Average Error: 1.5 → 1.5
Time: 39.7s
Precision: 64
\[\frac{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{1.0}{2} \cdot \frac{\sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)} - b}{a}\]
\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{1.0}{2} \cdot \frac{\sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)} - b}{a}
double f(double a, double b, double c) {
        double r980466 = b;
        double r980467 = -r980466;
        double r980468 = r980466 * r980466;
        double r980469 = 4.0;
        double r980470 = /* ERROR: no posit support in C */;
        double r980471 = a;
        double r980472 = c;
        double r980473 = r980471 * r980472;
        double r980474 = r980470 * r980473;
        double r980475 = r980468 - r980474;
        double r980476 = sqrt(r980475);
        double r980477 = r980467 + r980476;
        double r980478 = 2.0;
        double r980479 = /* ERROR: no posit support in C */;
        double r980480 = r980479 * r980471;
        double r980481 = r980477 / r980480;
        return r980481;
}
↓
double f(double a, double b, double c) {
        double r980482 = 1.0;
        double r980483 = 2.0;
        double r980484 = r980482 / r980483;
        double r980485 = b;
        double r980486 = r980485 * r980485;
        double r980487 = a;
        double r980488 = 4.0;
        double r980489 = c;
        double r980490 = r980488 * r980489;
        double r980491 = r980487 * r980490;
        double r980492 = r980486 - r980491;
        double r980493 = sqrt(r980492);
        double r980494 = r980493 - r980485;
        double r980495 = r980494 / r980487;
        double r980496 = r980484 * r980495;
        return r980496;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 1.5

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

  2. Simplified1.5

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

  4. Applied *p16-rgt-identity-expand1.5

    \[\leadsto \frac{\left(\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(c \cdot a\right) \cdot \left(4\right)\right)\right)}\right) - b\right)}{\left(\color{blue}{\left(\left(2\right) \cdot \left(1.0\right)\right)} \cdot a\right)}\]

  5. Applied associate-*l*1.5

    \[\leadsto \frac{\left(\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(c \cdot a\right) \cdot \left(4\right)\right)\right)}\right) - b\right)}{\color{blue}{\left(\left(2\right) \cdot \left(\left(1.0\right) \cdot a\right)\right)}}\]

  6. Applied p16-*-un-lft-identity1.5

    \[\leadsto \frac{\color{blue}{\left(\left(1.0\right) \cdot \left(\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(c \cdot a\right) \cdot \left(4\right)\right)\right)}\right) - b\right)\right)}}{\left(\left(2\right) \cdot \left(\left(1.0\right) \cdot a\right)\right)}\]

  7. Applied p16-times-frac1.5

    \[\leadsto \color{blue}{\left(\frac{\left(1.0\right)}{\left(2\right)}\right) \cdot \left(\frac{\left(\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(c \cdot a\right) \cdot \left(4\right)\right)\right)}\right) - b\right)}{\left(\left(1.0\right) \cdot a\right)}\right)}\]

  8. Simplified1.5

    \[\leadsto \left(\frac{\left(1.0\right)}{\left(2\right)}\right) \cdot \color{blue}{\left(\frac{\left(\left(\sqrt{\left(\left(b \cdot b\right) - \left(a \cdot \left(\left(4\right) \cdot c\right)\right)\right)}\right) - b\right)}{a}\right)}\]

  9. Final simplification1.5

    \[\leadsto \frac{1.0}{2} \cdot \frac{\sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)} - b}{a}\]

Reproduce

herbie shell --seed 2019158 
(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)))