jeff quadratic root 1

Percentage Accurate: 71.8% → 90.6%

Time: 20.5s

Alternatives: 11

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_0}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* (* 4.0 a) c)))))
   (if (>= b 0.0) (/ (- (- b) t_0) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) t_0)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - t_0) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + t_0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - ((4.0d0 * a) * c)))
    if (b >= 0.0d0) then
        tmp = (-b - t_0) / (2.0d0 * a)
    else
        tmp = (2.0d0 * c) / (-b + t_0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - t_0) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + t_0);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - ((4.0 * a) * c)))
	tmp = 0
	if b >= 0.0:
		tmp = (-b - t_0) / (2.0 * a)
	else:
		tmp = (2.0 * c) / (-b + t_0)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(Float64(-b) - t_0) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (-b - t_0) / (2.0 * a);
	else
		tmp = (2.0 * c) / (-b + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Initial Program: 71.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_0}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* (* 4.0 a) c)))))
   (if (>= b 0.0) (/ (- (- b) t_0) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) t_0)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - t_0) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + t_0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - ((4.0d0 * a) * c)))
    if (b >= 0.0d0) then
        tmp = (-b - t_0) / (2.0d0 * a)
    else
        tmp = (2.0d0 * c) / (-b + t_0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - t_0) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + t_0);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - ((4.0 * a) * c)))
	tmp = 0
	if b >= 0.0:
		tmp = (-b - t_0) / (2.0 * a)
	else:
		tmp = (2.0 * c) / (-b + t_0)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(Float64(-b) - t_0) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (-b - t_0) / (2.0 * a);
	else
		tmp = (2.0 * c) / (-b + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 90.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \frac{a}{\frac{b}{c}}\\ t_1 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ t_2 := \left(-b\right) - b\\ \mathbf{if}\;b \leq -5 \cdot 10^{+103}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{t_2}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{t_2}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{+93}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{t_1 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, b, t_0\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt[3]{{\left(\frac{t_0}{c}\right)}^{3}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* 2.0 (/ a (/ b c))))
        (t_1 (sqrt (- (* b b) (* c (* a 4.0)))))
        (t_2 (- (- b) b)))
   (if (<= b -5e+103)
     (if (>= b 0.0) (/ t_2 (* a 2.0)) (/ (* 2.0 c) t_2))
     (if (<= b 2.9e+93)
       (if (>= b 0.0) (/ (- (- b) t_1) (* a 2.0)) (/ (* 2.0 c) (- t_1 b)))
       (if (>= b 0.0)
         (/ (fma -2.0 b t_0) (* a 2.0))
         (/ 2.0 (cbrt (pow (/ t_0 c) 3.0))))))))

C

double code(double a, double b, double c) {
	double t_0 = 2.0 * (a / (b / c));
	double t_1 = sqrt(((b * b) - (c * (a * 4.0))));
	double t_2 = -b - b;
	double tmp_1;
	if (b <= -5e+103) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_2 / (a * 2.0);
		} else {
			tmp_2 = (2.0 * c) / t_2;
		}
		tmp_1 = tmp_2;
	} else if (b <= 2.9e+93) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_1) / (a * 2.0);
		} else {
			tmp_3 = (2.0 * c) / (t_1 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = fma(-2.0, b, t_0) / (a * 2.0);
	} else {
		tmp_1 = 2.0 / cbrt(pow((t_0 / c), 3.0));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(2.0 * Float64(a / Float64(b / c)))
	t_1 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	t_2 = Float64(Float64(-b) - b)
	tmp_1 = 0.0
	if (b <= -5e+103)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(t_2 / Float64(a * 2.0));
		else
			tmp_2 = Float64(Float64(2.0 * c) / t_2);
		end
		tmp_1 = tmp_2;
	elseif (b <= 2.9e+93)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(-b) - t_1) / Float64(a * 2.0));
		else
			tmp_3 = Float64(Float64(2.0 * c) / Float64(t_1 - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(fma(-2.0, b, t_0) / Float64(a * 2.0));
	else
		tmp_1 = Float64(2.0 / cbrt((Float64(t_0 / c) ^ 3.0)));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(2.0 * N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[((-b) - b), $MachinePrecision]}, If[LessEqual[b, -5e+103], If[GreaterEqual[b, 0.0], N[(t$95$2 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / t$95$2), $MachinePrecision]], If[LessEqual[b, 2.9e+93], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$1), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[(t$95$1 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * b + t$95$0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[Power[N[(t$95$0 / c), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -5e103

   1. Initial program 51.5%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Simplified 50.4%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]

   4. Taylor expanded in b around inf 50.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{b}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ \end{array} \]

   5. Taylor expanded in b around -inf 87.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \left(-1 \cdot b + 2 \cdot \frac{a \cdot c}{b}\right)}{c}}\\ \end{array} \]

   6. Taylor expanded in a around 0 97.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{-1 \cdot b - b}\\ \end{array} \]
   7. Step-by-step derivation

      1. associate-*r/ 97.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot b - b}\\ \end{array} \]

      2. mul-1-neg 97.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\left(-b\right) - b}\\ \end{array} \]

   8. Simplified 97.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \end{array} \]

   if -5e103 < b < 2.8999999999999998e93

   1. Initial program 92.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   if 2.8999999999999998e93 < b

   1. Initial program 58.5%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Simplified 58.5%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]

   4. Taylor expanded in b around inf 89.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{-2 \cdot b + 2 \cdot \frac{a \cdot c}{b}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ \end{array} \]
   5. Step-by-step derivation

      1. fma-def 89.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\mathsf{fma}\left(-2, b, 2 \cdot \frac{a \cdot c}{b}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ \end{array} \]

      2. associate-/l* 97.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, b, 2 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ \end{array} \]

   6. Simplified 97.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{\mathsf{fma}\left(-2, b, 2 \cdot \frac{a}{\frac{b}{c}}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ \end{array} \]
   7. Step-by-step derivation

      1. add-cbrt-cube 97.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, b, 2 \cdot \frac{a}{\frac{b}{c}}\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{\sqrt[3]{\left(\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}\right) \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}}}\\ \end{array} \]

      2. pow3 97.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, b, 2 \cdot \frac{a}{\frac{b}{c}}\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{2}}{\sqrt[3]{{\left(\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}\right)}^{3}}}\\ \end{array} \]

   8. Applied egg-rr 97.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, b, 2 \cdot \frac{a}{\frac{b}{c}}\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{\sqrt[3]{{\left(\frac{b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}{c}\right)}^{3}}}}\\ \end{array} \]

   9. Taylor expanded in b around -inf 97.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, b, 2 \cdot \frac{a}{\frac{b}{c}}\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt[3]{{\left(\frac{2 \cdot \frac{a \cdot c}{b}}{c}\right)}^{3}}}\\ \end{array} \]
   10. Step-by-step derivation

       1. *-commutative 97.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, b, 2 \cdot \frac{a}{\frac{b}{c}}\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt[3]{{\left(\frac{\frac{a \cdot c}{b} \cdot 2}{c}\right)}^{3}}}\\ \end{array} \]

       2. associate-/l* 97.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, b, 2 \cdot \frac{a}{\frac{b}{c}}\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt[3]{{\left(\frac{\frac{a}{\frac{b}{c}} \cdot 2}{c}\right)}^{3}}}\\ \end{array} \]

