Average Error: 1.5 → 1.5
Time: 48.7s
Precision: 64
\[\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{1.0}{2} \cdot \frac{\left(-b\right) - \sqrt{\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), 4, \left(a \cdot c\right)\right)\right)}}{a}\]
\frac{\left(\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{\left(-b\right) - \sqrt{\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), 4, \left(a \cdot c\right)\right)\right)}}{a}
double f(double a, double b, double c) {
        double r456583 = b;
        double r456584 = -r456583;
        double r456585 = r456583 * r456583;
        double r456586 = 4.0;
        double r456587 = /* ERROR: no posit support in C */;
        double r456588 = a;
        double r456589 = c;
        double r456590 = r456588 * r456589;
        double r456591 = r456587 * r456590;
        double r456592 = r456585 - r456591;
        double r456593 = sqrt(r456592);
        double r456594 = r456584 - r456593;
        double r456595 = 2.0;
        double r456596 = /* ERROR: no posit support in C */;
        double r456597 = r456596 * r456588;
        double r456598 = r456594 / r456597;
        return r456598;
}
↓
double f(double a, double b, double c) {
        double r456599 = 1.0;
        double r456600 = 2.0;
        double r456601 = r456599 / r456600;
        double r456602 = b;
        double r456603 = -r456602;
        double r456604 = r456602 * r456602;
        double r456605 = /*Error: no posit support in C */;
        double r456606 = 4.0;
        double r456607 = a;
        double r456608 = c;
        double r456609 = r456607 * r456608;
        double r456610 = /*Error: no posit support in C */;
        double r456611 = /*Error: no posit support in C */;
        double r456612 = sqrt(r456611);
        double r456613 = r456603 - r456612;
        double r456614 = r456613 / r456607;
        double r456615 = r456601 * r456614;
        return r456615;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 1.5

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

  3. Applied introduce-quire1.5

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

  4. Applied insert-quire-fdp-sub1.5

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

  6. Applied *p16-lft-identity-expand1.5

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

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

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

  8. Applied distribute-lft-out--1.5

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

  9. Applied p16-times-frac1.5

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

  10. Final simplification1.5

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

Reproduce

herbie shell --seed 2019153 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  (/.p16 (-.p16 (neg.p16 b) (sqrt.p16 (-.p16 (*.p16 b b) (*.p16 (real->posit16 4) (*.p16 a c))))) (*.p16 (real->posit16 2) a)))