Average Error: 1.6 → 1.5
Time: 43.1s
Precision: 64
\[\frac{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{1.0}{2} \cdot \frac{\sqrt{\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), \left(c \cdot a\right), 4\right)\right)} - b}{a}\]
\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{\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), \left(c \cdot a\right), 4\right)\right)} - b}{a}
double f(double a, double b, double c) {
        double r566674 = b;
        double r566675 = -r566674;
        double r566676 = r566674 * r566674;
        double r566677 = 4.0;
        double r566678 = /* ERROR: no posit support in C */;
        double r566679 = a;
        double r566680 = c;
        double r566681 = r566679 * r566680;
        double r566682 = r566678 * r566681;
        double r566683 = r566676 - r566682;
        double r566684 = sqrt(r566683);
        double r566685 = r566675 + r566684;
        double r566686 = 2.0;
        double r566687 = /* ERROR: no posit support in C */;
        double r566688 = r566687 * r566679;
        double r566689 = r566685 / r566688;
        return r566689;
}
↓
double f(double a, double b, double c) {
        double r566690 = 1.0;
        double r566691 = 2.0;
        double r566692 = r566690 / r566691;
        double r566693 = b;
        double r566694 = r566693 * r566693;
        double r566695 = /*Error: no posit support in C */;
        double r566696 = c;
        double r566697 = a;
        double r566698 = r566696 * r566697;
        double r566699 = 4.0;
        double r566700 = /*Error: no posit support in C */;
        double r566701 = /*Error: no posit support in C */;
        double r566702 = sqrt(r566701);
        double r566703 = r566702 - r566693;
        double r566704 = r566703 / r566697;
        double r566705 = r566692 * r566704;
        return r566705;
}

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 introduce-quire1.6

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

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

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

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

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

  8. Applied p16-times-frac1.5

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

  9. Final simplification1.5

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

Reproduce

herbie shell --seed 2019153 
(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)))