   11. Simplified 97.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, b, 2 \cdot \frac{a}{\frac{b}{c}}\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt[3]{{\left(\frac{\frac{a}{\frac{b}{c}} \cdot 2}{c}\right)}^{3}}}\\ \end{array} \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+103}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{+93}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, b, 2 \cdot \frac{a}{\frac{b}{c}}\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt[3]{{\left(\frac{2 \cdot \frac{a}{\frac{b}{c}}}{c}\right)}^{3}}}\\ \end{array} \]

Alternative 2: 90.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\\ t_1 := \left(-b\right) - b\\ \mathbf{if}\;b \leq -2 \cdot 10^{+101}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{t_1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{t_1}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+94}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t_0 - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b + -2 \cdot \frac{a}{\frac{b}{c}}\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t_1}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* 4.0 (* a c))))) (t_1 (- (- b) b)))
   (if (<= b -2e+101)
     (if (>= b 0.0) (/ t_1 (* a 2.0)) (/ (* 2.0 c) t_1))
     (if (<= b 1.8e+94)
       (if (>= b 0.0) (/ (- (- b) t_0) (* a 2.0)) (/ 2.0 (/ (- t_0 b) c)))
       (if (>= b 0.0)
         (/ (- (- b) (+ b (* -2.0 (/ a (/ b c))))) (* a 2.0))
         (/ 2.0 (/ t_1 c)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (4.0 * (a * c))));
	double t_1 = -b - b;
	double tmp_1;
	if (b <= -2e+101) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1 / (a * 2.0);
		} else {
			tmp_2 = (2.0 * c) / t_1;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.8e+94) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_0) / (a * 2.0);
		} else {
			tmp_3 = 2.0 / ((t_0 - b) / c);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (-b - (b + (-2.0 * (a / (b / c))))) / (a * 2.0);
	} else {
		tmp_1 = 2.0 / (t_1 / c);
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = sqrt(((b * b) - (4.0d0 * (a * c))))
    t_1 = -b - b
    if (b <= (-2d+101)) then
        if (b >= 0.0d0) then
            tmp_2 = t_1 / (a * 2.0d0)
        else
            tmp_2 = (2.0d0 * c) / t_1
        end if
        tmp_1 = tmp_2
    else if (b <= 1.8d+94) then
        if (b >= 0.0d0) then
            tmp_3 = (-b - t_0) / (a * 2.0d0)
        else
            tmp_3 = 2.0d0 / ((t_0 - b) / c)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = (-b - (b + ((-2.0d0) * (a / (b / c))))) / (a * 2.0d0)
    else
        tmp_1 = 2.0d0 / (t_1 / c)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (4.0 * (a * c))));
	double t_1 = -b - b;
	double tmp_1;
	if (b <= -2e+101) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1 / (a * 2.0);
		} else {
			tmp_2 = (2.0 * c) / t_1;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.8e+94) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_0) / (a * 2.0);
		} else {
			tmp_3 = 2.0 / ((t_0 - b) / c);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (-b - (b + (-2.0 * (a / (b / c))))) / (a * 2.0);
	} else {
		tmp_1 = 2.0 / (t_1 / c);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (4.0 * (a * c))))
	t_1 = -b - b
	tmp_1 = 0
	if b <= -2e+101:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_1 / (a * 2.0)
		else:
			tmp_2 = (2.0 * c) / t_1
		tmp_1 = tmp_2
	elif b <= 1.8e+94:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (-b - t_0) / (a * 2.0)
		else:
			tmp_3 = 2.0 / ((t_0 - b) / c)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = (-b - (b + (-2.0 * (a / (b / c))))) / (a * 2.0)
	else:
		tmp_1 = 2.0 / (t_1 / c)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))
	t_1 = Float64(Float64(-b) - b)
	tmp_1 = 0.0
	if (b <= -2e+101)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(t_1 / Float64(a * 2.0));
		else
			tmp_2 = Float64(Float64(2.0 * c) / t_1);
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.8e+94)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(-b) - t_0) / Float64(a * 2.0));
		else
			tmp_3 = Float64(2.0 / Float64(Float64(t_0 - b) / c));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(-b) - Float64(b + Float64(-2.0 * Float64(a / Float64(b / c))))) / Float64(a * 2.0));
	else
		tmp_1 = Float64(2.0 / Float64(t_1 / c));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) - (4.0 * (a * c))));
	t_1 = -b - b;
	tmp_2 = 0.0;
	if (b <= -2e+101)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_1 / (a * 2.0);
		else
			tmp_3 = (2.0 * c) / t_1;
		end
		tmp_2 = tmp_3;
	elseif (b <= 1.8e+94)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (-b - t_0) / (a * 2.0);
		else
			tmp_4 = 2.0 / ((t_0 - b) / c);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = (-b - (b + (-2.0 * (a / (b / c))))) / (a * 2.0);
	else
		tmp_2 = 2.0 / (t_1 / c);
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-b) - b), $MachinePrecision]}, If[LessEqual[b, -2e+101], If[GreaterEqual[b, 0.0], N[(t$95$1 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / t$95$1), $MachinePrecision]], If[LessEqual[b, 1.8e+94], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$0 - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[((-b) - N[(b + N[(-2.0 * N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 / c), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2e101

   1. Initial program 51.5%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Simplified 50.4%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]

   4. Taylor expanded in b around inf 50.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{b}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ \end{array} \]

   5. Taylor expanded in b around -inf 87.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \left(-1 \cdot b + 2 \cdot \frac{a \cdot c}{b}\right)}{c}}\\ \end{array} \]

   6. Taylor expanded in a around 0 97.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{-1 \cdot b - b}\\ \end{array} \]
   7. Step-by-step derivation

      1. associate-*r/ 97.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot b - b}\\ \end{array} \]

      2. mul-1-neg 97.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\left(-b\right) - b}\\ \end{array} \]

   8. Simplified 97.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \end{array} \]

   if -2e101 < b < 1.79999999999999996e94

   1. Initial program 92.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Simplified 92.1%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]

   if 1.79999999999999996e94 < b

   1. Initial program 58.5%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Simplified 58.5%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]

   4. Taylor expanded in b around -inf 58.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

   5. Taylor expanded in b around inf 89.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\left(b + -2 \cdot \frac{a \cdot c}{b}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-/l* 97.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b + -2 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

   7. Simplified 97.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\left(b + -2 \cdot \frac{a}{\frac{b}{c}}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+101}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+94}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b + -2 \cdot \frac{a}{\frac{b}{c}}\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - b}{c}}\\ \end{array} \]

