jeff quadratic root 1

Average Error: 19.8 → 6.9

Time: 9.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \left(-b\right) - b\\ t_1 := \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\\ t_2 := \frac{c \cdot 2}{t_1 - b}\\ \mathbf{if}\;b \leq -1.4362804565109957 \cdot 10^{+116}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t_0}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.0415532491463108 \cdot 10^{+69}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{t_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

FPCore

(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 (- (- b) b))
        (t_1 (sqrt (+ (* b b) (* c (* a -4.0)))))
        (t_2 (/ (* c 2.0) (- t_1 b))))
   (if (<= b -1.4362804565109957e+116)
     (if (>= b 0.0) (/ (- (- b) t_1) (* a 2.0)) (/ (* c 2.0) t_0))
     (if (<= b 1.0415532491463108e+69)
       (if (>= b 0.0)
         (/ (- (- b) (sqrt (fma c (* a -4.0) (* b b)))) (* a 2.0))
         t_2)
       (if (>= b 0.0) (/ t_0 (* a 2.0)) t_2)))))

C

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 = -b - b;
	double t_1 = sqrt(((b * b) + (c * (a * -4.0))));
	double t_2 = (c * 2.0) / (t_1 - b);
	double tmp_1;
	if (b <= -1.4362804565109957e+116) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-b - t_1) / (a * 2.0);
		} else {
			tmp_2 = (c * 2.0) / t_0;
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.0415532491463108e+69) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - sqrt(fma(c, (a * -4.0), (b * b)))) / (a * 2.0);
		} else {
			tmp_3 = t_2;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = t_0 / (a * 2.0);
	} else {
		tmp_1 = t_2;
	}
	return tmp_1;
}

Julia

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 = Float64(Float64(-b) - b)
	t_1 = sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0))))
	t_2 = Float64(Float64(c * 2.0) / Float64(t_1 - b))
	tmp_1 = 0.0
	if (b <= -1.4362804565109957e+116)
		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) / t_0);
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.0415532491463108e+69)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(-b) - sqrt(fma(c, Float64(a * -4.0), Float64(b * b)))) / Float64(a * 2.0));
		else
			tmp_3 = t_2;
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(t_0 / Float64(a * 2.0));
	else
		tmp_1 = t_2;
	end
	return tmp_1
end

Wolfram

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[((-b) - b), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * 2.0), $MachinePrecision] / N[(t$95$1 - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.4362804565109957e+116], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$1), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / t$95$0), $MachinePrecision]], If[LessEqual[b, 1.0415532491463108e+69], If[GreaterEqual[b, 0.0], N[(N[((-b) - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], t$95$2], If[GreaterEqual[b, 0.0], N[(t$95$0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b < -1.4362804565109957e116

   1. Initial program 32.8

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

   2. Taylor expanded in b around -inf 2.0

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

   if -1.4362804565109957e116 < b < 1.0415532491463108e69

   1. Initial program 9.0

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

   2. Taylor expanded in b around 0 9.0

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

   3. Simplified 9.0

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

   if 1.0415532491463108e69 < b

   1. Initial program 40.1

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

   2. Taylor expanded in b around inf 5.7

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

4. Final simplification 6.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.4362804565109957 \cdot 10^{+116}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.0415532491463108 \cdot 10^{+69}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022148 
(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)))))))