quadp (p42, positive)

Average Error: 34.6 → 10.9

Time: 7.8s

Precision: binary64

Cost: 8002

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -7.102439644234899 \cdot 10^{-32}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 9.908180707161701 \cdot 10^{-62}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right) + b \cdot b} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.102439644234899e-32)
   (/ (- b) a)
   (if (<= b 9.908180707161701e-62)
     (/ (- (sqrt (+ (* a (* c -4.0)) (* b b))) b) (* a 2.0))
     (- (/ c b)))))

C

double code(double a, double b, double c) {
	return (-b + sqrt((b * b) - (4.0 * (a * c)))) / (2.0 * a);
}

↓

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.102439644234899e-32) {
		tmp = -b / a;
	} else if (b <= 9.908180707161701e-62) {
		tmp = (sqrt((a * (c * -4.0)) + (b * b)) - b) / (a * 2.0);
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.6
Target	21.3
Herbie	10.9

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

Alternatives

Alternative 1
Error	10.9
Cost	8002

\[\begin{array}{l} \mathbf{if}\;b \leq -7.102439644234899 \cdot 10^{-32}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.227805983638796 \cdot 10^{-61}:\\ \;\;\;\;\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Alternative 2
Error	13.1
Cost	7746

\[\begin{array}{l} \mathbf{if}\;b \leq -4.398136104036632 \cdot 10^{-48}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.8770875812605243 \cdot 10^{-64}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Alternative 3
Error	13.5
Cost	7618

\[\begin{array}{l} \mathbf{if}\;b \leq -5.476283559993845 \cdot 10^{-124}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 2.2988954179358907 \cdot 10^{-63}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Alternative 4
Error	13.5
Cost	7618

\[\begin{array}{l} \mathbf{if}\;b \leq -4.7430289154668276 \cdot 10^{-124}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.011243506933701 \cdot 10^{-60}:\\ \;\;\;\;\frac{0.5}{a} \cdot \sqrt{a \cdot \left(c \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Alternative 5
Error	22.4
Cost	577

\[\begin{array}{l} \mathbf{if}\;b \leq 1.2380567347774514 \cdot 10^{-278}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Alternative 6
Error	39.4
Cost	577

\[\begin{array}{l} \mathbf{if}\;b \leq 5.7901680992929 \cdot 10^{-312}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 7
Error	56.1
Cost	64

\[0\]

Alternative 8
Error	61.5
Cost	64

\[1\]

Derivation

1. Split input into 3 regimes

2. if b < -7.1024396442348992e-32

   1. Initial program 30.2

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

   2. Simplified 30.2

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}}\]

   3. Taylor expanded around -inf 8.7

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}}\]

   4. Simplified 8.7

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

   5. Simplified 8.7

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

   if -7.1024396442348992e-32 < b < 9.9081807071617013e-62

   1. Initial program 15.4

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

   2. Simplified 15.4

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}}\]
   3. Using strategy rm

   4. Applied sub-neg_binary64_753 15.4

      \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}} - b}{a \cdot 2}\]

   5. Simplified 15.5

      \[\leadsto \frac{\sqrt{b \cdot b + \color{blue}{a \cdot \left(c \cdot -4\right)}} - b}{a \cdot 2}\]

   6. Simplified 15.5

      \[\leadsto \color{blue}{\frac{\sqrt{a \cdot \left(c \cdot -4\right) + b \cdot b} - b}{a \cdot 2}}\]

   if 9.9081807071617013e-62 < b

   1. Initial program 54.2

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

   2. Simplified 54.2

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}}\]

   3. Taylor expanded around inf 8.3

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

   4. Simplified 8.3

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

   5. Simplified 8.3

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

4. Final simplification 10.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.102439644234899 \cdot 10^{-32}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 9.908180707161701 \cdot 10^{-62}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right) + b \cdot b} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected #f

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))