Average Error: 34.6 → 8.7
Time: 18.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1150955755735961567232:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -3.11539491799786956147131222652382589094 \cdot 10^{-213}:\\ \;\;\;\;\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b} \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.974261024048120880950549217298529943371 \cdot 10^{145}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -1150955755735961567232:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -3.11539491799786956147131222652382589094 \cdot 10^{-213}:\\
\;\;\;\;\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b} \cdot \frac{1}{2 \cdot a}\\

\mathbf{elif}\;b \le 1.974261024048120880950549217298529943371 \cdot 10^{145}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r84477 = b;
        double r84478 = -r84477;
        double r84479 = r84477 * r84477;
        double r84480 = 4.0;
        double r84481 = a;
        double r84482 = c;
        double r84483 = r84481 * r84482;
        double r84484 = r84480 * r84483;
        double r84485 = r84479 - r84484;
        double r84486 = sqrt(r84485);
        double r84487 = r84478 - r84486;
        double r84488 = 2.0;
        double r84489 = r84488 * r84481;
        double r84490 = r84487 / r84489;
        return r84490;
}
↓
double f(double a, double b, double c) {
        double r84491 = b;
        double r84492 = -1.1509557557359616e+21;
        bool r84493 = r84491 <= r84492;
        double r84494 = -1.0;
        double r84495 = c;
        double r84496 = r84495 / r84491;
        double r84497 = r84494 * r84496;
        double r84498 = -3.1153949179978696e-213;
        bool r84499 = r84491 <= r84498;
        double r84500 = 4.0;
        double r84501 = a;
        double r84502 = r84501 * r84495;
        double r84503 = r84500 * r84502;
        double r84504 = -r84503;
        double r84505 = fma(r84491, r84491, r84504);
        double r84506 = sqrt(r84505);
        double r84507 = r84506 - r84491;
        double r84508 = r84503 / r84507;
        double r84509 = 1.0;
        double r84510 = 2.0;
        double r84511 = r84510 * r84501;
        double r84512 = r84509 / r84511;
        double r84513 = r84508 * r84512;
        double r84514 = 1.974261024048121e+145;
        bool r84515 = r84491 <= r84514;
        double r84516 = -r84491;
        double r84517 = r84491 * r84491;
        double r84518 = r84517 - r84503;
        double r84519 = sqrt(r84518);
        double r84520 = r84516 - r84519;
        double r84521 = r84520 / r84511;
        double r84522 = 1.0;
        double r84523 = r84491 / r84501;
        double r84524 = r84496 - r84523;
        double r84525 = r84522 * r84524;
        double r84526 = r84515 ? r84521 : r84525;
        double r84527 = r84499 ? r84513 : r84526;
        double r84528 = r84493 ? r84497 : r84527;
        return r84528;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.6
Target20.9
Herbie8.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.1509557557359616e+21

    1. Initial program 56.3

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

    2. Taylor expanded around -inf 4.5

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

    if -1.1509557557359616e+21 < b < -3.1153949179978696e-213

    1. Initial program 31.5

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

    3. Applied flip--31.5

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

    4. Simplified17.7

      \[\leadsto \frac{\frac{\color{blue}{0 + \left(a \cdot c\right) \cdot 4}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]

    5. Simplified17.7

      \[\leadsto \frac{\frac{0 + \left(a \cdot c\right) \cdot 4}{\color{blue}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}}}{2 \cdot a}\]
    6. Using strategy rm

    7. Applied div-inv17.8

      \[\leadsto \color{blue}{\frac{0 + \left(a \cdot c\right) \cdot 4}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b} \cdot \frac{1}{2 \cdot a}}\]

    if -3.1153949179978696e-213 < b < 1.974261024048121e+145

    1. Initial program 9.9

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

    if 1.974261024048121e+145 < b

    1. Initial program 60.1

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

    2. Taylor expanded around inf 2.3

      \[\leadsto \color{blue}{1 \cdot \frac{c}{b} - 1 \cdot \frac{b}{a}}\]

    3. Simplified2.3

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1150955755735961567232:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -3.11539491799786956147131222652382589094 \cdot 10^{-213}:\\ \;\;\;\;\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b} \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.974261024048120880950549217298529943371 \cdot 10^{145}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))