jeff quadratic root 2

Percentage Accurate: 72.4% → 90.1%

Time: 11.4s

Alternatives: 14

Speedup: 1.1×

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: 72.4% 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: 90.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+142}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-2\right) \cdot c}{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(4 \cdot {\left(\frac{\frac{c}{b} \cdot a}{b}\right)}^{2} - 4\right) \cdot \left(-b\right)}{\left(\frac{-2}{b} \cdot a\right) \cdot \frac{c}{b} - 2}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+34}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}{a} - \frac{b}{a}\right) \cdot 0.5\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{-2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+142)
   (if (>= b 0.0)
     (/ (* (- 2.0) c) (+ b (sqrt (- (* b b) (* (* 4.0 a) c)))))
     (/
      (/
       (* (- (* 4.0 (pow (/ (* (/ c b) a) b) 2.0)) 4.0) (- b))
       (- (* (* (/ -2.0 b) a) (/ c b)) 2.0))
      (* 2.0 a)))
   (if (<= b 6e+34)
     (if (>= b 0.0)
       (/ (* -2.0 c) (+ (sqrt (fma (* c a) -4.0 (* b b))) b))
       (* (- (/ (sqrt (fma -4.0 (* c a) (* b b))) a) (/ b a)) 0.5))
     (/ (* -2.0 c) (* -2.0 (- (* a (/ c b)) b))))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1e+142) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-2.0 * c) / (b + sqrt(((b * b) - ((4.0 * a) * c))));
		} else {
			tmp_2 = ((((4.0 * pow((((c / b) * a) / b), 2.0)) - 4.0) * -b) / ((((-2.0 / b) * a) * (c / b)) - 2.0)) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 6e+34) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-2.0 * c) / (sqrt(fma((c * a), -4.0, (b * b))) + b);
		} else {
			tmp_3 = ((sqrt(fma(-4.0, (c * a), (b * b))) / a) - (b / a)) * 0.5;
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = (-2.0 * c) / (-2.0 * ((a * (c / b)) - b));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -1e+142)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(Float64(-2.0) * c) / Float64(b + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))));
		else
			tmp_2 = Float64(Float64(Float64(Float64(Float64(4.0 * (Float64(Float64(Float64(c / b) * a) / b) ^ 2.0)) - 4.0) * Float64(-b)) / Float64(Float64(Float64(Float64(-2.0 / b) * a) * Float64(c / b)) - 2.0)) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 6e+34)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-2.0 * c) / Float64(sqrt(fma(Float64(c * a), -4.0, Float64(b * b))) + b));
		else
			tmp_3 = Float64(Float64(Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) / a) - Float64(b / a)) * 0.5);
		end
		tmp_1 = tmp_3;
	else
		tmp_1 = Float64(Float64(-2.0 * c) / Float64(-2.0 * Float64(Float64(a * Float64(c / b)) - b)));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+142], If[GreaterEqual[b, 0.0], N[(N[((-2.0) * c), $MachinePrecision] / N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(4.0 * N[Power[N[(N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 4.0), $MachinePrecision] * (-b)), $MachinePrecision] / N[(N[(N[(N[(-2.0 / b), $MachinePrecision] * a), $MachinePrecision] * N[(c / b), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 6e+34], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * c), $MachinePrecision] / N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], N[(N[(-2.0 * c), $MachinePrecision] / N[(-2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.00000000000000005e142

   1. Initial program 28.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 96.0

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

   5. Applied rewrites 96.0%

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

   7. Applied rewrites 96.0%

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

   if -1.00000000000000005e142 < b < 6.00000000000000037e34

   1. Initial program 85.3%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 85.3%

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

   7. Applied rewrites 85.3%

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

   if 6.00000000000000037e34 < b

   1. Initial program 55.7%

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

   4. Applied rewrites 55.7%

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

   5. Taylor expanded in a around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. if-same N/A

         \[\leadsto \color{blue}{-2 \cdot \frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}} \]

      8. +-commutative N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

   7. Applied rewrites 55.7%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{-2 \cdot c}{2 \cdot \color{blue}{b}} \]
   9. Step-by-step derivation

   10. Applied rewrites 93.3%

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

   11. Taylor expanded in a around 0

       \[\leadsto \frac{-2 \cdot c}{-2 \cdot \frac{a \cdot c}{b} + \color{blue}{2 \cdot b}} \]
   12. Step-by-step derivation

   13. Applied rewrites 93.7%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+142}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-2\right) \cdot c}{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(4 \cdot {\left(\frac{\frac{c}{b} \cdot a}{b}\right)}^{2} - 4\right) \cdot \left(-b\right)}{\left(\frac{-2}{b} \cdot a\right) \cdot \frac{c}{b} - 2}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+34}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}{a} - \frac{b}{a}\right) \cdot 0.5\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{-2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \frac{c}{b}\\ \mathbf{if}\;b \leq -6.6 \cdot 10^{+141}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-2\right) \cdot c}{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-2}{b}, t\_0, 2\right) \cdot \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+34}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}{a} - \frac{b}{a}\right) \cdot 0.5\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{-2 \cdot \left(t\_0 - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (/ c b))))
   (if (<= b -6.6e+141)
     (if (>= b 0.0)
       (/ (* (- 2.0) c) (+ b (sqrt (- (* b b) (* (* 4.0 a) c)))))
       (/ (* (fma (/ -2.0 b) t_0 2.0) (- b)) (* 2.0 a)))
     (if (<= b 6e+34)
       (if (>= b 0.0)
         (/ (* -2.0 c) (+ (sqrt (fma (* c a) -4.0 (* b b))) b))
         (* (- (/ (sqrt (fma -4.0 (* c a) (* b b))) a) (/ b a)) 0.5))
       (/ (* -2.0 c) (* -2.0 (- t_0 b)))))))

