jeff quadratic root 2

Percentage Accurate: 72.2% → 90.6%

Time: 24.1s

Alternatives: 9

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: 72.2% 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: 90.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\\ t_1 := b + \mathsf{fma}\left(a \cdot \frac{c}{b}, -2, b\right)\\ \mathbf{if}\;b \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, 2, a \cdot \frac{c \cdot -2}{b}\right)}{-2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.85 \cdot 10^{+107}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \left(b - 2 \cdot \frac{c \cdot a}{b}\right)}{-2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma b b (* a (* c -4.0)))))
        (t_1 (+ b (fma (* a (/ c b)) -2.0 b))))
   (if (<= b -2e+125)
     (if (>= b 0.0)
       (* c (/ -2.0 t_1))
       (/ (fma b 2.0 (* a (/ (* c -2.0) b))) (* -2.0 a)))
     (if (<= b 2.85e+107)
       (if (>= b 0.0) (/ (* c 2.0) (- (- b) t_0)) (/ (- t_0 b) (* a 2.0)))
       (if (>= b 0.0)
         (/ (* c -2.0) t_1)
         (/ (+ b (- b (* 2.0 (/ (* c a) b)))) (* -2.0 a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma(b, b, (a * (c * -4.0))));
	double t_1 = b + fma((a * (c / b)), -2.0, b);
	double tmp_1;
	if (b <= -2e+125) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / t_1);
		} else {
			tmp_2 = fma(b, 2.0, (a * ((c * -2.0) / b))) / (-2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 2.85e+107) {
		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) / t_1;
	} else {
		tmp_1 = (b + (b - (2.0 * ((c * a) / b)))) / (-2.0 * a);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(b, b, Float64(a * Float64(c * -4.0))))
	t_1 = Float64(b + fma(Float64(a * Float64(c / b)), -2.0, b))
	tmp_1 = 0.0
	if (b <= -2e+125)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * Float64(-2.0 / t_1));
		else
			tmp_2 = Float64(fma(b, 2.0, Float64(a * Float64(Float64(c * -2.0) / b))) / Float64(-2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 2.85e+107)
		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) / t_1);
	else
		tmp_1 = Float64(Float64(b + Float64(b - Float64(2.0 * Float64(Float64(c * a) / b)))) / Float64(-2.0 * a));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(b + N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] * -2.0 + b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2e+125], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(b * 2.0 + N[(a * N[(N[(c * -2.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 2.85e+107], 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] / t$95$1), $MachinePrecision], N[(N[(b + N[(b - N[(2.0 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.9999999999999998e125

