jeff quadratic root 2

Percentage Accurate: 72.3% → 90.7%

Time: 31.6s

Alternatives: 8

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + t\_0}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

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

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 = (2.0 * c) / (-b - t_0);
	} else {
		tmp = (-b + t_0) / (2.0 * a);
	}
	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 = (2.0d0 * c) / (-b - t_0)
    else
        tmp = (-b + t_0) / (2.0d0 * a)
    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 = (2.0 * c) / (-b - t_0);
	} else {
		tmp = (-b + t_0) / (2.0 * a);
	}
	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 = (2.0 * c) / (-b - t_0)
	else:
		tmp = (-b + t_0) / (2.0 * a)
	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(2.0 * c) / Float64(Float64(-b) - t_0));
	else
		tmp = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
	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 = (2.0 * c) / (-b - t_0);
	else
		tmp = (-b + t_0) / (2.0 * a);
	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[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Initial Program: 72.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + t\_0}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

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

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 = (2.0 * c) / (-b - t_0);
	} else {
		tmp = (-b + t_0) / (2.0 * a);
	}
	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 = (2.0d0 * c) / (-b - t_0)
    else
        tmp = (-b + t_0) / (2.0d0 * a)
    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 = (2.0 * c) / (-b - t_0);
	} else {
		tmp = (-b + t_0) / (2.0 * a);
	}
	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 = (2.0 * c) / (-b - t_0)
	else:
		tmp = (-b + t_0) / (2.0 * a)
	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(2.0 * c) / Float64(Float64(-b) - t_0));
	else
		tmp = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
	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 = (2.0 * c) / (-b - t_0);
	else
		tmp = (-b + t_0) / (2.0 * a);
	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[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 90.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ \mathbf{if}\;b \leq -1.85 \cdot 10^{+152}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{-2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{+111}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0))))))
   (if (<= b -1.85e+152)
     (if (>= b 0.0) (* c -2.0) (/ (+ b b) (* -2.0 a)))
     (if (<= b 1e+111)
       (if (>= b 0.0) (/ (* c 2.0) (- (- b) t_0)) (/ (- t_0 b) (* a 2.0)))
       (if (>= b 0.0)
         (/ (* c 2.0) (* 2.0 (fma a (/ c b) (- b))))
         (* b (+ (/ c (pow b 2.0)) (/ -1.0 a))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double tmp_1;
	if (b <= -1.85e+152) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * -2.0;
		} else {
			tmp_2 = (b + b) / (-2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1e+111) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c * 2.0) / (-b - t_0);
		} else {
			tmp_3 = (t_0 - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c * 2.0) / (2.0 * fma(a, (c / b), -b));
	} else {
		tmp_1 = b * ((c / pow(b, 2.0)) + (-1.0 / a));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))
	tmp_1 = 0.0
	if (b <= -1.85e+152)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * -2.0);
		else
			tmp_2 = Float64(Float64(b + b) / Float64(-2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1e+111)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(c * 2.0) / Float64(Float64(-b) - t_0));
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(a * 2.0));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c * 2.0) / Float64(2.0 * fma(a, Float64(c / b), Float64(-b))));
	else
		tmp_1 = Float64(b * Float64(Float64(c / (b ^ 2.0)) + Float64(-1.0 / a)));
	end
	return tmp_1
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]}, If[LessEqual[b, -1.85e+152], If[GreaterEqual[b, 0.0], N[(c * -2.0), $MachinePrecision], N[(N[(b + b), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1e+111], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[(2.0 * N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.84999999999999998e152

