Average Error: 1.5 → 1.5
Time: 48.0s
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 r443541 = b;
        double r443542 = -r443541;
        double r443543 = r443541 * r443541;
        double r443544 = 4.0;
        double r443545 = /* ERROR: no posit support in C */;
        double r443546 = a;
        double r443547 = c;
        double r443548 = r443546 * r443547;
        double r443549 = r443545 * r443548;
        double r443550 = r443543 - r443549;
        double r443551 = sqrt(r443550);
        double r443552 = r443542 - r443551;
        double r443553 = 2.0;
        double r443554 = /* ERROR: no posit support in C */;
        double r443555 = r443554 * r443546;
        double r443556 = r443552 / r443555;
        return r443556;
}
↓
double f(double a, double b, double c) {
        double r443557 = 1.0;
        double r443558 = 2.0;
        double r443559 = r443557 / r443558;
        double r443560 = b;
        double r443561 = -r443560;
        double r443562 = r443560 * r443560;
        double r443563 = /*Error: no posit support in C */;
        double r443564 = 4.0;
        double r443565 = a;
        double r443566 = r443564 * r443565;
        double r443567 = c;
        double r443568 = /*Error: no posit support in C */;
        double r443569 = /*Error: no posit support in C */;
        double r443570 = sqrt(r443569);
        double r443571 = r443561 - r443570;
        double r443572 = r443571 / r443565;
        double r443573 = r443559 * r443572;
        return r443573;
}

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-*-un-lft-identity1.5

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

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

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

  10. Applied distribute-lft-out--1.5

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

  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 +o rules:numerics
(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)))