quad2p (problem 3.2.1, positive)

Average Error: 34.2 → 9.7

Time: 18.3s

Precision: binary64

Cost: 14092

Math

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

↓

\[\begin{array}{l} t_0 := a \cdot \left(-c\right)\\ \mathbf{if}\;b_2 \leq -2.7 \cdot 10^{+148}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.12 \cdot 10^{-205}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 3.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{t_0}{b_2 + \mathsf{hypot}\left(b_2, \sqrt{t_0}\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

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 (* a (- c))))
   (if (<= b_2 -2.7e+148)
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
     (if (<= b_2 1.12e-205)
       (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
       (if (<= b_2 3.5e-43)
         (/ (/ t_0 (+ b_2 (hypot b_2 (sqrt t_0)))) a)
         (* (/ c b_2) -0.5))))))

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 = a * -c;
	double tmp;
	if (b_2 <= -2.7e+148) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 1.12e-205) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else if (b_2 <= 3.5e-43) {
		tmp = (t_0 / (b_2 + hypot(b_2, sqrt(t_0)))) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	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 = a * -c;
	double tmp;
	if (b_2 <= -2.7e+148) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 1.12e-205) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else if (b_2 <= 3.5e-43) {
		tmp = (t_0 / (b_2 + Math.hypot(b_2, Math.sqrt(t_0)))) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	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 = a * -c
	tmp = 0
	if b_2 <= -2.7e+148:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 1.12e-205:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
	elif b_2 <= 3.5e-43:
		tmp = (t_0 / (b_2 + math.hypot(b_2, math.sqrt(t_0)))) / a
	else:
		tmp = (c / b_2) * -0.5
	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(a * Float64(-c))
	tmp = 0.0
	if (b_2 <= -2.7e+148)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 1.12e-205)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	elseif (b_2 <= 3.5e-43)
		tmp = Float64(Float64(t_0 / Float64(b_2 + hypot(b_2, sqrt(t_0)))) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	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 = a * -c;
	tmp = 0.0;
	if (b_2 <= -2.7e+148)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 1.12e-205)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	elseif (b_2 <= 3.5e-43)
		tmp = (t_0 / (b_2 + hypot(b_2, sqrt(t_0)))) / a;
	else
		tmp = (c / b_2) * -0.5;
	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[(a * (-c)), $MachinePrecision]}, If[LessEqual[b$95$2, -2.7e+148], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.12e-205], 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, 3.5e-43], N[(N[(t$95$0 / N[(b$95$2 + N[Sqrt[b$95$2 ^ 2 + N[Sqrt[t$95$0], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -2.70000000000000019e148

   1. Initial program 62.5

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

   2. Taylor expanded in b_2 around -inf 2.1

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]

   if -2.70000000000000019e148 < b_2 < 1.1200000000000001e-205

   1. Initial program 9.8

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

   if 1.1200000000000001e-205 < b_2 < 3.49999999999999997e-43

   1. Initial program 26.5

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

   2. Applied egg-rr 27.1

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

   3. Taylor expanded in b_2 around 0 23.3

      \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \left(c \cdot a\right)}}{b_2 + \mathsf{hypot}\left(b_2, \sqrt{a \cdot \left(-c\right)}\right)}}{a} \]

   4. Simplified 23.3

      \[\leadsto \frac{\frac{\color{blue}{c \cdot \left(-a\right)}}{b_2 + \mathsf{hypot}\left(b_2, \sqrt{a \cdot \left(-c\right)}\right)}}{a} \]
      Proof
      (*.f64 c (neg.f64 a)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 c a))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 c a))): 0 points increase in error, 0 points decrease in error

   if 3.49999999999999997e-43 < b_2

   1. Initial program 54.3

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

   2. Taylor expanded in b_2 around inf 7.7

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

4. Final simplification 9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.7 \cdot 10^{+148}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.12 \cdot 10^{-205}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 3.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{a \cdot \left(-c\right)}{b_2 + \mathsf{hypot}\left(b_2, \sqrt{a \cdot \left(-c\right)}\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternatives

Alternative 1
Error	13.6
Cost	35792

\[\begin{array}{l} t_0 := \frac{c}{b_2} \cdot -0.5\\ t_1 := \frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-280}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	13.5
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.55 \cdot 10^{-80}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 4.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 3
Error	22.4
Cost	836

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 0:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2} + b_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 4
Error	22.3
Cost	836

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 0:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 5
Error	39.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.55 \cdot 10^{-263}:\\ \;\;\;\;b_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]

Alternative 6
Error	22.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 4.5 \cdot 10^{-281}:\\ \;\;\;\;b_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b_2}{c}}\\ \end{array} \]

Alternative 7
Error	22.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 4.5 \cdot 10^{-281}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b_2}{c}}\\ \end{array} \]

Alternative 8
Error	22.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 4.5 \cdot 10^{-281}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b_2}\\ \end{array} \]

Alternative 9
Error	22.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 4.5 \cdot 10^{-281}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 10
Error	53.2
Cost	388

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.55 \cdot 10^{-263}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]

Alternative 11
Error	56.3
Cost	192

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

Reproduce

herbie shell --seed 2022291 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))