Average Error: 1.6 → 1.6
Time: 29.4s
Precision: 64
\[\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
\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{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
double f(double a, double b, double c) {
        double r1943374 = b;
        double r1943375 = -r1943374;
        double r1943376 = r1943374 * r1943374;
        double r1943377 = 4.0;
        double r1943378 = /* ERROR: no posit support in C */;
        double r1943379 = a;
        double r1943380 = c;
        double r1943381 = r1943379 * r1943380;
        double r1943382 = r1943378 * r1943381;
        double r1943383 = r1943376 - r1943382;
        double r1943384 = sqrt(r1943383);
        double r1943385 = r1943375 - r1943384;
        double r1943386 = 2.0;
        double r1943387 = /* ERROR: no posit support in C */;
        double r1943388 = r1943387 * r1943379;
        double r1943389 = r1943385 / r1943388;
        return r1943389;
}
↓
double f(double a, double b, double c) {
        double r1943390 = b;
        double r1943391 = -r1943390;
        double r1943392 = r1943390 * r1943390;
        double r1943393 = 4.0;
        double r1943394 = a;
        double r1943395 = r1943393 * r1943394;
        double r1943396 = c;
        double r1943397 = r1943395 * r1943396;
        double r1943398 = r1943392 - r1943397;
        double r1943399 = sqrt(r1943398);
        double r1943400 = r1943391 - r1943399;
        double r1943401 = 2.0;
        double r1943402 = r1943401 * r1943394;
        double r1943403 = r1943400 / r1943402;
        return r1943403;
}

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 associate-*r*1.6

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

  4. Final simplification1.6

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

Reproduce

herbie shell --seed 2019130 +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)))