Average Error: 20.0 → 6.7
Time: 8.1s
Precision: binary64
\[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
↓
\[\begin{array}{l} t_0 := \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\\ t_1 := \frac{\left(-b\right) - t_0}{a \cdot 2}\\ t_2 := \frac{c \cdot 2}{t_0 - b}\\ \mathbf{if}\;b \leq -1.7 \cdot 10^{+143}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{+111}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-\frac{b + b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
(FPCore (a b c)
 :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)))))))
↓
(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* b b) (* c (* a -4.0)))))
        (t_1 (/ (- (- b) t_0) (* a 2.0)))
        (t_2 (/ (* c 2.0) (- t_0 b))))
   (if (<= b -1.7e+143)
     (if (>= b 0.0) t_1 (/ (- c) b))
     (if (<= b 8.6e+111)
       (if (>= b 0.0) t_1 t_2)
       (if (>= b 0.0) (- (/ (+ b b) (* a 2.0))) t_2)))))
double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + sqrt(((b * b) - ((4.0 * a) * c))));
	}
	return tmp;
}
↓
double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) + (c * (a * -4.0))));
	double t_1 = (-b - t_0) / (a * 2.0);
	double t_2 = (c * 2.0) / (t_0 - b);
	double tmp_1;
	if (b <= -1.7e+143) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = -c / b;
		}
		tmp_1 = tmp_2;
	} else if (b <= 8.6e+111) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = t_2;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -((b + b) / (a * 2.0));
	} else {
		tmp_1 = t_2;
	}
	return tmp_1;
}

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b >= 0.0d0) then
        tmp = (-b - sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
    else
        tmp = (2.0d0 * c) / (-b + sqrt(((b * b) - ((4.0d0 * a) * c))))
    end if
    code = tmp
end function
↓
real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = sqrt(((b * b) + (c * (a * (-4.0d0)))))
    t_1 = (-b - t_0) / (a * 2.0d0)
    t_2 = (c * 2.0d0) / (t_0 - b)
    if (b <= (-1.7d+143)) then
        if (b >= 0.0d0) then
            tmp_2 = t_1
        else
            tmp_2 = -c / b
        end if
        tmp_1 = tmp_2
    else if (b <= 8.6d+111) then
        if (b >= 0.0d0) then
            tmp_3 = t_1
        else
            tmp_3 = t_2
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = -((b + b) / (a * 2.0d0))
    else
        tmp_1 = t_2
    end if
    code = tmp_1
end function

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + Math.sqrt(((b * b) - ((4.0 * a) * c))));
	}
	return tmp;
}
↓
public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) + (c * (a * -4.0))));
	double t_1 = (-b - t_0) / (a * 2.0);
	double t_2 = (c * 2.0) / (t_0 - b);
	double tmp_1;
	if (b <= -1.7e+143) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_1;
		} else {
			tmp_2 = -c / b;
		}
		tmp_1 = tmp_2;
	} else if (b <= 8.6e+111) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_1;
		} else {
			tmp_3 = t_2;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = -((b + b) / (a * 2.0));
	} else {
		tmp_1 = t_2;
	}
	return tmp_1;
}

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = (-b - math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)
	else:
		tmp = (2.0 * c) / (-b + math.sqrt(((b * b) - ((4.0 * a) * c))))
	return tmp
↓
def code(a, b, c):
	t_0 = math.sqrt(((b * b) + (c * (a * -4.0))))
	t_1 = (-b - t_0) / (a * 2.0)
	t_2 = (c * 2.0) / (t_0 - b)
	tmp_1 = 0
	if b <= -1.7e+143:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_1
		else:
			tmp_2 = -c / b
		tmp_1 = tmp_2
	elif b <= 8.6e+111:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = t_1
		else:
			tmp_3 = t_2
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = -((b + b) / (a * 2.0))
	else:
		tmp_1 = t_2
	return tmp_1

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))));
	end
	return tmp
end
↓
function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0))))
	t_1 = Float64(Float64(Float64(-b) - t_0) / Float64(a * 2.0))
	t_2 = Float64(Float64(c * 2.0) / Float64(t_0 - b))
	tmp_1 = 0.0
	if (b <= -1.7e+143)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_1;
		else
			tmp_2 = Float64(Float64(-c) / b);
		end
		tmp_1 = tmp_2;
	elseif (b <= 8.6e+111)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = t_2;
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(-Float64(Float64(b + b) / Float64(a * 2.0)));
	else
		tmp_1 = t_2;
	end
	return tmp_1
end

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (-b - sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
	else
		tmp = (2.0 * c) / (-b + sqrt(((b * b) - ((4.0 * a) * c))));
	end
	tmp_2 = tmp;
end
↓
function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) + (c * (a * -4.0))));
	t_1 = (-b - t_0) / (a * 2.0);
	t_2 = (c * 2.0) / (t_0 - b);
	tmp_2 = 0.0;
	if (b <= -1.7e+143)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_1;
		else
			tmp_3 = -c / b;
		end
		tmp_2 = tmp_3;
	elseif (b <= 8.6e+111)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = t_1;
		else
			tmp_4 = t_2;
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = -((b + b) / (a * 2.0));
	else
		tmp_2 = t_2;
	end
	tmp_5 = tmp_2;
end

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], 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]]
↓
code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[((-b) - t$95$0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * 2.0), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.7e+143], If[GreaterEqual[b, 0.0], t$95$1, N[((-c) / b), $MachinePrecision]], If[LessEqual[b, 8.6e+111], If[GreaterEqual[b, 0.0], t$95$1, t$95$2], If[GreaterEqual[b, 0.0], (-N[(N[(b + b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), t$95$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}
↓
\begin{array}{l}
t_0 := \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\\
t_1 := \frac{\left(-b\right) - t_0}{a \cdot 2}\\
t_2 := \frac{c \cdot 2}{t_0 - b}\\
\mathbf{if}\;b \leq -1.7 \cdot 10^{+143}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \geq 0:\\
\;\;\;\;t_1\\

\mathbf{else}:\\
\;\;\;\;\frac{-c}{b}\\


\end{array}\\

\mathbf{elif}\;b \leq 8.6 \cdot 10^{+111}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \geq 0:\\
\;\;\;\;t_1\\

\mathbf{else}:\\
\;\;\;\;t_2\\


\end{array}\\

\mathbf{elif}\;b \geq 0:\\
\;\;\;\;-\frac{b + b}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;t_2\\


\end{array}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if b < -1.69999999999999991e143

    1. Initial program 36.0

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

    2. Taylor expanded in b around -inf 1.5

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

    if -1.69999999999999991e143 < b < 8.59999999999999987e111

    1. Initial program 8.6

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

    if 8.59999999999999987e111 < b

    1. Initial program 51.2

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

    2. Taylor expanded in b around inf 4.6

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

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{+143}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{+111}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-\frac{b + b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022156 
(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)))))))