Alternative 3: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ t_1 := \left(-b\right) - b\\ \mathbf{if}\;b \leq -2.2 \cdot 10^{+104}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{t_1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{t_1}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{+94}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{t_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b + -2 \cdot \frac{a}{\frac{b}{c}}\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t_1}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0))))) (t_1 (- (- b) b)))
   (if (<= b -2.2e+104)
     (if (>= b 0.0) (/ t_1 (* a 2.0)) (/ (* 2.0 c) t_1))
     (if (<= b 1e+94)
       (if (>= b 0.0) (/ (- (- b) t_0) (* a 2.0)) (/ (* 2.0 c) (- t_0 b)))
       (if (>= b 0.0)
         (/ (- (- b) (+ b (* -2.0 (/ a (/ b c))))) (* a 2.0))
         (/ 2.0 (/ t_1 c)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double t_1 = -b - b;
	double tmp_1;
	if (b <= -2.2e+104) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1 / (a * 2.0);
		} else {
			tmp_2 = (2.0 * c) / t_1;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1e+94) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_0) / (a * 2.0);
		} else {
			tmp_3 = (2.0 * c) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (-b - (b + (-2.0 * (a / (b / c))))) / (a * 2.0);
	} else {
		tmp_1 = 2.0 / (t_1 / c);
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = sqrt(((b * b) - (c * (a * 4.0d0))))
    t_1 = -b - b
    if (b <= (-2.2d+104)) then
        if (b >= 0.0d0) then
            tmp_2 = t_1 / (a * 2.0d0)
        else
            tmp_2 = (2.0d0 * c) / t_1
        end if
        tmp_1 = tmp_2
    else if (b <= 1d+94) then
        if (b >= 0.0d0) then
            tmp_3 = (-b - t_0) / (a * 2.0d0)
        else
            tmp_3 = (2.0d0 * c) / (t_0 - b)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = (-b - (b + ((-2.0d0) * (a / (b / c))))) / (a * 2.0d0)
    else
        tmp_1 = 2.0d0 / (t_1 / c)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (c * (a * 4.0))));
	double t_1 = -b - b;
	double tmp_1;
	if (b <= -2.2e+104) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1 / (a * 2.0);
		} else {
			tmp_2 = (2.0 * c) / t_1;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1e+94) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_0) / (a * 2.0);
		} else {
			tmp_3 = (2.0 * c) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (-b - (b + (-2.0 * (a / (b / c))))) / (a * 2.0);
	} else {
		tmp_1 = 2.0 / (t_1 / c);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (c * (a * 4.0))))
	t_1 = -b - b
	tmp_1 = 0
	if b <= -2.2e+104:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_1 / (a * 2.0)
		else:
			tmp_2 = (2.0 * c) / t_1
		tmp_1 = tmp_2
	elif b <= 1e+94:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (-b - t_0) / (a * 2.0)
		else:
			tmp_3 = (2.0 * c) / (t_0 - b)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = (-b - (b + (-2.0 * (a / (b / c))))) / (a * 2.0)
	else:
		tmp_1 = 2.0 / (t_1 / c)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	t_1 = Float64(Float64(-b) - b)
	tmp_1 = 0.0
	if (b <= -2.2e+104)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(t_1 / Float64(a * 2.0));
		else
			tmp_2 = Float64(Float64(2.0 * c) / t_1);
		end
		tmp_1 = tmp_2;
	elseif (b <= 1e+94)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(-b) - t_0) / Float64(a * 2.0));
		else
			tmp_3 = Float64(Float64(2.0 * c) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(-b) - Float64(b + Float64(-2.0 * Float64(a / Float64(b / c))))) / Float64(a * 2.0));
	else
		tmp_1 = Float64(2.0 / Float64(t_1 / c));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	t_1 = -b - b;
	tmp_2 = 0.0;
	if (b <= -2.2e+104)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_1 / (a * 2.0);
		else
			tmp_3 = (2.0 * c) / t_1;
		end
		tmp_2 = tmp_3;
	elseif (b <= 1e+94)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (-b - t_0) / (a * 2.0);
		else
			tmp_4 = (2.0 * c) / (t_0 - b);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = (-b - (b + (-2.0 * (a / (b / c))))) / (a * 2.0);
	else
		tmp_2 = 2.0 / (t_1 / c);
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-b) - b), $MachinePrecision]}, If[LessEqual[b, -2.2e+104], If[GreaterEqual[b, 0.0], N[(t$95$1 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / t$95$1), $MachinePrecision]], If[LessEqual[b, 1e+94], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[((-b) - N[(b + N[(-2.0 * N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 / c), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.2e104

   1. Initial program 51.5%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Simplified 50.4%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]

   4. Taylor expanded in b around inf 50.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{b}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ \end{array} \]

   5. Taylor expanded in b around -inf 87.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \left(-1 \cdot b + 2 \cdot \frac{a \cdot c}{b}\right)}{c}}\\ \end{array} \]

   6. Taylor expanded in a around 0 97.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{-1 \cdot b - b}\\ \end{array} \]
   7. Step-by-step derivation

      1. associate-*r/ 97.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot b - b}\\ \end{array} \]

      2. mul-1-neg 97.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\left(-b\right) - b}\\ \end{array} \]

   8. Simplified 97.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \end{array} \]

   if -2.2e104 < b < 1e94

   1. Initial program 92.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   if 1e94 < b

   1. Initial program 58.5%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Simplified 58.5%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]

   4. Taylor expanded in b around -inf 58.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

   5. Taylor expanded in b around inf 89.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\left(b + -2 \cdot \frac{a \cdot c}{b}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-/l* 97.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b + -2 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

   7. Simplified 97.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\left(b + -2 \cdot \frac{a}{\frac{b}{c}}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{+104}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{+94}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b + -2 \cdot \frac{a}{\frac{b}{c}}\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - b}{c}}\\ \end{array} \]

Alternative 4: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-b\right) - b\\ t_1 := \frac{t_0}{a \cdot 2}\\ t_2 := \frac{2}{\frac{t_0}{c}}\\ \mathbf{if}\;b \leq -1.1 \cdot 10^{+104}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{t_0}\\ \end{array}\\ \mathbf{elif}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-112}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b + -2 \cdot \frac{a}{\frac{b}{c}}\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (- b) b)) (t_1 (/ t_0 (* a 2.0))) (t_2 (/ 2.0 (/ t_0 c))))
   (if (<= b -1.1e+104)
     (if (>= b 0.0) t_1 (/ (* 2.0 c) t_0))
     (if (<= b -1e-310)
       (if (>= b 0.0)
         t_1
         (/ 2.0 (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) c)))
       (if (<= b 2.9e-112)
         (if (>= b 0.0) (/ (- (- b) (sqrt (* c (* a -4.0)))) (* a 2.0)) t_2)
         (if (>= b 0.0)
           (/ (- (- b) (+ b (* -2.0 (/ a (/ b c))))) (* a 2.0))
           t_2))))))

