Average Error: 34.1 → 6.7
Time: 5.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.478383220944118 \cdot 10^{90}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 1.071982619004943 \cdot 10^{-308}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{1}{3 \cdot a}\\ \mathbf{elif}\;b \le 1.8676563684114658 \cdot 10^{102}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{c}{1}} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, 0 - 3 \cdot \left(a \cdot c\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -7.478383220944118 \cdot 10^{90}:\\
\;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\

\mathbf{elif}\;b \le 1.071982619004943 \cdot 10^{-308}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{1}{3 \cdot a}\\

\mathbf{elif}\;b \le 1.8676563684114658 \cdot 10^{102}:\\
\;\;\;\;\frac{1}{\frac{1}{\frac{c}{1}} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, 0 - 3 \cdot \left(a \cdot c\right)\right)}\right)}\\

\mathbf{else}:\\
\;\;\;\;-0.5 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r129502 = b;
        double r129503 = -r129502;
        double r129504 = r129502 * r129502;
        double r129505 = 3.0;
        double r129506 = a;
        double r129507 = r129505 * r129506;
        double r129508 = c;
        double r129509 = r129507 * r129508;
        double r129510 = r129504 - r129509;
        double r129511 = sqrt(r129510);
        double r129512 = r129503 + r129511;
        double r129513 = r129512 / r129507;
        return r129513;
}
↓
double f(double a, double b, double c) {
        double r129514 = b;
        double r129515 = -7.478383220944118e+90;
        bool r129516 = r129514 <= r129515;
        double r129517 = 0.5;
        double r129518 = c;
        double r129519 = r129518 / r129514;
        double r129520 = r129517 * r129519;
        double r129521 = 0.6666666666666666;
        double r129522 = a;
        double r129523 = r129514 / r129522;
        double r129524 = r129521 * r129523;
        double r129525 = r129520 - r129524;
        double r129526 = 1.0719826190049434e-308;
        bool r129527 = r129514 <= r129526;
        double r129528 = -r129514;
        double r129529 = r129514 * r129514;
        double r129530 = 3.0;
        double r129531 = r129530 * r129522;
        double r129532 = r129531 * r129518;
        double r129533 = r129529 - r129532;
        double r129534 = sqrt(r129533);
        double r129535 = r129528 + r129534;
        double r129536 = 1.0;
        double r129537 = r129536 / r129531;
        double r129538 = r129535 * r129537;
        double r129539 = 1.8676563684114658e+102;
        bool r129540 = r129514 <= r129539;
        double r129541 = r129518 / r129536;
        double r129542 = r129536 / r129541;
        double r129543 = 0.0;
        double r129544 = r129522 * r129518;
        double r129545 = r129530 * r129544;
        double r129546 = r129543 - r129545;
        double r129547 = fma(r129514, r129514, r129546);
        double r129548 = sqrt(r129547);
        double r129549 = r129528 - r129548;
        double r129550 = r129542 * r129549;
        double r129551 = r129536 / r129550;
        double r129552 = -0.5;
        double r129553 = r129552 * r129519;
        double r129554 = r129540 ? r129551 : r129553;
        double r129555 = r129527 ? r129538 : r129554;
        double r129556 = r129516 ? r129525 : r129555;
        return r129556;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b < -7.478383220944118e+90

    1. Initial program 44.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

    2. Taylor expanded around -inf 3.7

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}}\]

    if -7.478383220944118e+90 < b < 1.0719826190049434e-308

    1. Initial program 9.2

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
    2. Using strategy rm

    3. Applied div-inv9.2

      \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{1}{3 \cdot a}}\]

    if 1.0719826190049434e-308 < b < 1.8676563684114658e+102

    1. Initial program 33.5

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
    2. Using strategy rm

    3. Applied flip-+33.5

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

    4. Simplified17.3

      \[\leadsto \frac{\frac{\color{blue}{0 + 3 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]
    5. Using strategy rm

    6. Applied fma-neg17.3

      \[\leadsto \frac{\frac{0 + 3 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot a\right) \cdot c\right)}}}}{3 \cdot a}\]

    7. Simplified17.3

      \[\leadsto \frac{\frac{0 + 3 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{0 - 3 \cdot \left(a \cdot c\right)}\right)}}}{3 \cdot a}\]
    8. Using strategy rm

    9. Applied clear-num17.5

      \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\frac{0 + 3 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, 0 - 3 \cdot \left(a \cdot c\right)\right)}}}}}\]

    10. Simplified16.5

      \[\leadsto \frac{1}{\color{blue}{\frac{3 \cdot a}{3 \cdot \left(a \cdot c\right)} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, 0 - 3 \cdot \left(a \cdot c\right)\right)}\right)}}\]
    11. Using strategy rm

    12. Applied clear-num16.5

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{3 \cdot \left(a \cdot c\right)}{3 \cdot a}}} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, 0 - 3 \cdot \left(a \cdot c\right)\right)}\right)}\]

    13. Simplified9.2

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{c}{1}}} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, 0 - 3 \cdot \left(a \cdot c\right)\right)}\right)}\]

    if 1.8676563684114658e+102 < b

    1. Initial program 59.8

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

    2. Taylor expanded around inf 2.4

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.478383220944118 \cdot 10^{90}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 1.071982619004943 \cdot 10^{-308}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{1}{3 \cdot a}\\ \mathbf{elif}\;b \le 1.8676563684114658 \cdot 10^{102}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{c}{1}} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, 0 - 3 \cdot \left(a \cdot c\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))