Average Error: 1.6 → 1.5
Time: 17.5s
Precision: 64
\[\frac{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\]
\frac{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}{\left(\left(2\right) \cdot a\right)}
↓
\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}
double f(double a, double b, double c) {
        double r842464 = b;
        double r842465 = -r842464;
        double r842466 = r842464 * r842464;
        double r842467 = 4.0;
        double r842468 = /* ERROR: no posit support in C */;
        double r842469 = a;
        double r842470 = c;
        double r842471 = r842469 * r842470;
        double r842472 = r842468 * r842471;
        double r842473 = r842466 - r842472;
        double r842474 = sqrt(r842473);
        double r842475 = r842465 + r842474;
        double r842476 = 2.0;
        double r842477 = /* ERROR: no posit support in C */;
        double r842478 = r842477 * r842469;
        double r842479 = r842475 / r842478;
        return r842479;
}
↓
double f(double a, double b, double c) {
        double r842480 = b;
        double r842481 = r842480 * r842480;
        double r842482 = c;
        double r842483 = a;
        double r842484 = 4.0;
        double r842485 = r842483 * r842484;
        double r842486 = r842482 * r842485;
        double r842487 = r842481 - r842486;
        double r842488 = sqrt(r842487);
        double r842489 = r842488 - r842480;
        double r842490 = 2.0;
        double r842491 = r842483 * r842490;
        double r842492 = r842489 / r842491;
        return r842492;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 1.6

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

  2. Simplified1.6

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

  4. Applied associate-/r*1.5

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

  6. Applied associate-*l*1.5

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

  8. Applied associate-/l/1.5

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

  9. Final simplification1.5

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

Reproduce

herbie shell --seed 2019121 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  (/.p16 (+.p16 (neg.p16 b) (sqrt.p16 (-.p16 (*.p16 b b) (*.p16 (real->posit16 4) (*.p16 a c))))) (*.p16 (real->posit16 2) a)))