quadp (p42, positive)

Average Error: 33.4 → 17.7

Time: 31.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le 4.3423124612709 \cdot 10^{-163}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} - b}{a}}{2}\\ \mathbf{elif}\;b \le 3.1351721739575554 \cdot 10^{+98}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{\sqrt{b + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}}{a} \cdot \frac{1}{\sqrt{b + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{2 \cdot b}}{a}}{2}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r943465 = b;
        double r943466 = -r943465;
        double r943467 = r943465 * r943465;
        double r943468 = 4.0;
        double r943469 = a;
        double r943470 = c;
        double r943471 = r943469 * r943470;
        double r943472 = r943468 * r943471;
        double r943473 = r943467 - r943472;
        double r943474 = sqrt(r943473);
        double r943475 = r943466 + r943474;
        double r943476 = 2.0;
        double r943477 = r943476 * r943469;
        double r943478 = r943475 / r943477;
        return r943478;
}

↓

double f(double a, double b, double c) {
        double r943479 = b;
        double r943480 = 4.3423124612709e-163;
        bool r943481 = r943479 <= r943480;
        double r943482 = a;
        double r943483 = c;
        double r943484 = r943482 * r943483;
        double r943485 = -4.0;
        double r943486 = r943479 * r943479;
        double r943487 = fma(r943484, r943485, r943486);
        double r943488 = sqrt(r943487);
        double r943489 = r943488 - r943479;
        double r943490 = r943489 / r943482;
        double r943491 = 2.0;
        double r943492 = r943490 / r943491;
        double r943493 = 3.1351721739575554e+98;
        bool r943494 = r943479 <= r943493;
        double r943495 = 0.0;
        double r943496 = fma(r943484, r943485, r943495);
        double r943497 = r943479 + r943488;
        double r943498 = sqrt(r943497);
        double r943499 = r943496 / r943498;
        double r943500 = r943499 / r943482;
        double r943501 = 1.0;
        double r943502 = r943501 / r943498;
        double r943503 = r943500 * r943502;
        double r943504 = r943503 / r943491;
        double r943505 = r943491 * r943479;
        double r943506 = r943496 / r943505;
        double r943507 = r943506 / r943482;
        double r943508 = r943507 / r943491;
        double r943509 = r943494 ? r943504 : r943508;
        double r943510 = r943481 ? r943492 : r943509;
        return r943510;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.4
Target	20.3
Herbie	17.7

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

Derivation

1. Split input into 3 regimes

2. if b < 4.3423124612709e-163

   1. Initial program 20.2

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

   2. Simplified 20.2

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

   if 4.3423124612709e-163 < b < 3.1351721739575554e+98

   1. Initial program 38.3

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

   2. Simplified 38.3

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

   4. Applied flip-- 38.4

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

   5. Simplified 15.8

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

   7. Applied *-un-lft-identity 15.8

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

   8. Applied add-sqr-sqrt 16.0

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

   9. Applied *-un-lft-identity 16.0

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

   10. Applied times-frac 16.0

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

   11. Applied times-frac 15.4

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

   if 3.1351721739575554e+98 < b

   1. Initial program 58.3

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

   2. Simplified 58.3

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

   4. Applied flip-- 58.4

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

   5. Simplified 31.8

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

   6. Taylor expanded around 0 14.2

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

4. Final simplification 17.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 4.3423124612709 \cdot 10^{-163}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} - b}{a}}{2}\\ \mathbf{elif}\;b \le 3.1351721739575554 \cdot 10^{+98}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{\sqrt{b + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}}{a} \cdot \frac{1}{\sqrt{b + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{2 \cdot b}}{a}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0) (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))))

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