C

double code(double a, double b, double c) {
	double t_0 = -b - b;
	double t_1 = t_0 / (a * 2.0);
	double t_2 = 2.0 / (t_0 / c);
	double tmp_1;
	if (b <= -1.1e+104) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = (2.0 * c) / t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -1e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = 2.0 / ((sqrt(((b * b) - (4.0 * (a * c)))) - b) / c);
		}
		tmp_1 = tmp_3;
	} else if (b <= 2.9e-112) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (-b - sqrt((c * (a * -4.0)))) / (a * 2.0);
		} else {
			tmp_4 = t_2;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (-b - (b + (-2.0 * (a / (b / c))))) / (a * 2.0);
	} else {
		tmp_1 = t_2;
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    real(8) :: tmp_4
    t_0 = -b - b
    t_1 = t_0 / (a * 2.0d0)
    t_2 = 2.0d0 / (t_0 / c)
    if (b <= (-1.1d+104)) then
        if (b >= 0.0d0) then
            tmp_2 = t_1
        else
            tmp_2 = (2.0d0 * c) / t_0
        end if
        tmp_1 = tmp_2
    else if (b <= (-1d-310)) then
        if (b >= 0.0d0) then
            tmp_3 = t_1
        else
            tmp_3 = 2.0d0 / ((sqrt(((b * b) - (4.0d0 * (a * c)))) - b) / c)
        end if
        tmp_1 = tmp_3
    else if (b <= 2.9d-112) then
        if (b >= 0.0d0) then
            tmp_4 = (-b - sqrt((c * (a * (-4.0d0))))) / (a * 2.0d0)
        else
            tmp_4 = t_2
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = (-b - (b + ((-2.0d0) * (a / (b / c))))) / (a * 2.0d0)
    else
        tmp_1 = t_2
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -b - b;
	double t_1 = t_0 / (a * 2.0);
	double t_2 = 2.0 / (t_0 / c);
	double tmp_1;
	if (b <= -1.1e+104) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = (2.0 * c) / t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -1e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = 2.0 / ((Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / c);
		}
		tmp_1 = tmp_3;
	} else if (b <= 2.9e-112) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (-b - Math.sqrt((c * (a * -4.0)))) / (a * 2.0);
		} else {
			tmp_4 = t_2;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (-b - (b + (-2.0 * (a / (b / c))))) / (a * 2.0);
	} else {
		tmp_1 = t_2;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = -b - b
	t_1 = t_0 / (a * 2.0)
	t_2 = 2.0 / (t_0 / c)
	tmp_1 = 0
	if b <= -1.1e+104:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_1
		else:
			tmp_2 = (2.0 * c) / t_0
		tmp_1 = tmp_2
	elif b <= -1e-310:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = t_1
		else:
			tmp_3 = 2.0 / ((math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / c)
		tmp_1 = tmp_3
	elif b <= 2.9e-112:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = (-b - math.sqrt((c * (a * -4.0)))) / (a * 2.0)
		else:
			tmp_4 = t_2
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = (-b - (b + (-2.0 * (a / (b / c))))) / (a * 2.0)
	else:
		tmp_1 = t_2
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-b) - b)
	t_1 = Float64(t_0 / Float64(a * 2.0))
	t_2 = Float64(2.0 / Float64(t_0 / c))
	tmp_1 = 0.0
	if (b <= -1.1e+104)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(Float64(2.0 * c) / t_0);
		end
		tmp_1 = tmp_2;
	elseif (b <= -1e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = Float64(2.0 / Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / c));
		end
		tmp_1 = tmp_3;
	elseif (b <= 2.9e-112)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(Float64(-b) - sqrt(Float64(c * Float64(a * -4.0)))) / Float64(a * 2.0));
		else
			tmp_4 = t_2;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(-b) - Float64(b + Float64(-2.0 * Float64(a / Float64(b / c))))) / Float64(a * 2.0));
	else
		tmp_1 = t_2;
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = -b - b;
	t_1 = t_0 / (a * 2.0);
	t_2 = 2.0 / (t_0 / c);
	tmp_2 = 0.0;
	if (b <= -1.1e+104)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = (2.0 * c) / t_0;
		end
		tmp_2 = tmp_3;
	elseif (b <= -1e-310)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = t_1;
		else
			tmp_4 = 2.0 / ((sqrt(((b * b) - (4.0 * (a * c)))) - b) / c);
		end
		tmp_2 = tmp_4;
	elseif (b <= 2.9e-112)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = (-b - sqrt((c * (a * -4.0)))) / (a * 2.0);
		else
			tmp_5 = t_2;
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = (-b - (b + (-2.0 * (a / (b / c))))) / (a * 2.0);
	else
		tmp_2 = t_2;
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) - b), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 / N[(t$95$0 / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.1e+104], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(2.0 * c), $MachinePrecision] / t$95$0), $MachinePrecision]], If[LessEqual[b, -1e-310], If[GreaterEqual[b, 0.0], t$95$1, N[(2.0 / N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 2.9e-112], If[GreaterEqual[b, 0.0], N[(N[((-b) - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], t$95$2], If[GreaterEqual[b, 0.0], N[(N[((-b) - N[(b + N[(-2.0 * N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.1e104

   1. Initial program 51.5%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Simplified 50.4%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]

   4. Taylor expanded in b around inf 50.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{b}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ \end{array} \]

   5. Taylor expanded in b around -inf 87.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \left(-1 \cdot b + 2 \cdot \frac{a \cdot c}{b}\right)}{c}}\\ \end{array} \]

   6. Taylor expanded in a around 0 97.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{-1 \cdot b - b}\\ \end{array} \]
   7. Step-by-step derivation

      1. associate-*r/ 97.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot b - b}\\ \end{array} \]

      2. mul-1-neg 97.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\left(-b\right) - b}\\ \end{array} \]

   8. Simplified 97.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \end{array} \]

   if -1.1e104 < b < -9.999999999999969e-311

   1. Initial program 88.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Simplified 87.9%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]

   4. Taylor expanded in b around inf 87.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{b}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ \end{array} \]

   if -9.999999999999969e-311 < b < 2.89999999999999992e-112

   1. Initial program 90.9%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Simplified 90.9%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]

   4. Taylor expanded in b around -inf 90.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

   5. Taylor expanded in b around 0 87.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-*r* 87.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(-4 \cdot a\right) \cdot c}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

   7. Simplified 87.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(-4 \cdot a\right) \cdot c}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

   if 2.89999999999999992e-112 < b

   1. Initial program 77.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Simplified 77.0%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]

   4. Taylor expanded in b around -inf 77.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

   5. Taylor expanded in b around inf 79.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\left(b + -2 \cdot \frac{a \cdot c}{b}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-/l* 84.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b + -2 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

   7. Simplified 84.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\left(b + -2 \cdot \frac{a}{\frac{b}{c}}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{+104}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-112}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b + -2 \cdot \frac{a}{\frac{b}{c}}\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - b}{c}}\\ \end{array} \]

Alternative 5: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-b\right) - b\\ t_1 := \frac{2}{\frac{t_0}{c}}\\ t_2 := \frac{t_0}{a \cdot 2}\\ \mathbf{if}\;b \leq -3.7 \cdot 10^{-65}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{t_0}\\ \end{array}\\ \mathbf{elif}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.26 \cdot 10^{-110}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\left(-b\right) - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b + -2 \cdot \frac{a}{\frac{b}{c}}\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (- b) b)) (t_1 (/ 2.0 (/ t_0 c))) (t_2 (/ t_0 (* a 2.0))))
   (if (<= b -3.7e-65)
     (if (>= b 0.0) t_2 (/ (* 2.0 c) t_0))
     (if (<= b -1e-310)
       (if (>= b 0.0) t_2 (/ 2.0 (/ (- (sqrt (* c (* a -4.0))) b) c)))
       (if (<= b 2.26e-110)
         (if (>= b 0.0) (* (/ 0.5 a) (- (- b) (sqrt (* a (* c -4.0))))) t_1)
         (if (>= b 0.0)
           (/ (- (- b) (+ b (* -2.0 (/ a (/ b c))))) (* a 2.0))
           t_1))))))

