Cubic critical

Average Error: 34.8 → 10.2

Time: 11.2s

Precision: binary64

Cost: 8002

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -5.45892145913632 \cdot 10^{+153}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.2901844560369386 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.45892145913632e+153)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 1.2901844560369386e-39)
     (/ (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) 3.0) a)
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	return (-b + sqrt((b * b) - ((3.0 * a) * c))) / (3.0 * a);
}

↓

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.45892145913632e+153) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 1.2901844560369386e-39) {
		tmp = ((sqrt((b * b) - ((3.0 * a) * c)) - b) / 3.0) / a;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Alternatives

Alternative 1
Error	10.1
Cost	8002

\[\begin{array}{l} \mathbf{if}\;b \leq -1.152442509815712 \cdot 10^{+138}:\\ \;\;\;\;\frac{\frac{b \cdot -2}{3}}{a}\\ \mathbf{elif}\;b \leq 4.7425045090226674 \cdot 10^{-42}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Alternative 2
Error	10.2
Cost	8002

\[\begin{array}{l} \mathbf{if}\;b \leq -8.887918546354078 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{b \cdot -2}{3}}{a}\\ \mathbf{elif}\;b \leq 4.3034790776983455 \cdot 10^{-41}:\\ \;\;\;\;\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Alternative 3
Error	13.9
Cost	7554

\[\begin{array}{l} \mathbf{if}\;b \leq -3.216026123156104 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{b \cdot -2}{3}}{a}\\ \mathbf{elif}\;b \leq 2.7246677044628124 \cdot 10^{-51}:\\ \;\;\;\;-\frac{c}{\sqrt{a \cdot \left(c \cdot -3\right)}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Alternative 4
Error	22.4
Cost	769

\[\begin{array}{l} \mathbf{if}\;b \leq 7.39592424827649 \cdot 10^{-212}:\\ \;\;\;\;\frac{\frac{b \cdot -2}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Alternative 5
Error	22.5
Cost	769

\[\begin{array}{l} \mathbf{if}\;b \leq 4.641734365265361 \cdot 10^{-210}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Alternative 6
Error	22.5
Cost	641

\[\begin{array}{l} \mathbf{if}\;b \leq 4.5984243801846445 \cdot 10^{-211}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Alternative 7
Error	39.3
Cost	641

\[\begin{array}{l} \mathbf{if}\;b \leq 4.1858178371903 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 8
Error	56.2
Cost	64

\[0\]

Alternative 9
Error	61.6
Cost	64

\[1\]

Derivation

1. Split input into 3 regimes

2. if b < -5.45892145913632016e153

   1. Initial program 63.8

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

   2. Simplified 63.8

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}}\]

   3. Taylor expanded around -inf 2.4

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a}\]

   4. Simplified 2.4

      \[\leadsto \frac{\color{blue}{b \cdot -2}}{3 \cdot a}\]

   5. Simplified 2.4

      \[\leadsto \color{blue}{\frac{b \cdot -2}{3 \cdot a}}\]

   if -5.45892145913632016e153 < b < 1.29018445603693855e-39

   1. Initial program 14.2

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

   2. Simplified 14.2

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}}\]
   3. Using strategy rm

   4. Applied associate-/r*_binary64_3773 14.2

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3}}{a}}\]

   5. Simplified 14.2

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3}}{a}}\]

   if 1.29018445603693855e-39 < b

   1. Initial program 54.3

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

   2. Simplified 54.3

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}}\]

   3. Taylor expanded around inf 7.0

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]

   4. Simplified 7.0

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

4. Final simplification 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.45892145913632 \cdot 10^{+153}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.2901844560369386 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))