The quadratic formula (r2)

Average Error: 34.8 → 20.2

Time: 4.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -1.33564804854513 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{\left(a \cdot c\right) \cdot -2}{a}}{b}\\ \mathbf{elif}\;b \leq -4.6696733198681835 \cdot 10^{-235}:\\ \;\;\;\;\frac{\frac{\left(a \cdot c\right) \cdot -2}{a}}{b - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}\\ \mathbf{elif}\;b \leq 1.3355072207722054 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a \cdot 2}\\ \end{array}\]

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 -1.33564804854513e+154)
   (/ (/ (* (* a c) -2.0) a) b)
   (if (<= b -4.6696733198681835e-235)
     (/ (/ (* (* a c) -2.0) a) (- b (sqrt (- (* b b) (* (* a c) 4.0)))))
     (if (<= b 1.3355072207722054e+154)
       (/ (- (- b) (sqrt (- (* b b) (* (* a c) 4.0)))) (* a 2.0))
       (/ (- b) (* a 2.0))))))

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 <= -1.33564804854513e+154) {
		tmp = (((a * c) * -2.0) / a) / b;
	} else if (b <= -4.6696733198681835e-235) {
		tmp = (((a * c) * -2.0) / a) / (b - sqrt((b * b) - ((a * c) * 4.0)));
	} else if (b <= 1.3355072207722054e+154) {
		tmp = (-b - sqrt((b * b) - ((a * c) * 4.0))) / (a * 2.0);
	} else {
		tmp = -b / (a * 2.0);
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.8
Target	21.1
Herbie	20.2

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

Derivation

1. Split input into 4 regimes

2. if b < -1.33564804854513e154

   1. Initial program 64.0

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

   2. Simplified 64.0

      \[\leadsto \color{blue}{\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-0.5}{a}}\]
   3. Using strategy rm

   4. Applied flip-+_binary64 64.0

      \[\leadsto \color{blue}{\frac{b \cdot b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}} \cdot \frac{-0.5}{a}\]

   5. Applied associate-*l/_binary64 64.0

      \[\leadsto \color{blue}{\frac{\left(b \cdot b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-0.5}{a}}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

   6. Simplified 36.7

      \[\leadsto \frac{\color{blue}{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{-0.5}{a}}}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\]
   7. Using strategy rm

   8. Applied associate-*r/_binary64 36.7

      \[\leadsto \frac{\color{blue}{\frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot -0.5}{a}}}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\]

   9. Simplified 36.7

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

   10. Taylor expanded around 0 32.5

       \[\leadsto \frac{\frac{\left(a \cdot c\right) \cdot -2}{a}}{b - \color{blue}{0}}\]

   if -1.33564804854513e154 < b < -4.66967331986818345e-235

   1. Initial program 36.8

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

   2. Simplified 36.8

      \[\leadsto \color{blue}{\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-0.5}{a}}\]
   3. Using strategy rm

   4. Applied flip-+_binary64 36.9

      \[\leadsto \color{blue}{\frac{b \cdot b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}} \cdot \frac{-0.5}{a}\]

   5. Applied associate-*l/_binary64 36.9

      \[\leadsto \color{blue}{\frac{\left(b \cdot b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{-0.5}{a}}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

   6. Simplified 13.9

      \[\leadsto \frac{\color{blue}{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{-0.5}{a}}}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\]
   7. Using strategy rm

   8. Applied associate-*r/_binary64 13.8

      \[\leadsto \frac{\color{blue}{\frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot -0.5}{a}}}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\]

   9. Simplified 13.8

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

   if -4.66967331986818345e-235 < b < 1.3355072207722054e154

   1. Initial program 10.3

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

   if 1.3355072207722054e154 < b

   1. Initial program 64.0

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

   2. Taylor expanded around 0 52.2

      \[\leadsto \frac{\left(-b\right) - \color{blue}{0}}{2 \cdot a}\]
3. Recombined 4 regimes into one program.

4. Final simplification 20.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.33564804854513 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{\left(a \cdot c\right) \cdot -2}{a}}{b}\\ \mathbf{elif}\;b \leq -4.6696733198681835 \cdot 10^{-235}:\\ \;\;\;\;\frac{\frac{\left(a \cdot c\right) \cdot -2}{a}}{b - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}\\ \mathbf{elif}\;b \leq 1.3355072207722054 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020288 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

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

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