   1. Initial program 35.7%

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

   3. Simplified 35.9%

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

   5. Taylor expanded in b around -inf 98.2%

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

   6. Taylor expanded in c around 0 98.2%

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

      1. associate-*r/ 98.2%

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

      2. flip-+ 98.2%

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

      3. +-inverses 98.2%

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

      4. +-inverses 98.2%

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

   8. Applied egg-rr 98.2%

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

   10. Simplified 98.2%

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

   if -1.84999999999999998e152 < b < 9.99999999999999957e110

   1. Initial program 85.0%

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

   if 9.99999999999999957e110 < b

   1. Initial program 44.6%

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

   3. Taylor expanded in a around 0 85.0%

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

      1. distribute-lft-out-- 85.0%

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

      2. associate-/l* 95.9%

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

      3. fma-neg 95.9%

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

   5. Simplified 95.9%

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

   6. Taylor expanded in b around -inf 95.9%

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

      1. associate-*r* 95.9%

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

      2. mul-1-neg 95.9%

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

      3. +-commutative 95.9%

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

      4. mul-1-neg 95.9%

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

      5. unsub-neg 95.9%

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

   8. Simplified 95.9%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{+152}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{-2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{+111}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b + b}{-2 \cdot a}\\ \mathbf{if}\;b \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{2} \cdot \frac{2}{\mathsf{fma}\left(a, \frac{c}{b}, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\frac{c}{b + \mathsf{fma}\left(-2, c \cdot \frac{a}{b}, b\right)}}{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (+ b b) (* -2.0 a))))
   (if (<= b -1e+154)
     (if (>= b 0.0) (* c -2.0) t_0)
     (if (<= b -4e-310)
       (if (>= b 0.0)
         (* (/ c 2.0) (/ 2.0 (fma a (/ c b) b)))
         (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)))
       (if (>= b 0.0) (/ (/ c (+ b (fma -2.0 (* c (/ a b)) b))) -0.5) t_0)))))

C

double code(double a, double b, double c) {
	double t_0 = (b + b) / (-2.0 * a);
	double tmp_1;
	if (b <= -1e+154) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * -2.0;
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= -4e-310) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (c / 2.0) * (2.0 / fma(a, (c / b), b));
		} else {
			tmp_3 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c / (b + fma(-2.0, (c * (a / b)), b))) / -0.5;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b + b) / Float64(-2.0 * a))
	tmp_1 = 0.0
	if (b <= -1e+154)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * -2.0);
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b <= -4e-310)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(c / 2.0) * Float64(2.0 / fma(a, Float64(c / b), b)));
		else
			tmp_3 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / Float64(b + fma(-2.0, Float64(c * Float64(a / b)), b))) / -0.5);
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b + b), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1e+154], If[GreaterEqual[b, 0.0], N[(c * -2.0), $MachinePrecision], t$95$0], If[LessEqual[b, -4e-310], If[GreaterEqual[b, 0.0], N[(N[(c / 2.0), $MachinePrecision] * N[(2.0 / N[(a * N[(c / b), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c / N[(b + N[(-2.0 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -0.5), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.00000000000000004e154

   1. Initial program 35.7%

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

   3. Simplified 35.9%

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

   5. Taylor expanded in b around -inf 98.2%

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

   6. Taylor expanded in c around 0 98.2%

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

      1. associate-*r/ 98.2%

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

      2. flip-+ 98.2%

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

      3. +-inverses 98.2%

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

      4. +-inverses 98.2%

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

   8. Applied egg-rr 98.2%

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

   10. Simplified 98.2%

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

   if -1.00000000000000004e154 < b < -3.999999999999988e-310

   1. Initial program 92.3%

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

   3. Taylor expanded in a around 0 92.3%

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

      1. distribute-lft-out-- 92.3%

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

      2. associate-/l* 92.3%

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

      3. fma-neg 92.3%

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

   5. Simplified 92.3%

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

      1. *-commutative 92.3%

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

      2. times-frac 92.3%

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

      3. add-sqr-sqrt 92.3%

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

      4. sqrt-unprod 92.3%

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

      5. sqr-neg 92.3%

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

      6. sqrt-prod 92.3%

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

      7. add-sqr-sqrt 92.3%

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

   7. Applied egg-rr 92.3%

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

   if -3.999999999999988e-310 < b

   1. Initial program 65.0%

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

   3. Simplified 65.0%

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

   5. Taylor expanded in b around -inf 65.0%

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

   6. Taylor expanded in c around 0 54.8%

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

      1. associate-/l* 58.7%

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

   8. Simplified 58.7%

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

      1. clear-num 58.7%

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

      2. un-div-inv 59.0%

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

      3. div-inv 59.0%

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

      4. +-commutative 59.0%

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

      5. fma-define 59.0%

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

      6. clear-num 59.0%

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

      7. un-div-inv 59.0%

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

      8. metadata-eval 59.0%

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

   10. Applied egg-rr 59.0%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c}{\left(b + \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, b\right)\right) \cdot -0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a \cdot -2}\\ \end{array} \]
   11. Step-by-step derivation

       1. associate-/r* 59.0%

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

       2. associate-/r/ 59.1%

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

       3. *-commutative 59.1%

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

   12. Simplified 59.1%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{\frac{c}{b + \mathsf{fma}\left(-2, c \cdot \frac{a}{b}, b\right)}}{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a \cdot -2}\\ \end{array} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{-2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{2} \cdot \frac{2}{\mathsf{fma}\left(a, \frac{c}{b}, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\frac{c}{b + \mathsf{fma}\left(-2, c \cdot \frac{a}{b}, b\right)}}{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{-2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b + b}{-2 \cdot a}\\ \mathbf{if}\;b \leq -9.1 \cdot 10^{+151}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{-189}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\frac{c}{b + \mathsf{fma}\left(-2, c \cdot \frac{a}{b}, b\right)}}{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (+ b b) (* -2.0 a))))
   (if (<= b -9.1e+151)
     (if (>= b 0.0) (* c -2.0) t_0)
     (if (<= b 1e-189)
       (if (>= b 0.0)
         (/ b a)
         (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)))
       (if (>= b 0.0) (/ (/ c (+ b (fma -2.0 (* c (/ a b)) b))) -0.5) t_0)))))

