Average Error: 52.8 → 0.1
Time: 2.6m
Precision: 64
\[4.930380657631324 \cdot 10^{-32} \lt a \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt b \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt c \lt 2.028240960365167 \cdot 10^{+31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{c}{\frac{1}{4}}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(\left(a \cdot c\right) \cdot -4\right)\right)}} \cdot \frac{1}{2}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{c}{\frac{1}{4}}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(\left(a \cdot c\right) \cdot -4\right)\right)}} \cdot \frac{1}{2}
double f(double a, double b, double c) {
        double r9373387 = b;
        double r9373388 = -r9373387;
        double r9373389 = r9373387 * r9373387;
        double r9373390 = 4.0;
        double r9373391 = a;
        double r9373392 = r9373390 * r9373391;
        double r9373393 = c;
        double r9373394 = r9373392 * r9373393;
        double r9373395 = r9373389 - r9373394;
        double r9373396 = sqrt(r9373395);
        double r9373397 = r9373388 + r9373396;
        double r9373398 = 2.0;
        double r9373399 = r9373398 * r9373391;
        double r9373400 = r9373397 / r9373399;
        return r9373400;
}
↓
double f(double a, double b, double c) {
        double r9373401 = c;
        double r9373402 = 0.25;
        double r9373403 = r9373401 / r9373402;
        double r9373404 = b;
        double r9373405 = -r9373404;
        double r9373406 = a;
        double r9373407 = r9373406 * r9373401;
        double r9373408 = -4.0;
        double r9373409 = r9373407 * r9373408;
        double r9373410 = fma(r9373404, r9373404, r9373409);
        double r9373411 = sqrt(r9373410);
        double r9373412 = r9373405 - r9373411;
        double r9373413 = r9373403 / r9373412;
        double r9373414 = 0.5;
        double r9373415 = r9373413 * r9373414;
        return r9373415;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 52.8

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

  3. Applied flip-+52.8

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

  4. Simplified0.4

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

  5. Simplified0.4

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

  7. Applied *-un-lft-identity0.4

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

  8. Applied times-frac0.4

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

  9. Simplified0.4

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

  10. Simplified0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))