C

double code(double a, double b, double c) {
	double t_0 = -b - b;
	double t_1 = 2.0 / (t_0 / c);
	double t_2 = t_0 / (a * 2.0);
	double tmp_1;
	if (b <= -3.7e-65) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_2;
		} else {
			tmp_2 = (2.0 * c) / t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -1e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_2;
		} else {
			tmp_3 = 2.0 / ((sqrt((c * (a * -4.0))) - b) / c);
		}
		tmp_1 = tmp_3;
	} else if (b <= 2.26e-110) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (0.5 / a) * (-b - sqrt((a * (c * -4.0))));
		} else {
			tmp_4 = t_1;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (-b - (b + (-2.0 * (a / (b / c))))) / (a * 2.0);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    real(8) :: tmp_4
    t_0 = -b - b
    t_1 = 2.0d0 / (t_0 / c)
    t_2 = t_0 / (a * 2.0d0)
    if (b <= (-3.7d-65)) then
        if (b >= 0.0d0) then
            tmp_2 = t_2
        else
            tmp_2 = (2.0d0 * c) / t_0
        end if
        tmp_1 = tmp_2
    else if (b <= (-1d-310)) then
        if (b >= 0.0d0) then
            tmp_3 = t_2
        else
            tmp_3 = 2.0d0 / ((sqrt((c * (a * (-4.0d0)))) - b) / c)
        end if
        tmp_1 = tmp_3
    else if (b <= 2.26d-110) then
        if (b >= 0.0d0) then
            tmp_4 = (0.5d0 / a) * (-b - sqrt((a * (c * (-4.0d0)))))
        else
            tmp_4 = t_1
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = (-b - (b + ((-2.0d0) * (a / (b / c))))) / (a * 2.0d0)
    else
        tmp_1 = t_1
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -b - b;
	double t_1 = 2.0 / (t_0 / c);
	double t_2 = t_0 / (a * 2.0);
	double tmp_1;
	if (b <= -3.7e-65) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_2;
		} else {
			tmp_2 = (2.0 * c) / t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -1e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_2;
		} else {
			tmp_3 = 2.0 / ((Math.sqrt((c * (a * -4.0))) - b) / c);
		}
		tmp_1 = tmp_3;
	} else if (b <= 2.26e-110) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (0.5 / a) * (-b - Math.sqrt((a * (c * -4.0))));
		} else {
			tmp_4 = t_1;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (-b - (b + (-2.0 * (a / (b / c))))) / (a * 2.0);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = -b - b
	t_1 = 2.0 / (t_0 / c)
	t_2 = t_0 / (a * 2.0)
	tmp_1 = 0
	if b <= -3.7e-65:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_2
		else:
			tmp_2 = (2.0 * c) / t_0
		tmp_1 = tmp_2
	elif b <= -1e-310:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = t_2
		else:
			tmp_3 = 2.0 / ((math.sqrt((c * (a * -4.0))) - b) / c)
		tmp_1 = tmp_3
	elif b <= 2.26e-110:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = (0.5 / a) * (-b - math.sqrt((a * (c * -4.0))))
		else:
			tmp_4 = t_1
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = (-b - (b + (-2.0 * (a / (b / c))))) / (a * 2.0)
	else:
		tmp_1 = t_1
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-b) - b)
	t_1 = Float64(2.0 / Float64(t_0 / c))
	t_2 = Float64(t_0 / Float64(a * 2.0))
	tmp_1 = 0.0
	if (b <= -3.7e-65)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_2;
		else
			tmp_2 = Float64(Float64(2.0 * c) / t_0);
		end
		tmp_1 = tmp_2;
	elseif (b <= -1e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_2;
		else
			tmp_3 = Float64(2.0 / Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / c));
		end
		tmp_1 = tmp_3;
	elseif (b <= 2.26e-110)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(0.5 / a) * Float64(Float64(-b) - sqrt(Float64(a * Float64(c * -4.0)))));
		else
			tmp_4 = t_1;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(-b) - Float64(b + Float64(-2.0 * Float64(a / Float64(b / c))))) / Float64(a * 2.0));
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = -b - b;
	t_1 = 2.0 / (t_0 / c);
	t_2 = t_0 / (a * 2.0);
	tmp_2 = 0.0;
	if (b <= -3.7e-65)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_2;
		else
			tmp_3 = (2.0 * c) / t_0;
		end
		tmp_2 = tmp_3;
	elseif (b <= -1e-310)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = t_2;
		else
			tmp_4 = 2.0 / ((sqrt((c * (a * -4.0))) - b) / c);
		end
		tmp_2 = tmp_4;
	elseif (b <= 2.26e-110)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = (0.5 / a) * (-b - sqrt((a * (c * -4.0))));
		else
			tmp_5 = t_1;
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = (-b - (b + (-2.0 * (a / (b / c))))) / (a * 2.0);
	else
		tmp_2 = t_1;
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) - b), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(t$95$0 / c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.7e-65], If[GreaterEqual[b, 0.0], t$95$2, N[(N[(2.0 * c), $MachinePrecision] / t$95$0), $MachinePrecision]], If[LessEqual[b, -1e-310], If[GreaterEqual[b, 0.0], t$95$2, N[(2.0 / N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 2.26e-110], If[GreaterEqual[b, 0.0], N[(N[(0.5 / a), $MachinePrecision] * N[((-b) - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1], If[GreaterEqual[b, 0.0], N[(N[((-b) - N[(b + N[(-2.0 * N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -3.7e-65

   1. Initial program 68.8%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Simplified 67.9%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]

   4. Taylor expanded in b around inf 67.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{b}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ \end{array} \]

   5. Taylor expanded in b around -inf 88.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \left(-1 \cdot b + 2 \cdot \frac{a \cdot c}{b}\right)}{c}}\\ \end{array} \]

   6. Taylor expanded in a around 0 95.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{-1 \cdot b - b}\\ \end{array} \]
   7. Step-by-step derivation

      1. associate-*r/ 95.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot b - b}\\ \end{array} \]

      2. mul-1-neg 95.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\left(-b\right) - b}\\ \end{array} \]

   8. Simplified 95.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \end{array} \]

   if -3.7e-65 < b < -9.999999999999969e-311

   1. Initial program 78.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Simplified 77.9%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]

   4. Taylor expanded in b around inf 77.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{b}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ \end{array} \]

   5. Taylor expanded in b around 0 64.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{c}}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-*r* 26.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(-4 \cdot a\right) \cdot c}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

   7. Simplified 64.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{\left(-4 \cdot a\right) \cdot c}}{c}}\\ \end{array} \]

   if -9.999999999999969e-311 < b < 2.26000000000000012e-110

   1. Initial program 90.9%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Simplified 90.9%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]

   4. Taylor expanded in b around -inf 90.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

   5. Taylor expanded in b around 0 87.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-*r* 87.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(-4 \cdot a\right) \cdot c}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

   7. Simplified 87.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(-4 \cdot a\right) \cdot c}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]
   8. Step-by-step derivation

      1. div-sub 87.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-b}{a \cdot 2} - \frac{\sqrt{\left(-4 \cdot a\right) \cdot c}}{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

      2. *-commutative 87.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a \cdot 2} - \frac{\sqrt{\color{blue}{c \cdot \left(-4 \cdot a\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

      3. *-commutative 87.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a \cdot 2} - \frac{\sqrt{c \cdot \color{blue}{\left(a \cdot -4\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

