Average Error: 28.6 → 0.3
Time: 2.2m
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{c \cdot -2}{\sqrt{\mathsf{fma}\left(b, b, \left(\left(a \cdot -4\right) \cdot c\right)\right)} + b}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{c \cdot -2}{\sqrt{\mathsf{fma}\left(b, b, \left(\left(a \cdot -4\right) \cdot c\right)\right)} + b}
double f(double a, double b, double c) {
        double r10335610 = b;
        double r10335611 = -r10335610;
        double r10335612 = r10335610 * r10335610;
        double r10335613 = 4.0;
        double r10335614 = a;
        double r10335615 = r10335613 * r10335614;
        double r10335616 = c;
        double r10335617 = r10335615 * r10335616;
        double r10335618 = r10335612 - r10335617;
        double r10335619 = sqrt(r10335618);
        double r10335620 = r10335611 + r10335619;
        double r10335621 = 2.0;
        double r10335622 = r10335621 * r10335614;
        double r10335623 = r10335620 / r10335622;
        return r10335623;
}
↓
double f(double a, double b, double c) {
        double r10335624 = c;
        double r10335625 = -2.0;
        double r10335626 = r10335624 * r10335625;
        double r10335627 = b;
        double r10335628 = a;
        double r10335629 = -4.0;
        double r10335630 = r10335628 * r10335629;
        double r10335631 = r10335630 * r10335624;
        double r10335632 = fma(r10335627, r10335627, r10335631);
        double r10335633 = sqrt(r10335632);
        double r10335634 = r10335633 + r10335627;
        double r10335635 = r10335626 / r10335634;
        return r10335635;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 28.6

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

  2. Simplified28.5

    \[\leadsto \color{blue}{\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot \left(-4 \cdot a\right)\right)\right)} - b}{2}}{a}}\]
  3. Using strategy rm

  4. Applied flip--28.7

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

  5. Simplified0.5

    \[\leadsto \frac{\frac{\frac{\color{blue}{c \cdot \left(-4 \cdot a\right)}}{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot \left(-4 \cdot a\right)\right)\right)} + b}}{2}}{a}\]
  6. Using strategy rm

  7. Applied *-un-lft-identity0.5

    \[\leadsto \frac{\frac{\frac{c \cdot \left(-4 \cdot a\right)}{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot \left(-4 \cdot a\right)\right)\right)} + b}}{2}}{\color{blue}{1 \cdot a}}\]

  8. Applied *-un-lft-identity0.5

    \[\leadsto \frac{\color{blue}{1 \cdot \frac{\frac{c \cdot \left(-4 \cdot a\right)}{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot \left(-4 \cdot a\right)\right)\right)} + b}}{2}}}{1 \cdot a}\]

  9. Applied times-frac0.5

    \[\leadsto \color{blue}{\frac{1}{1} \cdot \frac{\frac{\frac{c \cdot \left(-4 \cdot a\right)}{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot \left(-4 \cdot a\right)\right)\right)} + b}}{2}}{a}}\]

  10. Simplified0.5

    \[\leadsto \color{blue}{1} \cdot \frac{\frac{\frac{c \cdot \left(-4 \cdot a\right)}{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot \left(-4 \cdot a\right)\right)\right)} + b}}{2}}{a}\]

  11. Simplified0.3

    \[\leadsto 1 \cdot \color{blue}{\frac{-2 \cdot c}{b + \sqrt{\mathsf{fma}\left(b, b, \left(\left(a \cdot -4\right) \cdot c\right)\right)}}}\]

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))