Average Error: 1.6 → 1.5
Time: 25.5s
Precision: 64
\[\frac{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2}}{a}\]
\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{\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2}}{a}
double f(double a, double b, double c) {
        double r360762 = b;
        double r360763 = -r360762;
        double r360764 = r360762 * r360762;
        double r360765 = 4.0;
        double r360766 = /* ERROR: no posit support in C */;
        double r360767 = a;
        double r360768 = c;
        double r360769 = r360767 * r360768;
        double r360770 = r360766 * r360769;
        double r360771 = r360764 - r360770;
        double r360772 = sqrt(r360771);
        double r360773 = r360763 + r360772;
        double r360774 = 2.0;
        double r360775 = /* ERROR: no posit support in C */;
        double r360776 = r360775 * r360767;
        double r360777 = r360773 / r360776;
        return r360777;
}
↓
double f(double a, double b, double c) {
        double r360778 = b;
        double r360779 = r360778 * r360778;
        double r360780 = c;
        double r360781 = a;
        double r360782 = 4.0;
        double r360783 = r360781 * r360782;
        double r360784 = r360780 * r360783;
        double r360785 = r360779 - r360784;
        double r360786 = sqrt(r360785);
        double r360787 = r360786 - r360778;
        double r360788 = 2.0;
        double r360789 = r360787 / r360788;
        double r360790 = r360789 / r360781;
        return r360790;
}

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*1.5

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

  7. Final simplification1.5

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

Reproduce

herbie shell --seed 2019156 
(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)))