C

double code(double a, double b, double c) {
	double t_0 = a * (c / b);
	double tmp_1;
	if (b <= -6.6e+141) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-2.0 * c) / (b + sqrt(((b * b) - ((4.0 * a) * c))));
		} else {
			tmp_2 = (fma((-2.0 / b), t_0, 2.0) * -b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 6e+34) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-2.0 * c) / (sqrt(fma((c * a), -4.0, (b * b))) + b);
		} else {
			tmp_3 = ((sqrt(fma(-4.0, (c * a), (b * b))) / a) - (b / a)) * 0.5;
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = (-2.0 * c) / (-2.0 * (t_0 - b));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(c / b))
	tmp_1 = 0.0
	if (b <= -6.6e+141)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(Float64(-2.0) * c) / Float64(b + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))));
		else
			tmp_2 = Float64(Float64(fma(Float64(-2.0 / b), t_0, 2.0) * Float64(-b)) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 6e+34)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-2.0 * c) / Float64(sqrt(fma(Float64(c * a), -4.0, Float64(b * b))) + b));
		else
			tmp_3 = Float64(Float64(Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) / a) - Float64(b / a)) * 0.5);
		end
		tmp_1 = tmp_3;
	else
		tmp_1 = Float64(Float64(-2.0 * c) / Float64(-2.0 * Float64(t_0 - b)));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -6.6e+141], If[GreaterEqual[b, 0.0], N[(N[((-2.0) * c), $MachinePrecision] / N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-2.0 / b), $MachinePrecision] * t$95$0 + 2.0), $MachinePrecision] * (-b)), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 6e+34], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * c), $MachinePrecision] / N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], N[(N[(-2.0 * c), $MachinePrecision] / N[(-2.0 * N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.5999999999999993e141

   1. Initial program 28.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 96.0

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

   5. Applied rewrites 96.0%

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

   if -6.5999999999999993e141 < b < 6.00000000000000037e34

   1. Initial program 85.3%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 85.3%

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

   7. Applied rewrites 85.3%

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

   if 6.00000000000000037e34 < b

   1. Initial program 55.7%

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

   4. Applied rewrites 55.7%

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

   5. Taylor expanded in a around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. if-same N/A

         \[\leadsto \color{blue}{-2 \cdot \frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}} \]

      8. +-commutative N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

   7. Applied rewrites 55.7%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{-2 \cdot c}{2 \cdot \color{blue}{b}} \]
   9. Step-by-step derivation

   10. Applied rewrites 93.3%

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

   11. Taylor expanded in a around 0

       \[\leadsto \frac{-2 \cdot c}{-2 \cdot \frac{a \cdot c}{b} + \color{blue}{2 \cdot b}} \]
   12. Step-by-step derivation

   13. Applied rewrites 93.7%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{+141}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-2\right) \cdot c}{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-2}{b}, a \cdot \frac{c}{b}, 2\right) \cdot \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+34}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}{a} - \frac{b}{a}\right) \cdot 0.5\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{-2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{+82}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+34}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}{a} - \frac{b}{a}\right) \cdot 0.5\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{-2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.2e+82)
   (if (>= b 0.0) (/ (- b) a) (/ (+ (- b) (- b)) (* 2.0 a)))
   (if (<= b 6e+34)
     (if (>= b 0.0)
       (/ (* -2.0 c) (+ (sqrt (fma (* c a) -4.0 (* b b))) b))
       (* (- (/ (sqrt (fma -4.0 (* c a) (* b b))) a) (/ b a)) 0.5))
     (/ (* -2.0 c) (* -2.0 (- (* a (/ c b)) b))))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -6.2e+82) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -b / a;
		} else {
			tmp_2 = (-b + -b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 6e+34) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-2.0 * c) / (sqrt(fma((c * a), -4.0, (b * b))) + b);
		} else {
			tmp_3 = ((sqrt(fma(-4.0, (c * a), (b * b))) / a) - (b / a)) * 0.5;
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = (-2.0 * c) / (-2.0 * ((a * (c / b)) - b));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -6.2e+82)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-b) / a);
		else
			tmp_2 = Float64(Float64(Float64(-b) + Float64(-b)) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 6e+34)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-2.0 * c) / Float64(sqrt(fma(Float64(c * a), -4.0, Float64(b * b))) + b));
		else
			tmp_3 = Float64(Float64(Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) / a) - Float64(b / a)) * 0.5);
		end
		tmp_1 = tmp_3;
	else
		tmp_1 = Float64(Float64(-2.0 * c) / Float64(-2.0 * Float64(Float64(a * Float64(c / b)) - b)));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.2e+82], If[GreaterEqual[b, 0.0], N[((-b) / a), $MachinePrecision], N[(N[((-b) + (-b)), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 6e+34], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * c), $MachinePrecision] / N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], N[(N[(-2.0 * c), $MachinePrecision] / N[(-2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.20000000000000065e82

   1. Initial program 44.4%

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

   3. Taylor expanded in a around 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-- N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 44.4

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

   5. Applied rewrites 44.4%

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

   6. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 95.2

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

   8. Applied rewrites 95.2%

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

   9. Taylor expanded in b around -inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 95.2

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

   11. Applied rewrites 95.2%

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

   if -6.20000000000000065e82 < b < 6.00000000000000037e34

   1. Initial program 83.9%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 83.9%

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

   7. Applied rewrites 83.9%

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

   if 6.00000000000000037e34 < b

   1. Initial program 55.7%

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

   4. Applied rewrites 55.7%

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

   5. Taylor expanded in a around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. if-same N/A

         \[\leadsto \color{blue}{-2 \cdot \frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}} \]

      8. +-commutative N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

   7. Applied rewrites 55.7%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{-2 \cdot c}{2 \cdot \color{blue}{b}} \]
   9. Step-by-step derivation

   10. Applied rewrites 93.3%

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

   11. Taylor expanded in a around 0

       \[\leadsto \frac{-2 \cdot c}{-2 \cdot \frac{a \cdot c}{b} + \color{blue}{2 \cdot b}} \]
   12. Step-by-step derivation

