Average Error: 1.5 → 1.5
Time: 36.2s
Precision: 64
\[\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{\frac{\left(-b\right) - \sqrt{\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), 4, \left(a \cdot c\right)\right)\right)}}{2}}{a}\]
\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{\frac{\left(-b\right) - \sqrt{\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), 4, \left(a \cdot c\right)\right)\right)}}{2}}{a}
double f(double a, double b, double c) {
        double r435525 = b;
        double r435526 = -r435525;
        double r435527 = r435525 * r435525;
        double r435528 = 4.0;
        double r435529 = /* ERROR: no posit support in C */;
        double r435530 = a;
        double r435531 = c;
        double r435532 = r435530 * r435531;
        double r435533 = r435529 * r435532;
        double r435534 = r435527 - r435533;
        double r435535 = sqrt(r435534);
        double r435536 = r435526 - r435535;
        double r435537 = 2.0;
        double r435538 = /* ERROR: no posit support in C */;
        double r435539 = r435538 * r435530;
        double r435540 = r435536 / r435539;
        return r435540;
}
↓
double f(double a, double b, double c) {
        double r435541 = b;
        double r435542 = -r435541;
        double r435543 = r435541 * r435541;
        double r435544 = /*Error: no posit support in C */;
        double r435545 = 4.0;
        double r435546 = a;
        double r435547 = c;
        double r435548 = r435546 * r435547;
        double r435549 = /*Error: no posit support in C */;
        double r435550 = /*Error: no posit support in C */;
        double r435551 = sqrt(r435550);
        double r435552 = r435542 - r435551;
        double r435553 = 2.0;
        double r435554 = r435552 / r435553;
        double r435555 = r435554 / r435546;
        return r435555;
}

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 associate-/r*1.5

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

  7. Final simplification1.5

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

Reproduce

herbie shell --seed 2019155 
(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)))