Average Error: 1.7 → 1.7
Time: 15.4s
Precision: 64
\[\frac{\left(\frac{\left(-b_2\right)}{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)}\right)}{a}\]
↓
\[\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\]
\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{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}
double f(double a, double b_2, double c) {
        double r989149 = b_2;
        double r989150 = -r989149;
        double r989151 = r989149 * r989149;
        double r989152 = a;
        double r989153 = c;
        double r989154 = r989152 * r989153;
        double r989155 = r989151 - r989154;
        double r989156 = sqrt(r989155);
        double r989157 = r989150 + r989156;
        double r989158 = r989157 / r989152;
        return r989158;
}
↓
double f(double a, double b_2, double c) {
        double r989159 = b_2;
        double r989160 = r989159 * r989159;
        double r989161 = c;
        double r989162 = a;
        double r989163 = r989161 * r989162;
        double r989164 = r989160 - r989163;
        double r989165 = sqrt(r989164);
        double r989166 = r989165 - r989159;
        double r989167 = r989166 / r989162;
        return r989167;
}

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. Final simplification1.7

    \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\]

Reproduce

herbie shell --seed 2019104 
(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))