jeff quadratic root 1

Percentage Accurate: 72.8% → 90.7%

Time: 18.9s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, \frac{c}{b}, -b\right)\\ t_1 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;b \leq -1.25 \cdot 10^{+142}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot t\_0}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{+108}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -0.5, \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{a \cdot -2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t\_1 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{t\_0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (/ c b) (- b))) (t_1 (sqrt (- (* b b) (* (* 4.0 a) c)))))
   (if (<= b -1.25e+142)
     (if (>= b 0.0) (/ (- (- b) t_1) (* a 2.0)) (/ (* c 2.0) (* 2.0 t_0)))
     (if (<= b 1.95e+108)
       (if (>= b 0.0)
         (fma (/ b a) -0.5 (/ (sqrt (fma b b (* c (* a -4.0)))) (* a -2.0)))
         (/ (* c 2.0) (- t_1 b)))
       (if (>= b 0.0) (/ t_0 a) (/ (* c 2.0) (- (- b) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (c / b), -b);
	double t_1 = sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp_1;
	if (b <= -1.25e+142) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-b - t_1) / (a * 2.0);
		} else {
			tmp_2 = (c * 2.0) / (2.0 * t_0);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.95e+108) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = fma((b / a), -0.5, (sqrt(fma(b, b, (c * (a * -4.0)))) / (a * -2.0)));
		} else {
			tmp_3 = (c * 2.0) / (t_1 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = t_0 / a;
	} else {
		tmp_1 = (c * 2.0) / (-b - b);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c / b), Float64(-b))
	t_1 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	tmp_1 = 0.0
	if (b <= -1.25e+142)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(Float64(-b) - t_1) / Float64(a * 2.0));
		else
			tmp_2 = Float64(Float64(c * 2.0) / Float64(2.0 * t_0));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.95e+108)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = fma(Float64(b / a), -0.5, Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) / Float64(a * -2.0)));
		else
			tmp_3 = Float64(Float64(c * 2.0) / Float64(t_1 - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(t_0 / a);
	else
		tmp_1 = Float64(Float64(c * 2.0) / Float64(Float64(-b) - b));
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.25e+142], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$1), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.95e+108], If[GreaterEqual[b, 0.0], N[(N[(b / a), $MachinePrecision] * -0.5 + N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(t$95$1 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(t$95$0 / a), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - b), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.25e142

   1. Initial program 35.2%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 95.0

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

   5. Applied rewrites 95.0%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 95.0%

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

   if -1.25e142 < b < 1.94999999999999992e108

   1. Initial program 85.3%

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

   4. Applied rewrites 85.3%

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

   if 1.94999999999999992e108 < b

   1. Initial program 56.0%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 56.0

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

   5. Applied rewrites 56.0%

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

   6. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 90.8

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

   8. Applied rewrites 90.8%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. mul-1-neg N/A

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

       3. sub-neg N/A

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

       4. lower-/.f64 N/A

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

       5. sub-neg N/A

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

       6. mul-1-neg N/A

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

       7. associate-/l* N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. mul-1-neg N/A

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

       11. lower-neg.f64 91.1

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

   11. Applied rewrites 91.1%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{+142}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{+108}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -0.5, \frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}{a \cdot -2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ t_1 := \frac{\left(-b\right) - t\_0}{a \cdot 2}\\ t_2 := \mathsf{fma}\left(a, \frac{c}{b}, -b\right)\\ \mathbf{if}\;b \leq -1.25 \cdot 10^{+142}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot t\_2}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{+108}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{t\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* (* 4.0 a) c))))
        (t_1 (/ (- (- b) t_0) (* a 2.0)))
        (t_2 (fma a (/ c b) (- b))))
   (if (<= b -1.25e+142)
     (if (>= b 0.0) t_1 (/ (* c 2.0) (* 2.0 t_2)))
     (if (<= b 1.95e+108)
       (if (>= b 0.0) t_1 (/ (* c 2.0) (- t_0 b)))
       (if (>= b 0.0) (/ t_2 a) (/ (* c 2.0) (- (- b) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	double t_1 = (-b - t_0) / (a * 2.0);
	double t_2 = fma(a, (c / b), -b);
	double tmp_1;
	if (b <= -1.25e+142) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = (c * 2.0) / (2.0 * t_2);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.95e+108) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = (c * 2.0) / (t_0 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = t_2 / a;
	} else {
		tmp_1 = (c * 2.0) / (-b - b);
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	t_1 = Float64(Float64(Float64(-b) - t_0) / Float64(a * 2.0))
	t_2 = fma(a, Float64(c / b), Float64(-b))
	tmp_1 = 0.0
	if (b <= -1.25e+142)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(Float64(c * 2.0) / Float64(2.0 * t_2));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.95e+108)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = Float64(Float64(c * 2.0) / Float64(t_0 - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(t_2 / a);
	else
		tmp_1 = Float64(Float64(c * 2.0) / Float64(Float64(-b) - b));
	end
	return tmp_1
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]}, Block[{t$95$1 = N[(N[((-b) - t$95$0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision]}, If[LessEqual[b, -1.25e+142], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(c * 2.0), $MachinePrecision] / N[(2.0 * t$95$2), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.95e+108], If[GreaterEqual[b, 0.0], t$95$1, N[(N[(c * 2.0), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(t$95$2 / a), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - b), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.25e142

   1. Initial program 46.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 97.4

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 97.5%

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

   if -1.25e142 < b < 1.94999999999999992e108

   1. Initial program 86.9%

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

   if 1.94999999999999992e108 < b

   1. Initial program 53.0%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 53.0

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

   5. Applied rewrites 53.0%

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

   6. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 96.4

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

   8. Applied rewrites 96.4%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. mul-1-neg N/A

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

       3. sub-neg N/A

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

       4. lower-/.f64 N/A

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

       5. sub-neg N/A

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

       6. mul-1-neg N/A

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

       7. associate-/l* N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. mul-1-neg N/A

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

       11. lower-neg.f64 96.7

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

   11. Applied rewrites 96.7%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{+142}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, -b\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{+108}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024228 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  :precision binary64
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))