   1. Initial program 49.1%

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

   3. Simplified 49.3%

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

   5. Taylor expanded in c around 0 49.3%

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

      1. +-commutative 49.3%

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

      2. *-commutative 49.3%

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

      3. fma-define 49.3%

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

      4. associate-/l* 49.3%

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

   7. Simplified 49.3%

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

   8. Taylor expanded in b around -inf 88.3%

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

      1. +-commutative 88.3%

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

      2. *-commutative 88.3%

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

      3. fma-define 88.3%

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

      4. *-commutative 88.3%

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

      5. associate-/l* 95.5%

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

      6. associate-*l* 95.5%

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

      7. *-commutative 95.5%

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

      8. associate-*r/ 95.5%

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

   10. Simplified 95.5%

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

   if -1.9999999999999998e125 < b < 2.84999999999999986e107

   1. Initial program 89.7%

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

   3. Simplified 89.7%

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

   if 2.84999999999999986e107 < b

   1. Initial program 50.2%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in c around 0 88.9%

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

      1. +-commutative 88.9%

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

      2. *-commutative 88.9%

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

      3. fma-define 88.9%

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

      4. associate-/l* 99.8%

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

   7. Simplified 99.8%

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

      1. associate-*r/ 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in b around -inf 100.0%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \mathsf{fma}\left(a \cdot \frac{c}{b}, -2, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, 2, a \cdot \frac{c \cdot -2}{b}\right)}{-2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.85 \cdot 10^{+107}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + \mathsf{fma}\left(a \cdot \frac{c}{b}, -2, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \left(b - 2 \cdot \frac{c \cdot a}{b}\right)}{-2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\\ t_1 := b + \mathsf{fma}\left(a \cdot \frac{c}{b}, -2, b\right)\\ \mathbf{if}\;b \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, 2, a \cdot \frac{c \cdot -2}{b}\right)}{-2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{+107}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + t\_0}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b - t\_0}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \left(b - 2 \cdot \frac{c \cdot a}{b}\right)}{-2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma c (* a -4.0) (* b b))))
        (t_1 (+ b (fma (* a (/ c b)) -2.0 b))))
   (if (<= b -2e+125)
     (if (>= b 0.0)
       (* c (/ -2.0 t_1))
       (/ (fma b 2.0 (* a (/ (* c -2.0) b))) (* -2.0 a)))
     (if (<= b 3.7e+107)
       (if (>= b 0.0) (* -2.0 (/ c (+ b t_0))) (* -0.5 (/ (- b t_0) a)))
       (if (>= b 0.0)
         (/ (* c -2.0) t_1)
         (/ (+ b (- b (* 2.0 (/ (* c a) b)))) (* -2.0 a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma(c, (a * -4.0), (b * b)));
	double t_1 = b + fma((a * (c / b)), -2.0, b);
	double tmp_1;
	if (b <= -2e+125) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / t_1);
		} else {
			tmp_2 = fma(b, 2.0, (a * ((c * -2.0) / b))) / (-2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 3.7e+107) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -2.0 * (c / (b + t_0));
		} else {
			tmp_3 = -0.5 * ((b - t_0) / a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c * -2.0) / t_1;
	} else {
		tmp_1 = (b + (b - (2.0 * ((c * a) / b)))) / (-2.0 * a);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(c, Float64(a * -4.0), Float64(b * b)))
	t_1 = Float64(b + fma(Float64(a * Float64(c / b)), -2.0, b))
	tmp_1 = 0.0
	if (b <= -2e+125)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * Float64(-2.0 / t_1));
		else
			tmp_2 = Float64(fma(b, 2.0, Float64(a * Float64(Float64(c * -2.0) / b))) / Float64(-2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 3.7e+107)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(-2.0 * Float64(c / Float64(b + t_0)));
		else
			tmp_3 = Float64(-0.5 * Float64(Float64(b - t_0) / a));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c * -2.0) / t_1);
	else
		tmp_1 = Float64(Float64(b + Float64(b - Float64(2.0 * Float64(Float64(c * a) / b)))) / Float64(-2.0 * a));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(b + N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] * -2.0 + b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2e+125], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(b * 2.0 + N[(a * N[(N[(c * -2.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 3.7e+107], If[GreaterEqual[b, 0.0], N[(-2.0 * N[(c / N[(b + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(b - t$95$0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c * -2.0), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(b + N[(b - N[(2.0 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.9999999999999998e125