   9. Applied egg-rr 87.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-b}{a \cdot 2} - \frac{\sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]
   10. Step-by-step derivation

       1. div-sub 87.2%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

       2. *-lft-identity 87.2%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

       3. associate-*l/ 86.9%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{1}{a \cdot 2} \cdot \left(\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

       4. *-commutative 86.9%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{1}{\color{blue}{2 \cdot a}} \cdot \left(\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

       5. associate-/r* 86.9%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{\frac{1}{2}}{a}} \cdot \left(\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

       6. metadata-eval 86.9%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{0.5}}{a} \cdot \left(\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

       7. associate-*r* 86.9%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\left(-b\right) - \sqrt{\color{blue}{\left(c \cdot a\right) \cdot -4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

       8. *-commutative 86.9%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\left(-b\right) - \sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

       9. associate-*l* 86.9%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\left(-b\right) - \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

   11. Simplified 86.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{0.5}{a} \cdot \left(\left(-b\right) - \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

   if 2.26000000000000012e-110 < b

   1. Initial program 77.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Simplified 77.0%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]

   4. Taylor expanded in b around -inf 77.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

   5. Taylor expanded in b around inf 79.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\left(b + -2 \cdot \frac{a \cdot c}{b}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-/l* 84.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b + -2 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

   7. Simplified 84.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\left(b + -2 \cdot \frac{a}{\frac{b}{c}}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-65}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.26 \cdot 10^{-110}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\left(-b\right) - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b + -2 \cdot \frac{a}{\frac{b}{c}}\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - b}{c}}\\ \end{array} \]

Alternative 6: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-b\right) - b\\ t_1 := \frac{2}{\frac{t_0}{c}}\\ t_2 := \frac{t_0}{a \cdot 2}\\ t_3 := \sqrt{c \cdot \left(a \cdot -4\right)}\\ \mathbf{if}\;b \leq -1.05 \cdot 10^{-63}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{t_0}\\ \end{array}\\ \mathbf{elif}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t_3 - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \leq 9.1 \cdot 10^{-110}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_3}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b + -2 \cdot \frac{a}{\frac{b}{c}}\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (- b) b))
        (t_1 (/ 2.0 (/ t_0 c)))
        (t_2 (/ t_0 (* a 2.0)))
        (t_3 (sqrt (* c (* a -4.0)))))
   (if (<= b -1.05e-63)
     (if (>= b 0.0) t_2 (/ (* 2.0 c) t_0))
     (if (<= b -1e-310)
       (if (>= b 0.0) t_2 (/ 2.0 (/ (- t_3 b) c)))
       (if (<= b 9.1e-110)
         (if (>= b 0.0) (/ (- (- b) t_3) (* a 2.0)) t_1)
         (if (>= b 0.0)
           (/ (- (- b) (+ b (* -2.0 (/ a (/ b c))))) (* a 2.0))
           t_1))))))

C

double code(double a, double b, double c) {
	double t_0 = -b - b;
	double t_1 = 2.0 / (t_0 / c);
	double t_2 = t_0 / (a * 2.0);
	double t_3 = sqrt((c * (a * -4.0)));
	double tmp_1;
	if (b <= -1.05e-63) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_2;
		} else {
			tmp_2 = (2.0 * c) / t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -1e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_2;
		} else {
			tmp_3 = 2.0 / ((t_3 - b) / c);
		}
		tmp_1 = tmp_3;
	} else if (b <= 9.1e-110) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (-b - t_3) / (a * 2.0);
		} else {
			tmp_4 = t_1;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (-b - (b + (-2.0 * (a / (b / c))))) / (a * 2.0);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    real(8) :: tmp_4
    t_0 = -b - b
    t_1 = 2.0d0 / (t_0 / c)
    t_2 = t_0 / (a * 2.0d0)
    t_3 = sqrt((c * (a * (-4.0d0))))
    if (b <= (-1.05d-63)) then
        if (b >= 0.0d0) then
            tmp_2 = t_2
        else
            tmp_2 = (2.0d0 * c) / t_0
        end if
        tmp_1 = tmp_2
    else if (b <= (-1d-310)) then
        if (b >= 0.0d0) then
            tmp_3 = t_2
        else
            tmp_3 = 2.0d0 / ((t_3 - b) / c)
        end if
        tmp_1 = tmp_3
    else if (b <= 9.1d-110) then
        if (b >= 0.0d0) then
            tmp_4 = (-b - t_3) / (a * 2.0d0)
        else
            tmp_4 = t_1
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = (-b - (b + ((-2.0d0) * (a / (b / c))))) / (a * 2.0d0)
    else
        tmp_1 = t_1
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -b - b;
	double t_1 = 2.0 / (t_0 / c);
	double t_2 = t_0 / (a * 2.0);
	double t_3 = Math.sqrt((c * (a * -4.0)));
	double tmp_1;
	if (b <= -1.05e-63) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_2;
		} else {
			tmp_2 = (2.0 * c) / t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -1e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_2;
		} else {
			tmp_3 = 2.0 / ((t_3 - b) / c);
		}
		tmp_1 = tmp_3;
	} else if (b <= 9.1e-110) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (-b - t_3) / (a * 2.0);
		} else {
			tmp_4 = t_1;
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (-b - (b + (-2.0 * (a / (b / c))))) / (a * 2.0);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = -b - b
	t_1 = 2.0 / (t_0 / c)
	t_2 = t_0 / (a * 2.0)
	t_3 = math.sqrt((c * (a * -4.0)))
	tmp_1 = 0
	if b <= -1.05e-63:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_2
		else:
			tmp_2 = (2.0 * c) / t_0
		tmp_1 = tmp_2
	elif b <= -1e-310:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = t_2
		else:
			tmp_3 = 2.0 / ((t_3 - b) / c)
		tmp_1 = tmp_3
	elif b <= 9.1e-110:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = (-b - t_3) / (a * 2.0)
		else:
			tmp_4 = t_1
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = (-b - (b + (-2.0 * (a / (b / c))))) / (a * 2.0)
	else:
		tmp_1 = t_1
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-b) - b)
	t_1 = Float64(2.0 / Float64(t_0 / c))
	t_2 = Float64(t_0 / Float64(a * 2.0))
	t_3 = sqrt(Float64(c * Float64(a * -4.0)))
	tmp_1 = 0.0
	if (b <= -1.05e-63)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_2;
		else
			tmp_2 = Float64(Float64(2.0 * c) / t_0);
		end
		tmp_1 = tmp_2;
	elseif (b <= -1e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_2;
		else
			tmp_3 = Float64(2.0 / Float64(Float64(t_3 - b) / c));
		end
		tmp_1 = tmp_3;
	elseif (b <= 9.1e-110)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(Float64(-b) - t_3) / Float64(a * 2.0));
		else
			tmp_4 = t_1;
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(-b) - Float64(b + Float64(-2.0 * Float64(a / Float64(b / c))))) / Float64(a * 2.0));
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = -b - b;
	t_1 = 2.0 / (t_0 / c);
	t_2 = t_0 / (a * 2.0);
	t_3 = sqrt((c * (a * -4.0)));
	tmp_2 = 0.0;
	if (b <= -1.05e-63)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_2;
		else
			tmp_3 = (2.0 * c) / t_0;
		end
		tmp_2 = tmp_3;
	elseif (b <= -1e-310)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = t_2;
		else
			tmp_4 = 2.0 / ((t_3 - b) / c);
		end
		tmp_2 = tmp_4;
	elseif (b <= 9.1e-110)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = (-b - t_3) / (a * 2.0);
		else
			tmp_5 = t_1;
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = (-b - (b + (-2.0 * (a / (b / c))))) / (a * 2.0);
	else
		tmp_2 = t_1;
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) - b), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(t$95$0 / c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.05e-63], If[GreaterEqual[b, 0.0], t$95$2, N[(N[(2.0 * c), $MachinePrecision] / t$95$0), $MachinePrecision]], If[LessEqual[b, -1e-310], If[GreaterEqual[b, 0.0], t$95$2, N[(2.0 / N[(N[(t$95$3 - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 9.1e-110], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$3), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], t$95$1], If[GreaterEqual[b, 0.0], N[(N[((-b) - N[(b + N[(-2.0 * N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.05e-63

