = ((b_2 / a) * -2.0) + (0.5 * (c / b_2))
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-311)
		tmp = Float64(Float64(Float64(b_2 / a) * -2.0) + Float64(0.5 * Float64(c / b_2)));
	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 <= -5e-311)
		tmp = ((b_2 / a) * -2.0) + (0.5 * (c / b_2));
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-311], N[(N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -5.00000000000023e-311

   1. Initial program 72.0%

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

      1. +-commutative 72.0%

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

      2. unsub-neg 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in b_2 around -inf 76.4%

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

   if -5.00000000000023e-311 < b_2

   1. Initial program 33.8%

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

      1. +-commutative 33.8%

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

      2. unsub-neg 33.8%

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

   3. Simplified 33.8%

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

   5. Taylor expanded in b_2 around inf 72.2%

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

      1. *-commutative 72.2%

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

      2. associate-*l/ 72.2%

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

   7. Simplified 72.2%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2 + 0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.1% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-311) (* (/ b_2 a) -2.0) (/ 0.0 a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-311) {
		tmp = (b_2 / a) * -2.0;
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d-311)) then
        tmp = (b_2 / a) * (-2.0d0)
    else
        tmp = 0.0d0 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-311) {
		tmp = (b_2 / a) * -2.0;
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-311:
		tmp = (b_2 / a) * -2.0
	else:
		tmp = 0.0 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-311)
		tmp = Float64(Float64(b_2 / a) * -2.0);
	else
		tmp = Float64(0.0 / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-311)
		tmp = (b_2 / a) * -2.0;
	else
		tmp = 0.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-311], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision], N[(0.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -5.00000000000023e-311

   1. Initial program 72.0%

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

      1. +-commutative 72.0%

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

      2. unsub-neg 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in b_2 around -inf 76.2%

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

   if -5.00000000000023e-311 < b_2

   1. Initial program 33.8%

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

      1. +-commutative 33.8%

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

      2. unsub-neg 33.8%

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

   3. Simplified 33.8%

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

      1. add-sqr-sqrt 30.4%

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

      2. pow2 30.4%

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

      3. pow1/2 30.4%

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

      4. sqrt-pow1 30.4%

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

      5. pow2 30.4%

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

      6. metadata-eval 30.4%

         \[\leadsto \frac{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2} - b\_2}{a} \]

   6. Applied egg-rr 30.4%

      \[\leadsto \frac{\color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}} - b\_2}{a} \]

   7. Taylor expanded in b_2 around inf 20.2%

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

      1. neg-mul-1 20.2%

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

      2. sub-neg 20.2%

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

      3. +-inverses 20.2%

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

   9. Simplified 20.2%

      \[\leadsto \frac{\color{blue}{0}}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.4% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-311) (* (/ b_2 a) -2.0) (* c (/ -0.5 b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-311) {
		tmp = (b_2 / a) * -2.0;
	} 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 <= (-5d-311)) then
        tmp = (b_2 / a) * (-2.0d0)
    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 <= -5e-311) {
		tmp = (b_2 / a) * -2.0;
	} else {
		tmp = c * (-0.5 / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-311:
		tmp = (b_2 / a) * -2.0
	else:
		tmp = c * (-0.5 / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-311)
		tmp = Float64(Float64(b_2 / a) * -2.0);
	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 <= -5e-311)
		tmp = (b_2 / a) * -2.0;
	else
		tmp = c * (-0.5 / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-311], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision], N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -5.00000000000023e-311

   1. Initial program 72.0%

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

      1. +-commutative 72.0%

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

      2. unsub-neg 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in b_2 around -inf 76.2%

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

   if -5.00000000000023e-311 < b_2

   1. Initial program 33.8%

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

      1. +-commutative 33.8%

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

      2. unsub-neg 33.8%

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

   3. Simplified 33.8%

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

      1. add-sqr-sqrt 30.4%

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

      2. pow2 30.4%

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

      3. pow1/2 30.4%

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

      4. sqrt-pow1 30.4%

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

      5. pow2 30.4%

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

      6. metadata-eval 30.4%

         \[\leadsto \frac{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2} - b\_2}{a} \]

   6. Applied egg-rr 30.4%

      \[\leadsto \frac{\color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}} - b\_2}{a} \]

