Average Error: 1.7 → 1.7
Time: 23.1s
Precision: 64
\[\frac{\left(\frac{\left(-b_2\right)}{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)}\right)}{a}\]
↓
\[\frac{\sqrt{\frac{b_2 \cdot b_2 + c \cdot a}{\frac{b_2 \cdot b_2 + c \cdot a}{b_2 \cdot b_2 - c \cdot a}}} - b_2}{a}\]
\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{\frac{b_2 \cdot b_2 + c \cdot a}{\frac{b_2 \cdot b_2 + c \cdot a}{b_2 \cdot b_2 - c \cdot a}}} - b_2}{a}
double f(double a, double b_2, double c) {
        double r309669 = b_2;
        double r309670 = -r309669;
        double r309671 = r309669 * r309669;
        double r309672 = a;
        double r309673 = c;
        double r309674 = r309672 * r309673;
        double r309675 = r309671 - r309674;
        double r309676 = sqrt(r309675);
        double r309677 = r309670 + r309676;
        double r309678 = r309677 / r309672;
        return r309678;
}
↓
double f(double a, double b_2, double c) {
        double r309679 = b_2;
        double r309680 = r309679 * r309679;
        double r309681 = c;
        double r309682 = a;
        double r309683 = r309681 * r309682;
        double r309684 = r309680 + r309683;
        double r309685 = r309680 - r309683;
        double r309686 = r309684 / r309685;
        double r309687 = r309684 / r309686;
        double r309688 = sqrt(r309687);
        double r309689 = r309688 - r309679;
        double r309690 = r309689 / r309682;
        return r309690;
}

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-flip--2.7

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

  6. Applied difference-of-squares2.6

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

  7. Applied associate-/l*1.7

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

  8. Final simplification1.7

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(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))