quad2p (problem 3.2.1, positive)

Average Error: 33.9 → 11.1

Time: 10.7s

Precision: binary64

Cost: 20172

Math

\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

↓

\[\begin{array}{l} t_0 := \frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{if}\;b_2 \leq -1.4 \cdot 10^{+114}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq -1 \cdot 10^{-148}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq -3.7 \cdot 10^{-227}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b_2, \sqrt{-a} \cdot \sqrt{c}\right) - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 4 \cdot 10^{-97}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

↓

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)))
   (if (<= b_2 -1.4e+114)
     (/ (* b_2 -2.0) a)
     (if (<= b_2 -1e-148)
       t_0
       (if (<= b_2 -3.7e-227)
         (/ (- (hypot b_2 (* (sqrt (- a)) (sqrt c))) b_2) a)
         (if (<= b_2 4e-97) t_0 (* -0.5 (/ c b_2))))))))

C

double code(double a, double b_2, double c) {
	return (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
}

↓

double code(double a, double b_2, double c) {
	double t_0 = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	double tmp;
	if (b_2 <= -1.4e+114) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= -1e-148) {
		tmp = t_0;
	} else if (b_2 <= -3.7e-227) {
		tmp = (hypot(b_2, (sqrt(-a) * sqrt(c))) - b_2) / a;
	} else if (b_2 <= 4e-97) {
		tmp = t_0;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 + Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

↓

public static double code(double a, double b_2, double c) {
	double t_0 = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	double tmp;
	if (b_2 <= -1.4e+114) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= -1e-148) {
		tmp = t_0;
	} else if (b_2 <= -3.7e-227) {
		tmp = (Math.hypot(b_2, (Math.sqrt(-a) * Math.sqrt(c))) - b_2) / a;
	} else if (b_2 <= 4e-97) {
		tmp = t_0;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	return (-b_2 + math.sqrt(((b_2 * b_2) - (a * c)))) / a

↓

def code(a, b_2, c):
	t_0 = (math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
	tmp = 0
	if b_2 <= -1.4e+114:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= -1e-148:
		tmp = t_0
	elif b_2 <= -3.7e-227:
		tmp = (math.hypot(b_2, (math.sqrt(-a) * math.sqrt(c))) - b_2) / a
	elif b_2 <= 4e-97:
		tmp = t_0
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

↓

function code(a, b_2, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a)
	tmp = 0.0
	if (b_2 <= -1.4e+114)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= -1e-148)
		tmp = t_0;
	elseif (b_2 <= -3.7e-227)
		tmp = Float64(Float64(hypot(b_2, Float64(sqrt(Float64(-a)) * sqrt(c))) - b_2) / a);
	elseif (b_2 <= 4e-97)
		tmp = t_0;
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
end

↓

function tmp_2 = code(a, b_2, c)
	t_0 = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	tmp = 0.0;
	if (b_2 <= -1.4e+114)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= -1e-148)
		tmp = t_0;
	elseif (b_2 <= -3.7e-227)
		tmp = (hypot(b_2, (sqrt(-a) * sqrt(c))) - b_2) / a;
	elseif (b_2 <= 4e-97)
		tmp = t_0;
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) + N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

↓

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[b$95$2, -1.4e+114], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, -1e-148], t$95$0, If[LessEqual[b$95$2, -3.7e-227], N[(N[(N[Sqrt[b$95$2 ^ 2 + N[(N[Sqrt[(-a)], $MachinePrecision] * N[Sqrt[c], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 4e-97], t$95$0, N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.4e114

   1. Initial program 50.0

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Simplified 50.0

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   3. Taylor expanded in b_2 around -inf 4.2

      \[\leadsto \frac{\color{blue}{-2 \cdot b_2}}{a} \]

   if -1.4e114 < b_2 < -9.99999999999999936e-149 or -3.69999999999999978e-227 < b_2 < 4.00000000000000014e-97

   1. Initial program 11.8

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Simplified 11.8

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   if -9.99999999999999936e-149 < b_2 < -3.69999999999999978e-227

   1. Initial program 16.8

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Simplified 16.8

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   3. Applied egg-rr 11.1

      \[\leadsto \frac{\color{blue}{\mathsf{hypot}\left(b_2, \sqrt{-a \cdot c}\right)} - b_2}{a} \]

   4. Applied egg-rr 32.5

      \[\leadsto \frac{\mathsf{hypot}\left(b_2, \color{blue}{\sqrt{-a} \cdot \sqrt{c}}\right) - b_2}{a} \]

   if 4.00000000000000014e-97 < b_2

   1. Initial program 51.6

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Simplified 51.6

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   3. Taylor expanded in b_2 around inf 10.2

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 11.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.4 \cdot 10^{+114}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq -1 \cdot 10^{-148}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq -3.7 \cdot 10^{-227}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b_2, \sqrt{-a} \cdot \sqrt{c}\right) - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 4 \cdot 10^{-97}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.3
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.4 \cdot 10^{+114}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 4 \cdot 10^{-97}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 2
Error	13.4
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -7.5 \cdot 10^{-84}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \mathbf{elif}\;b_2 \leq 4 \cdot 10^{-97}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 3
Error	36.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 5.2 \cdot 10^{-231}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b_2}\\ \end{array} \]

Alternative 4
Error	36.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 5.2 \cdot 10^{-231}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 5
Error	22.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 5.2 \cdot 10^{-231}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 6
Error	53.0
Cost	388

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 2.1 \cdot 10^{-200}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]

Alternative 7
Error	56.1
Cost	192

\[\frac{0}{a} \]

Reproduce

herbie shell --seed 2022216 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))