Average Error: 1.5 → 1.5
Time: 46.5s
Precision: 64
\[\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{1.0}{2} \cdot \frac{\left(-b\right) - \sqrt{\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), \left(4 \cdot a\right), c\right)\right)}}{a}\]
\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), \left(4 \cdot a\right), c\right)\right)}}{a}
double f(double a, double b, double c) {
        double r443551 = b;
        double r443552 = -r443551;
        double r443553 = r443551 * r443551;
        double r443554 = 4.0;
        double r443555 = /* ERROR: no posit support in C */;
        double r443556 = a;
        double r443557 = c;
        double r443558 = r443556 * r443557;
        double r443559 = r443555 * r443558;
        double r443560 = r443553 - r443559;
        double r443561 = sqrt(r443560);
        double r443562 = r443552 - r443561;
        double r443563 = 2.0;
        double r443564 = /* ERROR: no posit support in C */;
        double r443565 = r443564 * r443556;
        double r443566 = r443562 / r443565;
        return r443566;
}
↓
double f(double a, double b, double c) {
        double r443567 = 1.0;
        double r443568 = 2.0;
        double r443569 = r443567 / r443568;
        double r443570 = b;
        double r443571 = -r443570;
        double r443572 = r443570 * r443570;
        double r443573 = /*Error: no posit support in C */;
        double r443574 = 4.0;
        double r443575 = a;
        double r443576 = r443574 * r443575;
        double r443577 = c;
        double r443578 = /*Error: no posit support in C */;
        double r443579 = /*Error: no posit support in C */;
        double r443580 = sqrt(r443579);
        double r443581 = r443571 - r443580;
        double r443582 = r443581 / r443575;
        double r443583 = r443569 * r443582;
        return r443583;
}

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 associate-*r*1.5

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

  5. Applied introduce-quire1.5

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

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

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

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

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

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

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

  10. Applied distribute-rgt-out--1.5

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

  11. Applied p16-times-frac1.5

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

  12. Final simplification1.5

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

Reproduce

herbie shell --seed 2019152 
(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)))