   1. Initial program 49.1%

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

   3. Simplified 49.3%

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

   5. Taylor expanded in c around 0 49.3%

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

      1. +-commutative 49.3%

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

      2. *-commutative 49.3%

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

      3. fma-define 49.3%

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

      4. associate-/l* 49.3%

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

   7. Simplified 49.3%

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

   8. Taylor expanded in b around -inf 88.3%

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

      1. +-commutative 88.3%

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

      2. *-commutative 88.3%

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

      3. fma-define 88.3%

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

      4. *-commutative 88.3%

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

      5. associate-/l* 95.5%

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

      6. associate-*l* 95.5%

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

      7. *-commutative 95.5%

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

      8. associate-*r/ 95.5%

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

   10. Simplified 95.5%

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

   if -1.9999999999999998e125 < b < 3.7e107

   1. Initial program 89.7%

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

   3. Simplified 89.7%

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

   if 3.7e107 < b

   1. Initial program 50.2%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in c around 0 88.9%

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

      1. +-commutative 88.9%

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

      2. *-commutative 88.9%

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

      3. fma-define 88.9%

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

      4. associate-/l* 99.8%

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

   7. Simplified 99.8%

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

      1. associate-*r/ 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in b around -inf 100.0%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \mathsf{fma}\left(a \cdot \frac{c}{b}, -2, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, 2, a \cdot \frac{c \cdot -2}{b}\right)}{-2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{+107}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + \mathsf{fma}\left(a \cdot \frac{c}{b}, -2, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \left(b - 2 \cdot \frac{c \cdot a}{b}\right)}{-2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ t_1 := b + \mathsf{fma}\left(a \cdot \frac{c}{b}, -2, b\right)\\ \mathbf{if}\;b \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, 2, a \cdot \frac{c \cdot -2}{b}\right)}{-2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{+107}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \left(b - 2 \cdot \frac{c \cdot a}{b}\right)}{-2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* a 4.0)))))
        (t_1 (+ b (fma (* a (/ c b)) -2.0 b))))
   (if (<= b -2e+125)
     (if (>= b 0.0)
       (* c (/ -2.0 t_1))
       (/ (fma b 2.0 (* a (/ (* c -2.0) b))) (* -2.0 a)))
     (if (<= b 3e+107)
       (if (>= b 0.0) (/ (* c 2.0) (- (- b) t_0)) (/ (- t_0 b) (* a 2.0)))
       (if (>= b 0.0)
         (/ (* c -2.0) t_1)
         (/ (+ b (- b (* 2.0 (/ (* c a) b)))) (* -2.0 a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (a * 4.0))));
	double t_1 = b + fma((a * (c / b)), -2.0, b);
	double tmp_1;
	if (b <= -2e+125) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / t_1);
		} else {
			tmp_2 = fma(b, 2.0, (a * ((c * -2.0) / b))) / (-2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 3e+107) {
		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) / t_1;
	} else {
		tmp_1 = (b + (b - (2.0 * ((c * a) / b)))) / (-2.0 * a);
	}
	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(b + fma(Float64(a * Float64(c / b)), -2.0, b))
	tmp_1 = 0.0
	if (b <= -2e+125)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * Float64(-2.0 / t_1));
		else
			tmp_2 = Float64(fma(b, 2.0, Float64(a * Float64(Float64(c * -2.0) / b))) / Float64(-2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 3e+107)
		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) / t_1);
	else
		tmp_1 = Float64(Float64(b + Float64(b - Float64(2.0 * Float64(Float64(c * a) / b)))) / Float64(-2.0 * 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]}, Block[{t$95$1 = N[(b + N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] * -2.0 + b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2e+125], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(b * 2.0 + N[(a * N[(N[(c * -2.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 3e+107], 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] / t$95$1), $MachinePrecision], N[(N[(b + N[(b - N[(2.0 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.9999999999999998e125

   1. Initial program 49.1%

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

   3. Simplified 49.3%

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

   5. Taylor expanded in c around 0 49.3%

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

      1. +-commutative 49.3%

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

      2. *-commutative 49.3%

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

      3. fma-define 49.3%

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

      4. associate-/l* 49.3%

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

   7. Simplified 49.3%

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

   8. Taylor expanded in b around -inf 88.3%

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

      1. +-commutative 88.3%

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

      2. *-commutative 88.3%

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

      3. fma-define 88.3%

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

      4. *-commutative 88.3%

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

      5. associate-/l* 95.5%

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

      6. associate-*l* 95.5%

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

      7. *-commutative 95.5%

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

      8. associate-*r/ 95.5%

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

   10. Simplified 95.5%

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

   if -1.9999999999999998e125 < b < 3.00000000000000023e107

   1. Initial program 89.7%

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

   if 3.00000000000000023e107 < b

   1. Initial program 50.2%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in c around 0 88.9%

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

      1. +-commutative 88.9%

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

      2. *-commutative 88.9%

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

      3. fma-define 88.9%

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

      4. associate-/l* 99.8%

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

   7. Simplified 99.8%

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

      1. associate-*r/ 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in b around -inf 100.0%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \mathsf{fma}\left(a \cdot \frac{c}{b}, -2, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, 2, a \cdot \frac{c \cdot -2}{b}\right)}{-2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{+107}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + \mathsf{fma}\left(a \cdot \frac{c}{b}, -2, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \left(b - 2 \cdot \frac{c \cdot a}{b}\right)}{-2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.6 \cdot 10^{+107}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + \mathsf{fma}\left(a \cdot \frac{c}{b}, -2, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \left(b - 2 \cdot \frac{c \cdot a}{b}\right)}{-2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.6e+107)
   (if (>= b 0.0)
     (/ (* c 2.0) (- (- b) (sqrt (- (* b b) (* c (* a 4.0))))))
     (/ (* b -2.0) (* a 2.0)))
   (if (>= b 0.0)
     (/ (* c -2.0) (+ b (fma (* a (/ c b)) -2.0 b)))
     (/ (+ b (- b (* 2.0 (/ (* c a) b)))) (* -2.0 a)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= 3.6e+107) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c * 2.0) / (-b - sqrt(((b * b) - (c * (a * 4.0)))));
		} else {
			tmp_2 = (b * -2.0) / (a * 2.0);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (c * -2.0) / (b + fma((a * (c / b)), -2.0, b));
	} else {
		tmp_1 = (b + (b - (2.0 * ((c * a) / b)))) / (-2.0 * a);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= 3.6e+107)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(c * 2.0) / Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0))))));
		else
			tmp_2 = Float64(Float64(b * -2.0) / Float64(a * 2.0));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c * -2.0) / Float64(b + fma(Float64(a * Float64(c / b)), -2.0, b)));
	else
		tmp_1 = Float64(Float64(b + Float64(b - Float64(2.0 * Float64(Float64(c * a) / b)))) / Float64(-2.0 * a));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.6e+107], If[GreaterEqual[b, 0.0], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * -2.0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c * -2.0), $MachinePrecision] / N[(b + N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] * -2.0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + N[(b - N[(2.0 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.5999999999999998e107

