Average Error: 1.6 → 1.6
Time: 26.2s
Precision: 64
\[\frac{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\]
\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{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}
double f(double a, double b, double c) {
        double r393764 = b;
        double r393765 = -r393764;
        double r393766 = r393764 * r393764;
        double r393767 = 4.0;
        double r393768 = /* ERROR: no posit support in C */;
        double r393769 = a;
        double r393770 = c;
        double r393771 = r393769 * r393770;
        double r393772 = r393768 * r393771;
        double r393773 = r393766 - r393772;
        double r393774 = sqrt(r393773);
        double r393775 = r393765 + r393774;
        double r393776 = 2.0;
        double r393777 = /* ERROR: no posit support in C */;
        double r393778 = r393777 * r393769;
        double r393779 = r393775 / r393778;
        return r393779;
}
↓
double f(double a, double b, double c) {
        double r393780 = b;
        double r393781 = r393780 * r393780;
        double r393782 = c;
        double r393783 = a;
        double r393784 = 4.0;
        double r393785 = r393783 * r393784;
        double r393786 = r393782 * r393785;
        double r393787 = r393781 - r393786;
        double r393788 = sqrt(r393787);
        double r393789 = r393788 - r393780;
        double r393790 = 2.0;
        double r393791 = r393783 * r393790;
        double r393792 = r393789 / r393791;
        return r393792;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 1.6

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

  2. Simplified1.6

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

  4. Applied associate-/r*1.6

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

  6. Applied associate-*l*1.6

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

  8. Applied associate-/l/1.6

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

  9. Final simplification1.6

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

Reproduce

herbie shell --seed 2019155 
(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)))