   1. Initial program 68.8%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Simplified 67.9%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]

   4. Taylor expanded in b around inf 67.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{b}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ \end{array} \]

   5. Taylor expanded in b around -inf 88.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \left(-1 \cdot b + 2 \cdot \frac{a \cdot c}{b}\right)}{c}}\\ \end{array} \]

   6. Taylor expanded in a around 0 95.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{-1 \cdot b - b}\\ \end{array} \]
   7. Step-by-step derivation

      1. associate-*r/ 95.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot b - b}\\ \end{array} \]

      2. mul-1-neg 95.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\left(-b\right) - b}\\ \end{array} \]

   8. Simplified 95.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \end{array} \]

   if -1.05e-63 < b < -9.999999999999969e-311

   1. Initial program 78.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Simplified 77.9%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]

   4. Taylor expanded in b around inf 77.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{b}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ \end{array} \]

   5. Taylor expanded in b around 0 64.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{c}}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-*r* 26.3%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(-4 \cdot a\right) \cdot c}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

   7. Simplified 64.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{\left(-4 \cdot a\right) \cdot c}}{c}}\\ \end{array} \]

   if -9.999999999999969e-311 < b < 9.1000000000000004e-110

   1. Initial program 90.9%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Simplified 90.9%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]

   4. Taylor expanded in b around -inf 90.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

   5. Taylor expanded in b around 0 87.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-*r* 87.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(-4 \cdot a\right) \cdot c}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

   7. Simplified 87.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(-4 \cdot a\right) \cdot c}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

   if 9.1000000000000004e-110 < b

   1. Initial program 77.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Simplified 77.0%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]

   4. Taylor expanded in b around -inf 77.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

   5. Taylor expanded in b around inf 79.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\left(b + -2 \cdot \frac{a \cdot c}{b}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-/l* 84.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b + -2 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

   7. Simplified 84.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\left(b + -2 \cdot \frac{a}{\frac{b}{c}}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-63}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \leq 9.1 \cdot 10^{-110}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b + -2 \cdot \frac{a}{\frac{b}{c}}\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - b}{c}}\\ \end{array} \]

Alternative 7: 74.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{\frac{\left(-b\right) - b}{c}}\\ \mathbf{if}\;b \leq 9.5 \cdot 10^{-110}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\left(-b\right) - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b + -2 \cdot \frac{a}{\frac{b}{c}}\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ 2.0 (/ (- (- b) b) c))))
   (if (<= b 9.5e-110)
     (if (>= b 0.0) (* (/ 0.5 a) (- (- b) (sqrt (* a (* c -4.0))))) t_0)
     (if (>= b 0.0)
       (/ (- (- b) (+ b (* -2.0 (/ a (/ b c))))) (* a 2.0))
       t_0))))

C

double code(double a, double b, double c) {
	double t_0 = 2.0 / ((-b - b) / c);
	double tmp_1;
	if (b <= 9.5e-110) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (0.5 / a) * (-b - sqrt((a * (c * -4.0))));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (-b - (b + (-2.0 * (a / (b / c))))) / (a * 2.0);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = 2.0d0 / ((-b - b) / c)
    if (b <= 9.5d-110) then
        if (b >= 0.0d0) then
            tmp_2 = (0.5d0 / a) * (-b - sqrt((a * (c * (-4.0d0)))))
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (-b - (b + ((-2.0d0) * (a / (b / c))))) / (a * 2.0d0)
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = 2.0 / ((-b - b) / c);
	double tmp_1;
	if (b <= 9.5e-110) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (0.5 / a) * (-b - Math.sqrt((a * (c * -4.0))));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (-b - (b + (-2.0 * (a / (b / c))))) / (a * 2.0);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = 2.0 / ((-b - b) / c)
	tmp_1 = 0
	if b <= 9.5e-110:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (0.5 / a) * (-b - math.sqrt((a * (c * -4.0))))
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = (-b - (b + (-2.0 * (a / (b / c))))) / (a * 2.0)
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(2.0 / Float64(Float64(Float64(-b) - b) / c))
	tmp_1 = 0.0
	if (b <= 9.5e-110)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(0.5 / a) * Float64(Float64(-b) - sqrt(Float64(a * Float64(c * -4.0)))));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(-b) - Float64(b + Float64(-2.0 * Float64(a / Float64(b / c))))) / Float64(a * 2.0));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = 2.0 / ((-b - b) / c);
	tmp_2 = 0.0;
	if (b <= 9.5e-110)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (0.5 / a) * (-b - sqrt((a * (c * -4.0))));
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = (-b - (b + (-2.0 * (a / (b / c))))) / (a * 2.0);
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(2.0 / N[(N[((-b) - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 9.5e-110], If[GreaterEqual[b, 0.0], N[(N[(0.5 / a), $MachinePrecision] * N[((-b) - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[((-b) - N[(b + N[(-2.0 * N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < 9.50000000000000004e-110

   1. Initial program 74.1%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Simplified 73.6%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]

   4. Taylor expanded in b around -inf 76.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

   5. Taylor expanded in b around 0 76.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-*r* 76.1%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(-4 \cdot a\right) \cdot c}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

   7. Simplified 76.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\left(-4 \cdot a\right) \cdot c}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]
   8. Step-by-step derivation

      1. div-sub 76.1%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-b}{a \cdot 2} - \frac{\sqrt{\left(-4 \cdot a\right) \cdot c}}{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

      2. *-commutative 76.1%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a \cdot 2} - \frac{\sqrt{\color{blue}{c \cdot \left(-4 \cdot a\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

      3. *-commutative 76.1%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a \cdot 2} - \frac{\sqrt{c \cdot \color{blue}{\left(a \cdot -4\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

   9. Applied egg-rr 76.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-b}{a \cdot 2} - \frac{\sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]
   10. Step-by-step derivation

       1. div-sub 76.1%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

       2. *-lft-identity 76.1%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

       3. associate-*l/ 76.0%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{1}{a \cdot 2} \cdot \left(\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

       4. *-commutative 76.0%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{1}{\color{blue}{2 \cdot a}} \cdot \left(\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

       5. associate-/r* 76.0%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{\frac{1}{2}}{a}} \cdot \left(\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

       6. metadata-eval 76.0%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{0.5}}{a} \cdot \left(\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

       7. associate-*r* 76.0%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\left(-b\right) - \sqrt{\color{blue}{\left(c \cdot a\right) \cdot -4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

