Average Error: 1.6 → 1.5
Time: 31.7s
Precision: 64
\[\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{\left(-b\right) - \sqrt{\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), \left(4 \cdot a\right), c\right)\right)}}{2 \cdot a}\]
\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{\left(-b\right) - \sqrt{\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), \left(4 \cdot a\right), c\right)\right)}}{2 \cdot a}
double f(double a, double b, double c) {
        double r1121665 = b;
        double r1121666 = -r1121665;
        double r1121667 = r1121665 * r1121665;
        double r1121668 = 4.0;
        double r1121669 = /* ERROR: no posit support in C */;
        double r1121670 = a;
        double r1121671 = c;
        double r1121672 = r1121670 * r1121671;
        double r1121673 = r1121669 * r1121672;
        double r1121674 = r1121667 - r1121673;
        double r1121675 = sqrt(r1121674);
        double r1121676 = r1121666 - r1121675;
        double r1121677 = 2.0;
        double r1121678 = /* ERROR: no posit support in C */;
        double r1121679 = r1121678 * r1121670;
        double r1121680 = r1121676 / r1121679;
        return r1121680;
}
↓
double f(double a, double b, double c) {
        double r1121681 = b;
        double r1121682 = -r1121681;
        double r1121683 = r1121681 * r1121681;
        double r1121684 = /*Error: no posit support in C */;
        double r1121685 = 4.0;
        double r1121686 = a;
        double r1121687 = r1121685 * r1121686;
        double r1121688 = c;
        double r1121689 = /*Error: no posit support in C */;
        double r1121690 = /*Error: no posit support in C */;
        double r1121691 = sqrt(r1121690);
        double r1121692 = r1121682 - r1121691;
        double r1121693 = 2.0;
        double r1121694 = r1121693 * r1121686;
        double r1121695 = r1121692 / r1121694;
        return r1121695;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 1.6

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

  3. Applied -p16-rgt-identity-expand1.6

    \[\leadsto \frac{\left(\color{blue}{\left(\left(-b\right) - \left(0.0\right)\right)} - \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]

  4. Applied associate--l-1.6

    \[\leadsto \frac{\color{blue}{\left(\left(-b\right) - \left(\frac{\left(0.0\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)\right)}}{\left(\left(2\right) \cdot a\right)}\]

  5. Simplified1.5

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

  7. Applied introduce-quire1.5

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

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

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

  9. Final simplification1.5

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

Reproduce

herbie shell --seed 2019163 
(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)))