The quadratic formula (r2)

Average Error: 33.9 → 11.3

Time: 16.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.869662346631121401645595393947635525169 \cdot 10^{101}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r91436 = b;
        double r91437 = -r91436;
        double r91438 = r91436 * r91436;
        double r91439 = 4.0;
        double r91440 = a;
        double r91441 = c;
        double r91442 = r91440 * r91441;
        double r91443 = r91439 * r91442;
        double r91444 = r91438 - r91443;
        double r91445 = sqrt(r91444);
        double r91446 = r91437 - r91445;
        double r91447 = 2.0;
        double r91448 = r91447 * r91440;
        double r91449 = r91446 / r91448;
        return r91449;
}

↓

double f(double a, double b, double c) {
        double r91450 = b;
        double r91451 = -1.8696623466311214e+101;
        bool r91452 = r91450 <= r91451;
        double r91453 = -1.0;
        double r91454 = c;
        double r91455 = r91454 / r91450;
        double r91456 = r91453 * r91455;
        double r91457 = 7.455592343308264e-170;
        bool r91458 = r91450 <= r91457;
        double r91459 = 2.0;
        double r91460 = r91459 * r91454;
        double r91461 = 4.0;
        double r91462 = a;
        double r91463 = r91462 * r91454;
        double r91464 = r91461 * r91463;
        double r91465 = -r91464;
        double r91466 = fma(r91450, r91450, r91465);
        double r91467 = sqrt(r91466);
        double r91468 = r91467 - r91450;
        double r91469 = r91460 / r91468;
        double r91470 = 1.0;
        double r91471 = r91450 / r91462;
        double r91472 = r91455 - r91471;
        double r91473 = r91470 * r91472;
        double r91474 = r91458 ? r91469 : r91473;
        double r91475 = r91452 ? r91456 : r91474;
        return r91475;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.9
Target	21.1
Herbie	11.3

\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if b < -1.8696623466311214e+101

   1. Initial program 59.8

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

   2. Taylor expanded around -inf 2.5

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

   if -1.8696623466311214e+101 < b < 7.455592343308264e-170

   1. Initial program 28.9

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

   3. Applied flip-- 29.1

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

   4. Simplified 16.7

      \[\leadsto \frac{\frac{\color{blue}{0 + \left(a \cdot c\right) \cdot 4}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]

   5. Simplified 16.7

      \[\leadsto \frac{\frac{0 + \left(a \cdot c\right) \cdot 4}{\color{blue}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}}}{2 \cdot a}\]
   6. Using strategy rm

   7. Applied div-inv 16.7

      \[\leadsto \color{blue}{\frac{0 + \left(a \cdot c\right) \cdot 4}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b} \cdot \frac{1}{2 \cdot a}}\]
   8. Using strategy rm

   9. Applied associate-*l/ 16.2

      \[\leadsto \color{blue}{\frac{\left(0 + \left(a \cdot c\right) \cdot 4\right) \cdot \frac{1}{2 \cdot a}}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}}\]

   10. Simplified 16.1

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

   11. Taylor expanded around 0 11.1

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

   if 7.455592343308264e-170 < b

   1. Initial program 23.0

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

   2. Taylor expanded around inf 17.1

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

   3. Simplified 17.1

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 11.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.869662346631121401645595393947635525169 \cdot 10^{101}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))