Average Error: 28.4 → 0.4
Time: 1.1m
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\left(c \cdot a\right) \cdot -2}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(\left(-4 \cdot a\right) \cdot c\right)\right)} + b\right) \cdot a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\left(c \cdot a\right) \cdot -2}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(\left(-4 \cdot a\right) \cdot c\right)\right)} + b\right) \cdot a}
double f(double a, double b, double c) {
        double r7099370 = b;
        double r7099371 = -r7099370;
        double r7099372 = r7099370 * r7099370;
        double r7099373 = 4.0;
        double r7099374 = a;
        double r7099375 = r7099373 * r7099374;
        double r7099376 = c;
        double r7099377 = r7099375 * r7099376;
        double r7099378 = r7099372 - r7099377;
        double r7099379 = sqrt(r7099378);
        double r7099380 = r7099371 + r7099379;
        double r7099381 = 2.0;
        double r7099382 = r7099381 * r7099374;
        double r7099383 = r7099380 / r7099382;
        return r7099383;
}
↓
double f(double a, double b, double c) {
        double r7099384 = c;
        double r7099385 = a;
        double r7099386 = r7099384 * r7099385;
        double r7099387 = -2.0;
        double r7099388 = r7099386 * r7099387;
        double r7099389 = b;
        double r7099390 = -4.0;
        double r7099391 = r7099390 * r7099385;
        double r7099392 = r7099391 * r7099384;
        double r7099393 = fma(r7099389, r7099389, r7099392);
        double r7099394 = sqrt(r7099393);
        double r7099395 = r7099394 + r7099389;
        double r7099396 = r7099395 * r7099385;
        double r7099397 = r7099388 / r7099396;
        return r7099397;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 28.4

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

  2. Simplified28.4

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

  4. Applied *-un-lft-identity28.4

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

  5. Applied div-inv28.4

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

  6. Applied times-frac28.4

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

  7. Simplified28.4

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

  8. Simplified28.4

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

  10. Applied flip--28.5

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

  11. Applied frac-times28.5

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

  12. Simplified0.4

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

  14. Applied +-commutative0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2019120 +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)))