C

double code(double a, double b, double c) {
	double t_0 = (b + b) / (-2.0 * a);
	double tmp_1;
	if (b <= -9.1e+151) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * -2.0;
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1e-189) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = b / a;
		} else {
			tmp_3 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c / (b + fma(-2.0, (c * (a / b)), b))) / -0.5;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b + b) / Float64(-2.0 * a))
	tmp_1 = 0.0
	if (b <= -9.1e+151)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * -2.0);
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b <= 1e-189)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(b / a);
		else
			tmp_3 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c / Float64(b + fma(-2.0, Float64(c * Float64(a / b)), b))) / -0.5);
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b + b), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -9.1e+151], If[GreaterEqual[b, 0.0], N[(c * -2.0), $MachinePrecision], t$95$0], If[LessEqual[b, 1e-189], If[GreaterEqual[b, 0.0], N[(b / a), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c / N[(b + N[(-2.0 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -0.5), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.10000000000000003e151

   1. Initial program 35.7%

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

   3. Simplified 35.9%

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

   5. Taylor expanded in b around -inf 98.2%

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

   6. Taylor expanded in c around 0 98.2%

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

      1. associate-*r/ 98.2%

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

      2. flip-+ 98.2%

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

      3. +-inverses 98.2%

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

      4. +-inverses 98.2%

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

   8. Applied egg-rr 98.2%

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

   10. Simplified 98.2%

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

   if -9.10000000000000003e151 < b < 1.00000000000000007e-189

   1. Initial program 88.7%

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

   3. Taylor expanded in a around 0 77.3%

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

      1. distribute-lft-out-- 77.3%

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

      2. associate-/l* 77.3%

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

      3. fma-neg 77.3%

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

   5. Simplified 77.3%

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

   6. Taylor expanded in c around inf 77.3%

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

   if 1.00000000000000007e-189 < b

   1. Initial program 64.0%

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

   3. Simplified 63.9%

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

   5. Taylor expanded in b around -inf 63.9%

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

   6. Taylor expanded in c around 0 63.1%

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

      1. associate-/l* 67.6%

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

   8. Simplified 67.6%

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

      1. clear-num 67.6%

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

      2. un-div-inv 67.9%

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

      3. div-inv 67.9%

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

      4. +-commutative 67.9%

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

      5. fma-define 67.9%

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

      6. clear-num 67.9%

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

      7. un-div-inv 67.9%

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

      8. metadata-eval 67.9%

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

   10. Applied egg-rr 67.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c}{\left(b + \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, b\right)\right) \cdot -0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a \cdot -2}\\ \end{array} \]
   11. Step-by-step derivation

       1. associate-/r* 67.9%

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

       2. associate-/r/ 68.0%

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

       3. *-commutative 68.0%

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

   12. Simplified 68.0%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{\frac{c}{b + \mathsf{fma}\left(-2, c \cdot \frac{a}{b}, b\right)}}{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a \cdot -2}\\ \end{array} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.1 \cdot 10^{+151}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{-2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{-189}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\frac{c}{b + \mathsf{fma}\left(-2, c \cdot \frac{a}{b}, b\right)}}{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{-2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.1 \cdot 10^{+152}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{-2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.1e+152)
   (if (>= b 0.0) (* c -2.0) (/ (+ b b) (* -2.0 a)))
   (if (>= b 0.0)
     (/ (* c 2.0) (* 2.0 (fma a (/ c b) (- b))))
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -8.1e+152) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * -2.0;
		} else {
			tmp_2 = (b + b) / (-2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (c * 2.0) / (2.0 * fma(a, (c / b), -b));
	} else {
		tmp_1 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -8.1e+152)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * -2.0);
		else
			tmp_2 = Float64(Float64(b + b) / Float64(-2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c * 2.0) / Float64(2.0 * fma(a, Float64(c / b), Float64(-b))));
	else
		tmp_1 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.1e+152], If[GreaterEqual[b, 0.0], N[(c * -2.0), $MachinePrecision], N[(N[(b + b), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[(2.0 * N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -8.09999999999999998e152

