Average Error: 1.5 → 1.5
Time: 16.9s
Precision: 64
\[\frac{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{\sqrt{\frac{b \cdot b + \left(c \cdot a\right) \cdot 4}{\frac{b \cdot b + \left(c \cdot a\right) \cdot 4}{b \cdot b - \left(c \cdot a\right) \cdot 4}}} - b}{2 \cdot a}\]
\frac{\left(\frac{\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{\sqrt{\frac{b \cdot b + \left(c \cdot a\right) \cdot 4}{\frac{b \cdot b + \left(c \cdot a\right) \cdot 4}{b \cdot b - \left(c \cdot a\right) \cdot 4}}} - b}{2 \cdot a}
double f(double a, double b, double c) {
        double r512637 = b;
        double r512638 = -r512637;
        double r512639 = r512637 * r512637;
        double r512640 = 4.0;
        double r512641 = /* ERROR: no posit support in C */;
        double r512642 = a;
        double r512643 = c;
        double r512644 = r512642 * r512643;
        double r512645 = r512641 * r512644;
        double r512646 = r512639 - r512645;
        double r512647 = sqrt(r512646);
        double r512648 = r512638 + r512647;
        double r512649 = 2.0;
        double r512650 = /* ERROR: no posit support in C */;
        double r512651 = r512650 * r512642;
        double r512652 = r512648 / r512651;
        return r512652;
}
↓
double f(double a, double b, double c) {
        double r512653 = b;
        double r512654 = r512653 * r512653;
        double r512655 = c;
        double r512656 = a;
        double r512657 = r512655 * r512656;
        double r512658 = 4.0;
        double r512659 = r512657 * r512658;
        double r512660 = r512654 + r512659;
        double r512661 = r512654 - r512659;
        double r512662 = r512660 / r512661;
        double r512663 = r512660 / r512662;
        double r512664 = sqrt(r512663);
        double r512665 = r512664 - r512653;
        double r512666 = 2.0;
        double r512667 = r512666 * r512656;
        double r512668 = r512665 / r512667;
        return r512668;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 1.5

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

  2. Simplified1.5

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

  4. Applied p16-flip--2.6

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

  6. Applied difference-of-squares2.5

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

  7. Applied associate-/l*1.5

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

  8. Final simplification1.5

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

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"
  (/.p16 (+.p16 (neg.p16 b) (sqrt.p16 (-.p16 (*.p16 b b) (*.p16 (real->posit16 4) (*.p16 a c))))) (*.p16 (real->posit16 2) a)))