   1. Initial program 79.5%

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

   3. Taylor expanded in b around -inf 74.9%

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

      1. *-commutative 74.9%

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

   5. Simplified 74.9%

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

   if 3.5999999999999998e107 < b

   1. Initial program 50.2%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in c around 0 88.9%

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

      1. +-commutative 88.9%

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

      2. *-commutative 88.9%

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

      3. fma-define 88.9%

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

      4. associate-/l* 99.8%

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

   7. Simplified 99.8%

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

      1. associate-*r/ 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in b around -inf 100.0%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.6 \cdot 10^{+107}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{b + \mathsf{fma}\left(a \cdot \frac{c}{b}, -2, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \left(b - 2 \cdot \frac{c \cdot a}{b}\right)}{-2 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0)
   (* c (/ -2.0 (+ b (fma (* a (/ c b)) -2.0 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 + fma((a * (c / b)), -2.0, 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 + fma(Float64(a * Float64(c / b)), -2.0, b))));
	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[(c * N[(-2.0 / N[(b + N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] * -2.0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + b), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 73.4%

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

3. Simplified 73.4%

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

5. Taylor expanded in c around 0 68.5%

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

   1. +-commutative 68.5%

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

   2. *-commutative 68.5%

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

   3. fma-define 68.5%

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

   4. associate-/l* 70.7%

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

7. Simplified 70.7%

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

8. Taylor expanded in b around -inf 67.0%

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

9. Final simplification 67.0%

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

Alternative 6: 67.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

3. Simplified 73.4%

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

5. Taylor expanded in c around 0 68.5%

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

   1. +-commutative 68.5%

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

   2. *-commutative 68.5%

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

   3. fma-define 68.5%

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

   4. associate-/l* 70.7%

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

7. Simplified 70.7%

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

8. Taylor expanded in b around -inf 66.1%

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

   1. +-commutative 66.1%

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

   2. *-commutative 66.1%

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

   3. fma-define 66.1%

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

   4. *-commutative 66.1%

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

   5. associate-/l* 67.6%

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

   6. associate-*l* 67.6%

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

   7. *-commutative 67.6%

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

   8. associate-*r/ 67.6%

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

10. Simplified 67.6%

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

11. Final simplification 67.6%

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

Alternative 7: 67.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

3. Simplified 73.4%

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

5. Taylor expanded in c around 0 68.5%

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

   1. +-commutative 68.5%

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

   2. *-commutative 68.5%

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

   3. fma-define 68.5%

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

   4. associate-/l* 70.7%

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

7. Simplified 70.7%

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

   1. associate-*r/ 70.8%

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

9. Applied egg-rr 70.8%

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

10. Taylor expanded in b around -inf 66.2%

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

    1. +-commutative 66.1%

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

    2. *-commutative 66.1%

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

    3. fma-define 66.1%

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

    4. *-commutative 66.1%

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

    5. associate-/l* 67.6%

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

    6. associate-*l* 67.6%

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

    7. *-commutative 67.6%

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

    8. associate-*r/ 67.6%

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

12. Simplified 67.7%

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

13. Final simplification 67.7%

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

Alternative 8: 67.1% accurate, 10.1× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0) (* c (/ -1.0 b)) (/ (+ b b) (* -2.0 a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = c * (-1.0 / 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 * ((-1.0d0) / 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 * (-1.0 / b);
	} else {
		tmp = (b + b) / (-2.0 * a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = c * (-1.0 / 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(-1.0 / 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 * (-1.0 / 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[(-1.0 / b), $MachinePrecision]), $MachinePrecision], N[(N[(b + b), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 73.4%

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

3. Simplified 73.4%

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

5. Taylor expanded in c around 0 68.5%

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

   1. +-commutative 68.5%

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

   2. *-commutative 68.5%

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

   3. fma-define 68.5%

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

   4. associate-/l* 70.7%

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

7. Simplified 70.7%

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

8. Taylor expanded in b around -inf 67.0%

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

9. Taylor expanded in b around inf 66.7%

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

10. Final simplification 66.7%

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

Alternative 9: 67.2% 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 73.4%

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

3. Simplified 73.4%

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

5. Taylor expanded in c around 0 68.5%

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

   1. +-commutative 68.5%

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

   2. *-commutative 68.5%

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

   3. fma-define 68.5%

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

   4. associate-/l* 70.7%

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

7. Simplified 70.7%

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

8. Taylor expanded in b around -inf 67.0%

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

9. Taylor expanded in c around 0 66.8%

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

    1. mul-1-neg 66.8%

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

    2. distribute-neg-frac2 66.8%

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

11. Simplified 66.8%

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

12. Final simplification 66.8%

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

Reproduce

herbie shell --seed 2024050 
(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))))