   7. Taylor expanded in a around 0 72.2%

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

      1. associate-*r/ 72.2%

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

      2. *-commutative 72.2%

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

      3. associate-/l* 72.0%

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

   9. Simplified 72.0%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.5% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-311) (* (/ b_2 a) -2.0) (/ (* c -0.5) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-311) {
		tmp = (b_2 / a) * -2.0;
	} 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 <= (-5d-311)) then
        tmp = (b_2 / a) * (-2.0d0)
    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 <= -5e-311) {
		tmp = (b_2 / a) * -2.0;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-311:
		tmp = (b_2 / a) * -2.0
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-311)
		tmp = Float64(Float64(b_2 / a) * -2.0);
	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 <= -5e-311)
		tmp = (b_2 / a) * -2.0;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-311], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -5.00000000000023e-311

   1. Initial program 72.0%

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

      1. +-commutative 72.0%

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

      2. unsub-neg 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in b_2 around -inf 76.2%

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

   if -5.00000000000023e-311 < b_2

   1. Initial program 33.8%

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

      1. +-commutative 33.8%

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

      2. unsub-neg 33.8%

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

   3. Simplified 33.8%

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

   5. Taylor expanded in b_2 around inf 72.2%

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

      1. *-commutative 72.2%

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

      2. associate-*l/ 72.2%

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

   7. Simplified 72.2%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 24.2% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-311) (/ (- b_2) a) (/ 0.0 a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-311) {
		tmp = -b_2 / a;
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d-311)) then
        tmp = -b_2 / a
    else
        tmp = 0.0d0 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-311) {
		tmp = -b_2 / a;
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-311:
		tmp = -b_2 / a
	else:
		tmp = 0.0 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-311)
		tmp = Float64(Float64(-b_2) / a);
	else
		tmp = Float64(0.0 / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-311)
		tmp = -b_2 / a;
	else
		tmp = 0.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-311], N[((-b$95$2) / a), $MachinePrecision], N[(0.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -5.00000000000023e-311

   1. Initial program 72.0%

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

      1. +-commutative 72.0%

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

      2. unsub-neg 72.0%

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

   3. Simplified 72.0%

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

      1. add-sqr-sqrt 71.8%

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

      2. pow2 71.8%

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

      3. pow1/2 71.8%

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

      4. sqrt-pow1 71.8%

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

      5. pow2 71.8%

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

      6. metadata-eval 71.8%

         \[\leadsto \frac{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2} - b\_2}{a} \]

   6. Applied egg-rr 71.8%

      \[\leadsto \frac{\color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}} - b\_2}{a} \]

   7. Taylor expanded in b_2 around inf 36.7%

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

      1. neg-mul-1 36.7%

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

   9. Simplified 36.7%

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

   if -5.00000000000023e-311 < b_2

   1. Initial program 33.8%

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

      1. +-commutative 33.8%

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

      2. unsub-neg 33.8%

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

   3. Simplified 33.8%

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

      1. add-sqr-sqrt 30.4%

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

      2. pow2 30.4%

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

      3. pow1/2 30.4%

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

      4. sqrt-pow1 30.4%

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

      5. pow2 30.4%

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

      6. metadata-eval 30.4%

         \[\leadsto \frac{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2} - b\_2}{a} \]

   6. Applied egg-rr 30.4%

      \[\leadsto \frac{\color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}} - b\_2}{a} \]

   7. Taylor expanded in b_2 around inf 20.2%

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

      1. neg-mul-1 20.2%

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

      2. sub-neg 20.2%

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

      3. +-inverses 20.2%

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

   9. Simplified 20.2%

      \[\leadsto \frac{\color{blue}{0}}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 11.3% accurate, 37.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.5%

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

   1. +-commutative 52.5%

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

   2. unsub-neg 52.5%

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

3. Simplified 52.5%

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

   1. add-sqr-sqrt 50.6%

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

   2. pow2 50.6%

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

   3. pow1/2 50.6%

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

   4. sqrt-pow1 50.6%

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

   5. pow2 50.6%

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

   6. metadata-eval 50.6%

      \[\leadsto \frac{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2} - b\_2}{a} \]

6. Applied egg-rr 50.6%

   \[\leadsto \frac{\color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}} - b\_2}{a} \]

7. Taylor expanded in b_2 around inf 11.6%

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

   1. neg-mul-1 11.6%

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

   2. sub-neg 11.6%

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

   3. +-inverses 11.6%

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

9. Simplified 11.6%

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

10. Final simplification 11.6%

    \[\leadsto \frac{0}{a} \]
11. Add Preprocessing

Developer target: 99.7% 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 2024048 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (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))