   13. Applied rewrites 93.7%

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

Alternative 4: 90.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}\\ \mathbf{if}\;b \leq -6.2 \cdot 10^{+82}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+34}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{t\_0 + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{-2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* c a) -4.0 (* b b)))))
   (if (<= b -6.2e+82)
     (if (>= b 0.0) (/ (- b) a) (/ (+ (- b) (- b)) (* 2.0 a)))
     (if (<= b 6e+34)
       (if (>= b 0.0) (/ (* -2.0 c) (+ t_0 b)) (* (/ (- t_0 b) a) 0.5))
       (/ (* -2.0 c) (* -2.0 (- (* a (/ c b)) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((c * a), -4.0, (b * b)));
	double tmp_1;
	if (b <= -6.2e+82) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -b / a;
		} else {
			tmp_2 = (-b + -b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 6e+34) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-2.0 * c) / (t_0 + b);
		} else {
			tmp_3 = ((t_0 - b) / a) * 0.5;
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = (-2.0 * c) / (-2.0 * ((a * (c / b)) - b));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(c * a), -4.0, Float64(b * b)))
	tmp_1 = 0.0
	if (b <= -6.2e+82)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-b) / a);
		else
			tmp_2 = Float64(Float64(Float64(-b) + Float64(-b)) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 6e+34)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-2.0 * c) / Float64(t_0 + b));
		else
			tmp_3 = Float64(Float64(Float64(t_0 - b) / a) * 0.5);
		end
		tmp_1 = tmp_3;
	else
		tmp_1 = Float64(Float64(-2.0 * c) / Float64(-2.0 * Float64(Float64(a * Float64(c / b)) - b)));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -6.2e+82], If[GreaterEqual[b, 0.0], N[((-b) / a), $MachinePrecision], N[(N[((-b) + (-b)), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 6e+34], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * c), $MachinePrecision] / N[(t$95$0 + b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]], N[(N[(-2.0 * c), $MachinePrecision] / N[(-2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.20000000000000065e82

   1. Initial program 44.4%

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

   3. Taylor expanded in a around 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-- N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 44.4

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

   5. Applied rewrites 44.4%

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

   6. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 95.2

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

   8. Applied rewrites 95.2%

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

   9. Taylor expanded in b around -inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 95.2

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

   11. Applied rewrites 95.2%

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

   if -6.20000000000000065e82 < b < 6.00000000000000037e34

   1. Initial program 83.9%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 83.9%

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

   if 6.00000000000000037e34 < b

   1. Initial program 55.7%

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

   4. Applied rewrites 55.7%

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

   5. Taylor expanded in a around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. if-same N/A

         \[\leadsto \color{blue}{-2 \cdot \frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}} \]

      8. +-commutative N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

   7. Applied rewrites 55.7%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{-2 \cdot c}{2 \cdot \color{blue}{b}} \]
   9. Step-by-step derivation

   10. Applied rewrites 93.3%

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

   11. Taylor expanded in a around 0

       \[\leadsto \frac{-2 \cdot c}{-2 \cdot \frac{a \cdot c}{b} + \color{blue}{2 \cdot b}} \]
   12. Step-by-step derivation

   13. Applied rewrites 93.7%

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

Alternative 5: 89.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{+82}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+34}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}{a} \cdot 0.5\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{-2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.2e+82)
   (if (>= b 0.0) (/ (- b) a) (/ (+ (- b) (- b)) (* 2.0 a)))
   (if (<= b 6e+34)
     (if (>= b 0.0)
       (* c (/ -2.0 (+ (sqrt (fma -4.0 (* c a) (* b b))) b)))
       (* (/ (- (sqrt (fma (* c a) -4.0 (* b b))) b) a) 0.5))
     (/ (* -2.0 c) (* -2.0 (- (* a (/ c b)) b))))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -6.2e+82) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -b / a;
		} else {
			tmp_2 = (-b + -b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 6e+34) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = c * (-2.0 / (sqrt(fma(-4.0, (c * a), (b * b))) + b));
		} else {
			tmp_3 = ((sqrt(fma((c * a), -4.0, (b * b))) - b) / a) * 0.5;
		}
		tmp_1 = tmp_3;
	} else {
		tmp_1 = (-2.0 * c) / (-2.0 * ((a * (c / b)) - b));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -6.2e+82)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-b) / a);
		else
			tmp_2 = Float64(Float64(Float64(-b) + Float64(-b)) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 6e+34)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(c * Float64(-2.0 / Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) + b)));
		else
			tmp_3 = Float64(Float64(Float64(sqrt(fma(Float64(c * a), -4.0, Float64(b * b))) - b) / a) * 0.5);
		end
		tmp_1 = tmp_3;
	else
		tmp_1 = Float64(Float64(-2.0 * c) / Float64(-2.0 * Float64(Float64(a * Float64(c / b)) - b)));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.2e+82], If[GreaterEqual[b, 0.0], N[((-b) / a), $MachinePrecision], N[(N[((-b) + (-b)), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 6e+34], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.5), $MachinePrecision]], N[(N[(-2.0 * c), $MachinePrecision] / N[(-2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.20000000000000065e82

   1. Initial program 44.4%

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

   3. Taylor expanded in a around 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-- N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 44.4

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

   5. Applied rewrites 44.4%

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

   6. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 95.2

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

   8. Applied rewrites 95.2%

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

   9. Taylor expanded in b around -inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 95.2

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

   11. Applied rewrites 95.2%

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

   if -6.20000000000000065e82 < b < 6.00000000000000037e34

   1. Initial program 83.9%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 83.9%

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

   7. Applied rewrites 83.8%

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

   if 6.00000000000000037e34 < b

   1. Initial program 55.7%

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

   4. Applied rewrites 55.7%

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

   5. Taylor expanded in a around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. if-same N/A

         \[\leadsto \color{blue}{-2 \cdot \frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}} \]

      8. +-commutative N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

   7. Applied rewrites 55.7%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{-2 \cdot c}{2 \cdot \color{blue}{b}} \]
   9. Step-by-step derivation

   10. Applied rewrites 93.3%

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

   11. Taylor expanded in a around 0

       \[\leadsto \frac{-2 \cdot c}{-2 \cdot \frac{a \cdot c}{b} + \color{blue}{2 \cdot b}} \]
   12. Step-by-step derivation