   1. Initial program 35.7%

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

   3. Simplified 35.9%

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

   5. Taylor expanded in b around -inf 98.2%

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

   6. Taylor expanded in c around 0 98.2%

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

      1. associate-*r/ 98.2%

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

      2. flip-+ 98.2%

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

      3. +-inverses 98.2%

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

      4. +-inverses 98.2%

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

   8. Applied egg-rr 98.2%

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

   10. Simplified 98.2%

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

   if -8.09999999999999998e152 < b

   1. Initial program 76.1%

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

   3. Taylor expanded in a around 0 70.2%

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

      1. distribute-lft-out-- 70.2%

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

      2. associate-/l* 72.5%

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

      3. fma-neg 72.5%

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

   5. Simplified 72.5%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.1 \cdot 10^{+152}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{-2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{c}{b + \mathsf{fma}\left(-2, c \cdot \frac{a}{b}, b\right)}}{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{-2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0)
   (/ (/ c (+ b (fma -2.0 (* c (/ a b)) b))) -0.5)
   (/ (+ b b) (* -2.0 a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (c / (b + fma(-2.0, (c * (a / b)), b))) / -0.5;
	} else {
		tmp = (b + b) / (-2.0 * a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(c / Float64(b + fma(-2.0, Float64(c * Float64(a / b)), b))) / -0.5);
	else
		tmp = Float64(Float64(b + b) / Float64(-2.0 * a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(N[(c / N[(b + N[(-2.0 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -0.5), $MachinePrecision], N[(N[(b + b), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 67.9%

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

3. Simplified 67.9%

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

5. Taylor expanded in b around -inf 71.5%

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

6. Taylor expanded in c around 0 66.7%

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

   1. associate-/l* 68.5%

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

8. Simplified 68.5%

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

   1. clear-num 68.5%

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

   2. un-div-inv 68.6%

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

   3. div-inv 68.6%

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

   4. +-commutative 68.6%

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

   5. fma-define 68.6%

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

   6. clear-num 68.6%

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

   7. un-div-inv 68.6%

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

   8. metadata-eval 68.6%

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

10. Applied egg-rr 68.6%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c}{\left(b + \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, b\right)\right) \cdot -0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - -1 \cdot b}{a \cdot -2}\\ \end{array} \]
11. Step-by-step derivation

    1. associate-/r* 68.6%

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

    2. associate-/r/ 68.7%

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

    3. *-commutative 68.7%

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

12. Simplified 68.7%

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

13. Final simplification 68.7%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\frac{c}{b + \mathsf{fma}\left(-2, c \cdot \frac{a}{b}, b\right)}}{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{-2 \cdot a}\\ \end{array} \]
14. Add Preprocessing

Alternative 6: 36.4% accurate, 10.1× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = c * -2.0;
	} else {
		tmp = (b + b) / (-2.0 * a);
	}
	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 = c * (-2.0d0)
    else
        tmp = (b + b) / ((-2.0d0) * a)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

3. Simplified 67.9%

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

5. Taylor expanded in b around -inf 71.5%

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

6. Taylor expanded in c around 0 68.5%

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

   1. associate-*r/ 68.6%

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

   2. flip-+ 40.8%

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

   3. +-inverses 40.8%

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

   4. +-inverses 40.8%

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

8. Applied egg-rr 40.8%

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

10. Simplified 43.6%

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

11. Final simplification 43.6%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{-2 \cdot a}\\ \end{array} \]
12. Add Preprocessing

Alternative 7: 67.4% accurate, 10.1× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = c * (-2.0 / (b + b));
	} else {
		tmp = (b + b) / (-2.0 * a);
	}
	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 = c * ((-2.0d0) / (b + b))
    else
        tmp = (b + b) / ((-2.0d0) * a)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

3. Simplified 67.9%

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

5. Taylor expanded in b around -inf 71.5%

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

6. Taylor expanded in c around 0 68.5%

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

7. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{-2 \cdot a}\\ \end{array} \]
8. Add Preprocessing

Alternative 8: 67.5% accurate, 10.1× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = c / -b;
	} else {
		tmp = (b + b) / (-2.0 * a);
	}
	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 = c / -b
    else
        tmp = (b + b) / ((-2.0d0) * a)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

3. Simplified 67.9%

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

5. Taylor expanded in b around -inf 71.5%

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

6. Taylor expanded in c around 0 68.6%

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

   1. associate-*r/ 68.6%

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

   2. mul-1-neg 68.6%

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

8. Simplified 68.6%

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

9. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + b}{-2 \cdot a}\\ \end{array} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024060 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))