jeff quadratic root 2

Average Error: 19.4 → 8.8

Time: 8.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\\ t_1 := \sqrt{\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)}\\ t_2 := \left(-b\right) - b\\ t_3 := \frac{2 \cdot c}{\left(-b\right) - t_0}\\ \mathbf{if}\;b \leq -124785765.50715955:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 3.600333794359356 \cdot 10^{+83}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 \cdot t_1 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 - b}{2 \cdot a}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :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))))

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* c (* 4.0 a)))))
        (t_1 (sqrt (hypot (sqrt (* c (* a -4.0))) b)))
        (t_2 (- (- b) b))
        (t_3 (/ (* 2.0 c) (- (- b) t_0))))
   (if (<= b -124785765.50715955)
     (if (>= b 0.0) t_3 (/ t_2 (* 2.0 a)))
     (if (<= b 3.600333794359356e+83)
       (if (>= b 0.0) t_3 (/ (- (* t_1 t_1) b) (* 2.0 a)))
       (if (>= b 0.0) (/ (* 2.0 c) t_2) (/ (- t_0 b) (* 2.0 a)))))))

C

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

↓

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (c * (4.0 * a))));
	double t_1 = sqrt(hypot(sqrt((c * (a * -4.0))), b));
	double t_2 = -b - b;
	double t_3 = (2.0 * c) / (-b - t_0);
	double tmp_1;
	if (b <= -124785765.50715955) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_3;
		} else {
			tmp_2 = t_2 / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 3.600333794359356e+83) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_3;
		} else {
			tmp_3 = ((t_1 * t_1) - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / t_2;
	} else {
		tmp_1 = (t_0 - b) / (2.0 * a);
	}
	return tmp_1;
}

Java

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

↓

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (c * (4.0 * a))));
	double t_1 = Math.sqrt(Math.hypot(Math.sqrt((c * (a * -4.0))), b));
	double t_2 = -b - b;
	double t_3 = (2.0 * c) / (-b - t_0);
	double tmp_1;
	if (b <= -124785765.50715955) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_3;
		} else {
			tmp_2 = t_2 / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 3.600333794359356e+83) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_3;
		} else {
			tmp_3 = ((t_1 * t_1) - b) / (2.0 * a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (2.0 * c) / t_2;
	} else {
		tmp_1 = (t_0 - b) / (2.0 * a);
	}
	return tmp_1;
}

Python

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

↓

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (c * (4.0 * a))))
	t_1 = math.sqrt(math.hypot(math.sqrt((c * (a * -4.0))), b))
	t_2 = -b - b
	t_3 = (2.0 * c) / (-b - t_0)
	tmp_1 = 0
	if b <= -124785765.50715955:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = t_3
		else:
			tmp_2 = t_2 / (2.0 * a)
		tmp_1 = tmp_2
	elif b <= 3.600333794359356e+83:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = t_3
		else:
			tmp_3 = ((t_1 * t_1) - b) / (2.0 * a)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = (2.0 * c) / t_2
	else:
		tmp_1 = (t_0 - b) / (2.0 * a)
	return tmp_1

Julia

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

↓

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a))))
	t_1 = sqrt(hypot(sqrt(Float64(c * Float64(a * -4.0))), b))
	t_2 = Float64(Float64(-b) - b)
	t_3 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0))
	tmp_1 = 0.0
	if (b <= -124785765.50715955)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_3;
		else
			tmp_2 = Float64(t_2 / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 3.600333794359356e+83)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_3;
		else
			tmp_3 = Float64(Float64(Float64(t_1 * t_1) - b) / Float64(2.0 * a));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / t_2);
	else
		tmp_1 = Float64(Float64(t_0 - b) / Float64(2.0 * a));
	end
	return tmp_1
end

MATLAB

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

↓

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) - (c * (4.0 * a))));
	t_1 = sqrt(hypot(sqrt((c * (a * -4.0))), b));
	t_2 = -b - b;
	t_3 = (2.0 * c) / (-b - t_0);
	tmp_2 = 0.0;
	if (b <= -124785765.50715955)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = t_3;
		else
			tmp_3 = t_2 / (2.0 * a);
		end
		tmp_2 = tmp_3;
	elseif (b <= 3.600333794359356e+83)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = t_3;
		else
			tmp_4 = ((t_1 * t_1) - b) / (2.0 * a);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = (2.0 * c) / t_2;
	else
		tmp_2 = (t_0 - b) / (2.0 * a);
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := 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[((-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]]

↓

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[Sqrt[N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2 + b ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[((-b) - b), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -124785765.50715955], If[GreaterEqual[b, 0.0], t$95$3, N[(t$95$2 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 3.600333794359356e+83], If[GreaterEqual[b, 0.0], t$95$3, N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]]]]]]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b < -124785765.507159546

   1. Initial program 32.1

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

   2. Taylor expanded in b around -inf 8.3

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

   if -124785765.507159546 < b < 3.60033379435935626e83

   1. Initial program 9.6

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

   2. Applied add-sqr-sqrt_binary64 9.7

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

   3. Simplified 12.1

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

   4. Simplified 11.4

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

   if 3.60033379435935626e83 < b

   1. Initial program 28.0

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

   2. Taylor expanded in b around inf 3.8

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

4. Final simplification 8.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -124785765.50715955:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 3.600333794359356 \cdot 10^{+83}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)} \cdot \sqrt{\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022129 
(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))))