   13. Applied rewrites 93.7%

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

Alternative 6: 89.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.1 \cdot 10^{+82}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq -2.9 \cdot 10^{-265}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right)\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+34}:\\ \;\;\;\;\frac{-2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{-2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.1e+82)
   (if (>= b 0.0) (/ (- b) a) (/ (+ (- b) (- b)) (* 2.0 a)))
   (if (<= b -2.9e-265)
     (* (/ 0.5 a) (- (sqrt (fma (* -4.0 c) a (* b b))) b))
     (if (<= b 6e+34)
       (/ (* -2.0 c) (+ (sqrt (fma (* c a) -4.0 (* b b))) b))
       (/ (* -2.0 c) (* -2.0 (- (* a (/ c b)) b)))))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -4.1e+82) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -b / a;
		} else {
			tmp_2 = (-b + -b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= -2.9e-265) {
		tmp_1 = (0.5 / a) * (sqrt(fma((-4.0 * c), a, (b * b))) - b);
	} else if (b <= 6e+34) {
		tmp_1 = (-2.0 * c) / (sqrt(fma((c * a), -4.0, (b * b))) + b);
	} else {
		tmp_1 = (-2.0 * c) / (-2.0 * ((a * (c / b)) - b));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -4.1e+82)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-b) / a);
		else
			tmp_2 = Float64(Float64(Float64(-b) + Float64(-b)) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= -2.9e-265)
		tmp_1 = Float64(Float64(0.5 / a) * Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) - b));
	elseif (b <= 6e+34)
		tmp_1 = Float64(Float64(-2.0 * c) / Float64(sqrt(fma(Float64(c * a), -4.0, Float64(b * b))) + b));
	else
		tmp_1 = Float64(Float64(-2.0 * c) / Float64(-2.0 * Float64(Float64(a * Float64(c / b)) - b)));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.1e+82], If[GreaterEqual[b, 0.0], N[((-b) / a), $MachinePrecision], N[(N[((-b) + (-b)), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -2.9e-265], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6e+34], N[(N[(-2.0 * c), $MachinePrecision] / N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * c), $MachinePrecision] / N[(-2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -4.09999999999999995e82

   1. Initial program 44.4%

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

   3. Taylor expanded in a around 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-- N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 44.4

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

   5. Applied rewrites 44.4%

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

   6. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 95.2

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

   8. Applied rewrites 95.2%

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

   9. Taylor expanded in b around -inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 95.2

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

   11. Applied rewrites 95.2%

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

   if -4.09999999999999995e82 < b < -2.89999999999999975e-265

   1. Initial program 83.9%

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

   4. Applied rewrites 29.6%

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

   5. Taylor expanded in a around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. if-same N/A

         \[\leadsto \color{blue}{-2 \cdot \frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}} \]

      8. +-commutative N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

   7. Applied rewrites 43.1%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}} \]
   8. Step-by-step derivation

   9. Applied rewrites 42.6%

      \[\leadsto \frac{-2 \cdot c}{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right) - b \cdot b} \cdot \color{blue}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right)} \]

   10. Taylor expanded in a around 0

       \[\leadsto \frac{\frac{1}{2}}{a} \cdot \left(\color{blue}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}} - b\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 83.7%

       \[\leadsto \frac{0.5}{a} \cdot \left(\color{blue}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}} - b\right) \]

   if -2.89999999999999975e-265 < b < 6.00000000000000037e34

   1. Initial program 84.0%

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

   4. Applied rewrites 83.8%

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

   5. Taylor expanded in a around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. if-same N/A

         \[\leadsto \color{blue}{-2 \cdot \frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}} \]

      8. +-commutative N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

   7. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}} \]

   if 6.00000000000000037e34 < b

   1. Initial program 55.7%

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

   4. Applied rewrites 55.7%

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

   5. Taylor expanded in a around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. if-same N/A

         \[\leadsto \color{blue}{-2 \cdot \frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}} \]

      8. +-commutative N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

   7. Applied rewrites 55.7%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{-2 \cdot c}{2 \cdot \color{blue}{b}} \]
   9. Step-by-step derivation

   10. Applied rewrites 93.3%

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

   11. Taylor expanded in a around 0

       \[\leadsto \frac{-2 \cdot c}{-2 \cdot \frac{a \cdot c}{b} + \color{blue}{2 \cdot b}} \]
   12. Step-by-step derivation

   13. Applied rewrites 93.7%

       \[\leadsto \frac{-2 \cdot c}{-2 \cdot \color{blue}{\left(a \cdot \frac{c}{b} - b\right)}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 89.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\\ \mathbf{if}\;b \leq -4.1 \cdot 10^{+82}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq -2.35 \cdot 10^{-266}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(t\_0 - b\right)\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+34}:\\ \;\;\;\;c \cdot \frac{-2}{t\_0 + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{-2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 c) a (* b b)))))
   (if (<= b -4.1e+82)
     (if (>= b 0.0) (/ (- b) a) (/ (+ (- b) (- b)) (* 2.0 a)))
     (if (<= b -2.35e-266)
       (* (/ 0.5 a) (- t_0 b))
       (if (<= b 6e+34)
         (* c (/ -2.0 (+ t_0 b)))
         (/ (* -2.0 c) (* -2.0 (- (* a (/ c b)) b))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-4.0 * c), a, (b * b)));
	double tmp_1;
	if (b <= -4.1e+82) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -b / a;
		} else {
			tmp_2 = (-b + -b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= -2.35e-266) {
		tmp_1 = (0.5 / a) * (t_0 - b);
	} else if (b <= 6e+34) {
		tmp_1 = c * (-2.0 / (t_0 + b));
	} else {
		tmp_1 = (-2.0 * c) / (-2.0 * ((a * (c / b)) - b));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-4.0 * c), a, Float64(b * b)))
	tmp_1 = 0.0
	if (b <= -4.1e+82)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-b) / a);
		else
			tmp_2 = Float64(Float64(Float64(-b) + Float64(-b)) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= -2.35e-266)
		tmp_1 = Float64(Float64(0.5 / a) * Float64(t_0 - b));
	elseif (b <= 6e+34)
		tmp_1 = Float64(c * Float64(-2.0 / Float64(t_0 + b)));
	else
		tmp_1 = Float64(Float64(-2.0 * c) / Float64(-2.0 * Float64(Float64(a * Float64(c / b)) - b)));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -4.1e+82], If[GreaterEqual[b, 0.0], N[((-b) / a), $MachinePrecision], N[(N[((-b) + (-b)), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -2.35e-266], N[(N[(0.5 / a), $MachinePrecision] * N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6e+34], N[(c * N[(-2.0 / N[(t$95$0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * c), $MachinePrecision] / N[(-2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -4.09999999999999995e82

   1. Initial program 44.4%

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

   3. Taylor expanded in a around 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-- N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 44.4

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

   5. Applied rewrites 44.4%

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

   6. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 95.2

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

   8. Applied rewrites 95.2%

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

   9. Taylor expanded in b around -inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 95.2

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

   11. Applied rewrites 95.2%

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

   if -4.09999999999999995e82 < b < -2.35000000000000014e-266

   1. Initial program 83.9%

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

   4. Applied rewrites 29.6%

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

   5. Taylor expanded in a around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. if-same N/A

         \[\leadsto \color{blue}{-2 \cdot \frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}} \]