       8. *-commutative 76.0%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\left(-b\right) - \sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

       9. associate-*l* 76.0%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\left(-b\right) - \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

   11. Simplified 76.0%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{0.5}{a} \cdot \left(\left(-b\right) - \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

   if 9.50000000000000004e-110 < b

   1. Initial program 77.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Simplified 77.0%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]

   4. Taylor expanded in b around -inf 77.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

   5. Taylor expanded in b around inf 79.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\left(b + -2 \cdot \frac{a \cdot c}{b}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]
   6. Step-by-step derivation

      1. associate-/l* 84.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b + -2 \cdot \color{blue}{\frac{a}{\frac{b}{c}}}\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]

   7. Simplified 84.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\left(b + -2 \cdot \frac{a}{\frac{b}{c}}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + -1 \cdot b}{c}}\\ \end{array} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{-110}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\left(-b\right) - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - b}{c}}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b + -2 \cdot \frac{a}{\frac{b}{c}}\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - b}{c}}\\ \end{array} \]

Alternative 8: 35.9% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{-2 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0) (/ (- (- b) b) (* a 2.0)) (/ 2.0 (* -2.0 (/ a b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - b) / (a * 2.0);
	} else {
		tmp = 2.0 / (-2.0 * (a / b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b >= 0.0d0) then
        tmp = (-b - b) / (a * 2.0d0)
    else
        tmp = 2.0d0 / ((-2.0d0) * (a / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - b) / (a * 2.0);
	} else {
		tmp = 2.0 / (-2.0 * (a / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = (-b - b) / (a * 2.0)
	else:
		tmp = 2.0 / (-2.0 * (a / b))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(Float64(-b) - b) / Float64(a * 2.0));
	else
		tmp = Float64(2.0 / Float64(-2.0 * Float64(a / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (-b - b) / (a * 2.0);
	else
		tmp = 2.0 / (-2.0 * (a / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(N[((-b) - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(-2.0 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 75.2%

   \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

3. Simplified 74.9%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]

4. Taylor expanded in b around inf 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{b}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ \end{array} \]

5. Taylor expanded in b around inf 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{-2 \cdot \frac{a}{b}}}\\ \end{array} \]

6. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{-2 \cdot \frac{a}{b}}\\ \end{array} \]

Alternative 9: 36.4% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0) (/ (- (- b) b) (* a 2.0)) (/ 2.0 (* 2.0 (/ a b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - b) / (a * 2.0);
	} else {
		tmp = 2.0 / (2.0 * (a / b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b >= 0.0d0) then
        tmp = (-b - b) / (a * 2.0d0)
    else
        tmp = 2.0d0 / (2.0d0 * (a / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - b) / (a * 2.0);
	} else {
		tmp = 2.0 / (2.0 * (a / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = (-b - b) / (a * 2.0)
	else:
		tmp = 2.0 / (2.0 * (a / b))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(Float64(-b) - b) / Float64(a * 2.0));
	else
		tmp = Float64(2.0 / Float64(2.0 * Float64(a / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (-b - b) / (a * 2.0);
	else
		tmp = 2.0 / (2.0 * (a / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(N[((-b) - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(2.0 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 75.2%

   \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

3. Simplified 74.9%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]

4. Taylor expanded in b around inf 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{b}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ \end{array} \]

5. Taylor expanded in b around -inf 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \left(-1 \cdot b + 2 \cdot \frac{a \cdot c}{b}\right)}{c}}\\ \end{array} \]

6. Taylor expanded in b around 0 33.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{2 \cdot \frac{a}{b}}}\\ \end{array} \]

7. Final simplification 33.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{a}{b}}\\ \end{array} \]

Alternative 10: 67.8% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{b \cdot -2}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0) (/ (- (- b) b) (* a 2.0)) (/ 2.0 (/ (* b -2.0) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - b) / (a * 2.0);
	} else {
		tmp = 2.0 / ((b * -2.0) / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b >= 0.0d0) then
        tmp = (-b - b) / (a * 2.0d0)
    else
        tmp = 2.0d0 / ((b * (-2.0d0)) / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - b) / (a * 2.0);
	} else {
		tmp = 2.0 / ((b * -2.0) / c);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = (-b - b) / (a * 2.0)
	else:
		tmp = 2.0 / ((b * -2.0) / c)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(Float64(-b) - b) / Float64(a * 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(b * -2.0) / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (-b - b) / (a * 2.0);
	else
		tmp = 2.0 / ((b * -2.0) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(N[((-b) - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(b * -2.0), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 75.2%

   \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

3. Simplified 74.9%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]

4. Taylor expanded in b around inf 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{b}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ \end{array} \]

5. Taylor expanded in b around -inf 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{-2 \cdot \frac{b}{c}}}\\ \end{array} \]
6. Step-by-step derivation

   1. associate-*r/ 72.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{\frac{-2 \cdot b}{c}}}\\ \end{array} \]

   2. *-commutative 72.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{2}}{\frac{b \cdot -2}{c}}\\ \end{array} \]

7. Simplified 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{\frac{b \cdot -2}{c}}}\\ \end{array} \]

8. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{b \cdot -2}{c}}\\ \end{array} \]

Alternative 11: 68.3% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-b\right) - b\\ \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{t_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (- b) b)))
   (if (>= b 0.0) (/ t_0 (* a 2.0)) (/ (* 2.0 c) t_0))))

C

double code(double a, double b, double c) {
	double t_0 = -b - b;
	double tmp;
	if (b >= 0.0) {
		tmp = t_0 / (a * 2.0);
	} else {
		tmp = (2.0 * c) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -b - b
    if (b >= 0.0d0) then
        tmp = t_0 / (a * 2.0d0)
    else
        tmp = (2.0d0 * c) / t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -b - b;
	double tmp;
	if (b >= 0.0) {
		tmp = t_0 / (a * 2.0);
	} else {
		tmp = (2.0 * c) / t_0;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = -b - b
	tmp = 0
	if b >= 0.0:
		tmp = t_0 / (a * 2.0)
	else:
		tmp = (2.0 * c) / t_0
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-b) - b)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(t_0 / Float64(a * 2.0));
	else
		tmp = Float64(Float64(2.0 * c) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = -b - b;
	tmp = 0.0;
	if (b >= 0.0)
		tmp = t_0 / (a * 2.0);
	else
		tmp = (2.0 * c) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) - b), $MachinePrecision]}, If[GreaterEqual[b, 0.0], N[(t$95$0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Initial program 75.2%

   \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

3. Simplified 74.9%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ } \end{array}} \]

4. Taylor expanded in b around inf 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{b}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ \end{array} \]

5. Taylor expanded in b around -inf 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) + \left(-1 \cdot b + 2 \cdot \frac{a \cdot c}{b}\right)}{c}}\\ \end{array} \]

6. Taylor expanded in a around 0 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{-1 \cdot b - b}\\ \end{array} \]
7. Step-by-step derivation

   1. associate-*r/ 72.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-1 \cdot b - b}\\ \end{array} \]

   2. mul-1-neg 72.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\left(-b\right) - b}\\ \end{array} \]

8. Simplified 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \end{array} \]

9. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \end{array} \]

Reproduce

herbie shell --seed 2023319 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  :precision binary64
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))