Average Error: 1.7 → 1.8
Time: 21.6s
Precision: 64
\[\frac{\left(\frac{\left(-b_2\right)}{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)}\right)}{a}\]
↓
\[\frac{\frac{1.0}{\frac{1.0}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
\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{\frac{1.0}{\frac{1.0}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}
double f(double a, double b_2, double c) {
        double r917479 = b_2;
        double r917480 = -r917479;
        double r917481 = r917479 * r917479;
        double r917482 = a;
        double r917483 = c;
        double r917484 = r917482 * r917483;
        double r917485 = r917481 - r917484;
        double r917486 = sqrt(r917485);
        double r917487 = r917480 + r917486;
        double r917488 = r917487 / r917482;
        return r917488;
}
↓
double f(double a, double b_2, double c) {
        double r917489 = 1.0;
        double r917490 = b_2;
        double r917491 = r917490 * r917490;
        double r917492 = a;
        double r917493 = c;
        double r917494 = r917492 * r917493;
        double r917495 = r917491 - r917494;
        double r917496 = sqrt(r917495);
        double r917497 = r917496 - r917490;
        double r917498 = r917489 / r917497;
        double r917499 = r917489 / r917498;
        double r917500 = r917499 / r917492;
        return r917500;
}

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--2.9

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

  6. Applied p16-*-un-lft-identity2.9

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

  7. Applied p16-*-un-lft-identity2.9

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

  8. Applied associate-*l*2.9

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

  9. Applied distribute-lft-out--2.9

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

  10. Applied associate-/l*3.2

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

  11. Simplified1.8

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

  12. Final simplification1.8

    \[\leadsto \frac{\frac{1.0}{\frac{1.0}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]

Reproduce

herbie shell --seed 2019158 
(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))