      8. +-commutative N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

   7. Applied rewrites 43.1%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}} \]
   8. Step-by-step derivation

   9. Applied rewrites 42.6%

      \[\leadsto \frac{-2 \cdot c}{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right) - b \cdot b} \cdot \color{blue}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right)} \]

   10. Taylor expanded in a around 0

       \[\leadsto \frac{\frac{1}{2}}{a} \cdot \left(\color{blue}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}} - b\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 83.7%

       \[\leadsto \frac{0.5}{a} \cdot \left(\color{blue}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}} - b\right) \]

   if -2.35000000000000014e-266 < b < 6.00000000000000037e34

   1. Initial program 84.0%

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

   4. Applied rewrites 83.8%

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

   5. Taylor expanded in a around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. if-same N/A

         \[\leadsto \color{blue}{-2 \cdot \frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}} \]

      8. +-commutative N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

   7. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}} \]
   8. Step-by-step derivation

   9. Applied rewrites 83.8%

      \[\leadsto c \cdot \color{blue}{\frac{-2}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}} \]

   if 6.00000000000000037e34 < b

   1. Initial program 55.7%

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

   4. Applied rewrites 55.7%

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

   5. Taylor expanded in a around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. if-same N/A

         \[\leadsto \color{blue}{-2 \cdot \frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}} \]

      8. +-commutative N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

   7. Applied rewrites 55.7%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{-2 \cdot c}{2 \cdot \color{blue}{b}} \]
   9. Step-by-step derivation

   10. Applied rewrites 93.3%

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

   11. Taylor expanded in a around 0

       \[\leadsto \frac{-2 \cdot c}{-2 \cdot \frac{a \cdot c}{b} + \color{blue}{2 \cdot b}} \]
   12. Step-by-step derivation

   13. Applied rewrites 93.7%

       \[\leadsto \frac{-2 \cdot c}{-2 \cdot \color{blue}{\left(a \cdot \frac{c}{b} - b\right)}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 84.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.12 \cdot 10^{-11}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+34}:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{-2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.12e-11)
   (if (>= b 0.0) (/ (- b) a) (/ (+ (- b) (- b)) (* 2.0 a)))
   (if (<= b 6e+34)
     (* c (/ -2.0 (+ (sqrt (fma (* -4.0 c) a (* b b))) b)))
     (/ (* -2.0 c) (* -2.0 (- (* a (/ c b)) b))))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1.12e-11) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -b / a;
		} else {
			tmp_2 = (-b + -b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 6e+34) {
		tmp_1 = c * (-2.0 / (sqrt(fma((-4.0 * c), a, (b * b))) + b));
	} else {
		tmp_1 = (-2.0 * c) / (-2.0 * ((a * (c / b)) - b));
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -1.12e-11)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-b) / a);
		else
			tmp_2 = Float64(Float64(Float64(-b) + Float64(-b)) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 6e+34)
		tmp_1 = Float64(c * Float64(-2.0 / Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) + b)));
	else
		tmp_1 = Float64(Float64(-2.0 * c) / Float64(-2.0 * Float64(Float64(a * Float64(c / b)) - b)));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.12e-11], If[GreaterEqual[b, 0.0], N[((-b) / a), $MachinePrecision], N[(N[((-b) + (-b)), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 6e+34], N[(c * N[(-2.0 / N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * c), $MachinePrecision] / N[(-2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.1200000000000001e-11

   1. Initial program 55.2%

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

   3. Taylor expanded in a around 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-- N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.2

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 91.3

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

   8. Applied rewrites 91.3%

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

   9. Taylor expanded in b around -inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 91.3

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

   11. Applied rewrites 91.3%

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

   if -1.1200000000000001e-11 < b < 6.00000000000000037e34

   1. Initial program 82.4%

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

   4. Applied rewrites 73.0%

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

   5. Taylor expanded in a around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. if-same N/A

         \[\leadsto \color{blue}{-2 \cdot \frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}} \]

      8. +-commutative N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

   7. Applied rewrites 77.6%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}} \]
   8. Step-by-step derivation

   9. Applied rewrites 77.4%

      \[\leadsto c \cdot \color{blue}{\frac{-2}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}} \]

   if 6.00000000000000037e34 < b

   1. Initial program 55.7%

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

   4. Applied rewrites 55.7%

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

   5. Taylor expanded in a around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. if-same N/A

         \[\leadsto \color{blue}{-2 \cdot \frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}} \]

      8. +-commutative N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

   7. Applied rewrites 55.7%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{-2 \cdot c}{2 \cdot \color{blue}{b}} \]
   9. Step-by-step derivation

   10. Applied rewrites 93.3%

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

   11. Taylor expanded in a around 0

       \[\leadsto \frac{-2 \cdot c}{-2 \cdot \frac{a \cdot c}{b} + \color{blue}{2 \cdot b}} \]
   12. Step-by-step derivation

