Average Error: 1.7 → 1.8
Time: 15.7s
Precision: 64
\[\frac{\left(\frac{\left(-b_2\right)}{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)}\right)}{a}\]
↓
\[\frac{1.0}{a} \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)\]
\frac{\left(\frac{\left(-b_2\right)}{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)}\right)}{a}
↓
\frac{1.0}{a} \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)
double f(double a, double b_2, double c) {
        double r335790 = b_2;
        double r335791 = -r335790;
        double r335792 = r335790 * r335790;
        double r335793 = a;
        double r335794 = c;
        double r335795 = r335793 * r335794;
        double r335796 = r335792 - r335795;
        double r335797 = sqrt(r335796);
        double r335798 = r335791 + r335797;
        double r335799 = r335798 / r335793;
        return r335799;
}
↓
double f(double a, double b_2, double c) {
        double r335800 = 1.0;
        double r335801 = a;
        double r335802 = r335800 / r335801;
        double r335803 = b_2;
        double r335804 = r335803 * r335803;
        double r335805 = c;
        double r335806 = r335805 * r335801;
        double r335807 = r335804 - r335806;
        double r335808 = sqrt(r335807);
        double r335809 = r335808 - r335803;
        double r335810 = r335802 * r335809;
        return r335810;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Initial program 1.7

    \[\frac{\left(\frac{\left(-b_2\right)}{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)}\right)}{a}\]

  2. Simplified1.7

    \[\leadsto \color{blue}{\frac{\left(\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right) - b_2\right)}{a}}\]
  3. Using strategy rm

  4. Applied p16-*-un-lft-identity1.7

    \[\leadsto \frac{\color{blue}{\left(\left(1.0\right) \cdot \left(\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right) - b_2\right)\right)}}{a}\]

  5. Applied associate-/l*1.9

    \[\leadsto \color{blue}{\frac{\left(1.0\right)}{\left(\frac{a}{\left(\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right) - b_2\right)}\right)}}\]
  6. Using strategy rm

  7. Applied associate-/r/1.8

    \[\leadsto \color{blue}{\left(\frac{\left(1.0\right)}{a}\right) \cdot \left(\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right) - b_2\right)}\]

  8. Final simplification1.8

    \[\leadsto \frac{1.0}{a} \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)\]

Reproduce

herbie shell --seed 2019156 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/.p16 (+.p16 (neg.p16 b_2) (sqrt.p16 (-.p16 (*.p16 b_2 b_2) (*.p16 a c)))) a))