   13. Applied rewrites 93.7%

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

Alternative 9: 80.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.12 \cdot 10^{-11}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-64}:\\ \;\;\;\;\frac{-2 \cdot c}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{-2 \cdot \left(\frac{a}{b} \cdot c - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.12e-11)
   (if (>= b 0.0) (/ (- b) a) (/ (+ (- b) (- b)) (* 2.0 a)))
   (if (<= b 9.5e-64)
     (/ (* -2.0 c) (+ (sqrt (* (* a c) -4.0)) b))
     (/ (* -2.0 c) (* -2.0 (- (* (/ a b) c) b))))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1.12e-11) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -b / a;
		} else {
			tmp_2 = (-b + -b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 9.5e-64) {
		tmp_1 = (-2.0 * c) / (sqrt(((a * c) * -4.0)) + b);
	} else {
		tmp_1 = (-2.0 * c) / (-2.0 * (((a / b) * c) - b));
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    if (b <= (-1.12d-11)) then
        if (b >= 0.0d0) then
            tmp_2 = -b / a
        else
            tmp_2 = (-b + -b) / (2.0d0 * a)
        end if
        tmp_1 = tmp_2
    else if (b <= 9.5d-64) then
        tmp_1 = ((-2.0d0) * c) / (sqrt(((a * c) * (-4.0d0))) + b)
    else
        tmp_1 = ((-2.0d0) * c) / ((-2.0d0) * (((a / b) * c) - b))
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1.12e-11) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -b / a;
		} else {
			tmp_2 = (-b + -b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 9.5e-64) {
		tmp_1 = (-2.0 * c) / (Math.sqrt(((a * c) * -4.0)) + b);
	} else {
		tmp_1 = (-2.0 * c) / (-2.0 * (((a / b) * c) - b));
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if b <= -1.12e-11:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -b / a
		else:
			tmp_2 = (-b + -b) / (2.0 * a)
		tmp_1 = tmp_2
	elif b <= 9.5e-64:
		tmp_1 = (-2.0 * c) / (math.sqrt(((a * c) * -4.0)) + b)
	else:
		tmp_1 = (-2.0 * c) / (-2.0 * (((a / b) * c) - b))
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -1.12e-11)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-b) / a);
		else
			tmp_2 = Float64(Float64(Float64(-b) + Float64(-b)) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 9.5e-64)
		tmp_1 = Float64(Float64(-2.0 * c) / Float64(sqrt(Float64(Float64(a * c) * -4.0)) + b));
	else
		tmp_1 = Float64(Float64(-2.0 * c) / Float64(-2.0 * Float64(Float64(Float64(a / b) * c) - b)));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp_2 = 0.0;
	if (b <= -1.12e-11)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -b / a;
		else
			tmp_3 = (-b + -b) / (2.0 * a);
		end
		tmp_2 = tmp_3;
	elseif (b <= 9.5e-64)
		tmp_2 = (-2.0 * c) / (sqrt(((a * c) * -4.0)) + b);
	else
		tmp_2 = (-2.0 * c) / (-2.0 * (((a / b) * c) - b));
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.12e-11], If[GreaterEqual[b, 0.0], N[((-b) / a), $MachinePrecision], N[(N[((-b) + (-b)), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 9.5e-64], N[(N[(-2.0 * c), $MachinePrecision] / N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * c), $MachinePrecision] / N[(-2.0 * N[(N[(N[(a / b), $MachinePrecision] * c), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.1200000000000001e-11

   1. Initial program 55.2%

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

   3. Taylor expanded in a around 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-- N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.2

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 91.3

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

   8. Applied rewrites 91.3%

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

   9. Taylor expanded in b around -inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 91.3

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

   11. Applied rewrites 91.3%

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

   if -1.1200000000000001e-11 < b < 9.50000000000000043e-64

   1. Initial program 78.1%

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

   4. Applied rewrites 66.5%

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

   5. Taylor expanded in a around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. if-same N/A

         \[\leadsto \color{blue}{-2 \cdot \frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}} \]

      8. +-commutative N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

   7. Applied rewrites 72.2%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}} \]

   8. Taylor expanded in a around inf

      \[\leadsto \frac{-2 \cdot c}{\sqrt{-4 \cdot \left(a \cdot c\right)} + b} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.5%

       \[\leadsto \frac{-2 \cdot c}{\sqrt{\left(a \cdot c\right) \cdot -4} + b} \]

   if 9.50000000000000043e-64 < b

   1. Initial program 64.3%

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

   4. Applied rewrites 64.3%

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

   5. Taylor expanded in a around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. if-same N/A

         \[\leadsto \color{blue}{-2 \cdot \frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}} \]

      8. +-commutative N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

   7. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{-2 \cdot c}{-2 \cdot \frac{a \cdot c}{b} + \color{blue}{2 \cdot b}} \]
   9. Step-by-step derivation

   10. Applied rewrites 90.9%

       \[\leadsto \frac{-2 \cdot c}{\mathsf{fma}\left(a \cdot \frac{c}{b}, \color{blue}{-2}, 2 \cdot b\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 90.9%

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

Alternative 10: 80.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.12 \cdot 10^{-11}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-64}:\\ \;\;\;\;\frac{-2 \cdot c}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{-2 \cdot \left(a \cdot \frac{c}{b} - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.12e-11)
   (if (>= b 0.0) (/ (- b) a) (/ (+ (- b) (- b)) (* 2.0 a)))
   (if (<= b 9.5e-64)
     (/ (* -2.0 c) (+ (sqrt (* (* a c) -4.0)) b))
     (/ (* -2.0 c) (* -2.0 (- (* a (/ c b)) b))))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1.12e-11) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -b / a;
		} else {
			tmp_2 = (-b + -b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 9.5e-64) {
		tmp_1 = (-2.0 * c) / (sqrt(((a * c) * -4.0)) + b);
	} else {
		tmp_1 = (-2.0 * c) / (-2.0 * ((a * (c / b)) - b));
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    if (b <= (-1.12d-11)) then
        if (b >= 0.0d0) then
            tmp_2 = -b / a
        else
            tmp_2 = (-b + -b) / (2.0d0 * a)
        end if
        tmp_1 = tmp_2
    else if (b <= 9.5d-64) then
        tmp_1 = ((-2.0d0) * c) / (sqrt(((a * c) * (-4.0d0))) + b)
    else
        tmp_1 = ((-2.0d0) * c) / ((-2.0d0) * ((a * (c / b)) - b))
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1.12e-11) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -b / a;
		} else {
			tmp_2 = (-b + -b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 9.5e-64) {
		tmp_1 = (-2.0 * c) / (Math.sqrt(((a * c) * -4.0)) + b);
	} else {
		tmp_1 = (-2.0 * c) / (-2.0 * ((a * (c / b)) - b));
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if b <= -1.12e-11:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -b / a
		else:
			tmp_2 = (-b + -b) / (2.0 * a)
		tmp_1 = tmp_2
	elif b <= 9.5e-64:
		tmp_1 = (-2.0 * c) / (math.sqrt(((a * c) * -4.0)) + b)
	else:
		tmp_1 = (-2.0 * c) / (-2.0 * ((a * (c / b)) - b))
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -1.12e-11)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-b) / a);
		else
			tmp_2 = Float64(Float64(Float64(-b) + Float64(-b)) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 9.5e-64)
		tmp_1 = Float64(Float64(-2.0 * c) / Float64(sqrt(Float64(Float64(a * c) * -4.0)) + b));
	else
		tmp_1 = Float64(Float64(-2.0 * c) / Float64(-2.0 * Float64(Float64(a * Float64(c / b)) - b)));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp_2 = 0.0;
	if (b <= -1.12e-11)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -b / a;
		else
			tmp_3 = (-b + -b) / (2.0 * a);
		end
		tmp_2 = tmp_3;
	elseif (b <= 9.5e-64)
		tmp_2 = (-2.0 * c) / (sqrt(((a * c) * -4.0)) + b);
	else
		tmp_2 = (-2.0 * c) / (-2.0 * ((a * (c / b)) - b));
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.12e-11], If[GreaterEqual[b, 0.0], N[((-b) / a), $MachinePrecision], N[(N[((-b) + (-b)), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 9.5e-64], N[(N[(-2.0 * c), $MachinePrecision] / N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * c), $MachinePrecision] / N[(-2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.1200000000000001e-11

   1. Initial program 55.2%

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

   3. Taylor expanded in a around 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-- N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.2

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 91.3

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

   8. Applied rewrites 91.3%

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

   9. Taylor expanded in b around -inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 91.3

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

   11. Applied rewrites 91.3%

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

   if -1.1200000000000001e-11 < b < 9.50000000000000043e-64

   1. Initial program 78.1%

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

   4. Applied rewrites 66.5%

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

   5. Taylor expanded in a around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. if-same N/A

         \[\leadsto \color{blue}{-2 \cdot \frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}} \]

      8. +-commutative N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

   7. Applied rewrites 72.2%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}} \]

   8. Taylor expanded in a around inf

      \[\leadsto \frac{-2 \cdot c}{\sqrt{-4 \cdot \left(a \cdot c\right)} + b} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.5%

       \[\leadsto \frac{-2 \cdot c}{\sqrt{\left(a \cdot c\right) \cdot -4} + b} \]

   if 9.50000000000000043e-64 < b

   1. Initial program 64.3%

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

   4. Applied rewrites 64.3%

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

   5. Taylor expanded in a around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. if-same N/A

         \[\leadsto \color{blue}{-2 \cdot \frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}} \]

      8. +-commutative N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

   7. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{-2 \cdot c}{2 \cdot \color{blue}{b}} \]
   9. Step-by-step derivation

   10. Applied rewrites 90.4%

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

   11. Taylor expanded in a around 0

       \[\leadsto \frac{-2 \cdot c}{-2 \cdot \frac{a \cdot c}{b} + \color{blue}{2 \cdot b}} \]
   12. Step-by-step derivation

   13. Applied rewrites 90.9%

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

Alternative 11: 80.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.12 \cdot 10^{-11}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-64}:\\ \;\;\;\;\frac{-2 \cdot c}{\sqrt{\left(a \cdot c\right) \cdot -4} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{2 \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.12e-11)
   (if (>= b 0.0) (/ (- b) a) (/ (+ (- b) (- b)) (* 2.0 a)))
   (if (<= b 9.5e-64)
     (/ (* -2.0 c) (+ (sqrt (* (* a c) -4.0)) b))
     (/ (* -2.0 c) (* 2.0 b)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1.12e-11) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -b / a;
		} else {
			tmp_2 = (-b + -b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 9.5e-64) {
		tmp_1 = (-2.0 * c) / (sqrt(((a * c) * -4.0)) + b);
	} else {
		tmp_1 = (-2.0 * c) / (2.0 * b);
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    if (b <= (-1.12d-11)) then
        if (b >= 0.0d0) then
            tmp_2 = -b / a
        else
            tmp_2 = (-b + -b) / (2.0d0 * a)
        end if
        tmp_1 = tmp_2
    else if (b <= 9.5d-64) then
        tmp_1 = ((-2.0d0) * c) / (sqrt(((a * c) * (-4.0d0))) + b)
    else
        tmp_1 = ((-2.0d0) * c) / (2.0d0 * b)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if (b <= -1.12e-11) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -b / a;
		} else {
			tmp_2 = (-b + -b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 9.5e-64) {
		tmp_1 = (-2.0 * c) / (Math.sqrt(((a * c) * -4.0)) + b);
	} else {
		tmp_1 = (-2.0 * c) / (2.0 * b);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if b <= -1.12e-11:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -b / a
		else:
			tmp_2 = (-b + -b) / (2.0 * a)
		tmp_1 = tmp_2
	elif b <= 9.5e-64:
		tmp_1 = (-2.0 * c) / (math.sqrt(((a * c) * -4.0)) + b)
	else:
		tmp_1 = (-2.0 * c) / (2.0 * b)
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= -1.12e-11)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-b) / a);
		else
			tmp_2 = Float64(Float64(Float64(-b) + Float64(-b)) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 9.5e-64)
		tmp_1 = Float64(Float64(-2.0 * c) / Float64(sqrt(Float64(Float64(a * c) * -4.0)) + b));
	else
		tmp_1 = Float64(Float64(-2.0 * c) / Float64(2.0 * b));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp_2 = 0.0;
	if (b <= -1.12e-11)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -b / a;
		else
			tmp_3 = (-b + -b) / (2.0 * a);
		end
		tmp_2 = tmp_3;
	elseif (b <= 9.5e-64)
		tmp_2 = (-2.0 * c) / (sqrt(((a * c) * -4.0)) + b);
	else
		tmp_2 = (-2.0 * c) / (2.0 * b);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.12e-11], If[GreaterEqual[b, 0.0], N[((-b) / a), $MachinePrecision], N[(N[((-b) + (-b)), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 9.5e-64], N[(N[(-2.0 * c), $MachinePrecision] / N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * c), $MachinePrecision] / N[(2.0 * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.1200000000000001e-11

   1. Initial program 55.2%

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

   3. Taylor expanded in a around 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-- N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.2

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 91.3

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

   8. Applied rewrites 91.3%

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

   9. Taylor expanded in b around -inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 91.3

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

   11. Applied rewrites 91.3%

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

   if -1.1200000000000001e-11 < b < 9.50000000000000043e-64

   1. Initial program 78.1%

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

   4. Applied rewrites 66.5%

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

   5. Taylor expanded in a around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. if-same N/A

         \[\leadsto \color{blue}{-2 \cdot \frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}} \]

      8. +-commutative N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

   7. Applied rewrites 72.2%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}} \]

   8. Taylor expanded in a around inf

      \[\leadsto \frac{-2 \cdot c}{\sqrt{-4 \cdot \left(a \cdot c\right)} + b} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.5%

       \[\leadsto \frac{-2 \cdot c}{\sqrt{\left(a \cdot c\right) \cdot -4} + b} \]

   if 9.50000000000000043e-64 < b

   1. Initial program 64.3%

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

   4. Applied rewrites 64.3%

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

   5. Taylor expanded in a around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. if-same N/A

         \[\leadsto \color{blue}{-2 \cdot \frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}} \]

      8. +-commutative N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

   7. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{-2 \cdot c}{2 \cdot \color{blue}{b}} \]
   9. Step-by-step derivation

   10. Applied rewrites 90.4%

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

Alternative 12: 67.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.3 \cdot 10^{-308}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{2 \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.3e-308)
   (if (>= b 0.0) (/ (- b) a) (/ (+ (- b) (- b)) (* 2.0 a)))
   (/ (* -2.0 c) (* 2.0 b))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= 2.3e-308) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -b / a;
		} else {
			tmp_2 = (-b + -b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = (-2.0 * c) / (2.0 * b);
	}
	return tmp_1;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    if (b <= 2.3d-308) then
        if (b >= 0.0d0) then
            tmp_2 = -b / a
        else
            tmp_2 = (-b + -b) / (2.0d0 * a)
        end if
        tmp_1 = tmp_2
    else
        tmp_1 = ((-2.0d0) * c) / (2.0d0 * b)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if (b <= 2.3e-308) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -b / a;
		} else {
			tmp_2 = (-b + -b) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = (-2.0 * c) / (2.0 * b);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if b <= 2.3e-308:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = -b / a
		else:
			tmp_2 = (-b + -b) / (2.0 * a)
		tmp_1 = tmp_2
	else:
		tmp_1 = (-2.0 * c) / (2.0 * b)
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= 2.3e-308)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(-b) / a);
		else
			tmp_2 = Float64(Float64(Float64(-b) + Float64(-b)) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	else
		tmp_1 = Float64(Float64(-2.0 * c) / Float64(2.0 * b));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp_2 = 0.0;
	if (b <= 2.3e-308)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = -b / a;
		else
			tmp_3 = (-b + -b) / (2.0 * a);
		end
		tmp_2 = tmp_3;
	else
		tmp_2 = (-2.0 * c) / (2.0 * b);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.3e-308], If[GreaterEqual[b, 0.0], N[((-b) / a), $MachinePrecision], N[(N[((-b) + (-b)), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], N[(N[(-2.0 * c), $MachinePrecision] / N[(2.0 * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.2999999999999999e-308

   1. Initial program 61.0%

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

   3. Taylor expanded in a around 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-- N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 61.0

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

   5. Applied rewrites 61.0%

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

   6. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 72.4

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

   8. Applied rewrites 72.4%

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

   9. Taylor expanded in b around -inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 72.4

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

   11. Applied rewrites 72.4%

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

   if 2.2999999999999999e-308 < b

   1. Initial program 68.9%

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

   4. Applied rewrites 68.9%

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

   5. Taylor expanded in a around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. if-same N/A

         \[\leadsto \color{blue}{-2 \cdot \frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}} \]

      8. +-commutative N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

   7. Applied rewrites 68.9%

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{-2 \cdot c}{2 \cdot \color{blue}{b}} \]
   9. Step-by-step derivation

   10. Applied rewrites 69.2%

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

Alternative 13: 34.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{-2 \cdot c}{2 \cdot b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ (* -2.0 c) (* 2.0 b)))

C

double code(double a, double b, double c) {
	return (-2.0 * c) / (2.0 * b);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-2.0d0) * c) / (2.0d0 * b)
end function

Java

public static double code(double a, double b, double c) {
	return (-2.0 * c) / (2.0 * b);
}

Python

def code(a, b, c):
	return (-2.0 * c) / (2.0 * b)

Julia

function code(a, b, c)
	return Float64(Float64(-2.0 * c) / Float64(2.0 * b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-2.0 * c) / (2.0 * b);
end

Wolfram

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

Derivation

1. Initial program 65.1%

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

4. Applied rewrites 43.1%

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

5. Taylor expanded in a around 0

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

   1. fp-cancel-sub-sign-inv N/A

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. if-same N/A

      \[\leadsto \color{blue}{-2 \cdot \frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

   5. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}} \]

   8. +-commutative N/A

      \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

   9. lower-+.f64 N/A

      \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

7. Applied rewrites 47.0%

   \[\leadsto \color{blue}{\frac{-2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}} \]

8. Taylor expanded in a around 0

   \[\leadsto \frac{-2 \cdot c}{2 \cdot \color{blue}{b}} \]
9. Step-by-step derivation

10. Applied rewrites 37.2%

    \[\leadsto \frac{-2 \cdot c}{2 \cdot \color{blue}{b}} \]
11. Add Preprocessing

Alternative 14: 34.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{-2}{2 \cdot b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* c (/ -2.0 (* 2.0 b))))

C

double code(double a, double b, double c) {
	return c * (-2.0 / (2.0 * b));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c * ((-2.0d0) / (2.0d0 * b))
end function

Java

public static double code(double a, double b, double c) {
	return c * (-2.0 / (2.0 * b));
}

Python

def code(a, b, c):
	return c * (-2.0 / (2.0 * b))

Julia

function code(a, b, c)
	return Float64(c * Float64(-2.0 / Float64(2.0 * b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * (-2.0 / (2.0 * b));
end

Wolfram

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

Derivation

1. Initial program 65.1%

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

4. Applied rewrites 43.1%

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

5. Taylor expanded in a around 0

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

   1. fp-cancel-sub-sign-inv N/A

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. if-same N/A

      \[\leadsto \color{blue}{-2 \cdot \frac{c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

   5. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-2 \cdot c}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{-2 \cdot c}}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}} \]

   8. +-commutative N/A

      \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

   9. lower-+.f64 N/A

      \[\leadsto \frac{-2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} + b}} \]

7. Applied rewrites 47.0%

   \[\leadsto \color{blue}{\frac{-2 \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}} \]

8. Taylor expanded in a around 0

   \[\leadsto \frac{-2 \cdot c}{2 \cdot \color{blue}{b}} \]
9. Step-by-step derivation

10. Applied rewrites 37.2%

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

12. Applied rewrites 37.1%

    \[\leadsto c \cdot \color{blue}{\frac{-2}{2 \cdot b}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024326 
(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))))