Average Error: 34.4 → 10.4
Time: 5.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.1515494582665793 \cdot 10^{-119}:\\ \;\;\;\;1 \cdot \left(-1 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \le 4.2504918589151378 \cdot 10^{117}:\\ \;\;\;\;1 \cdot \frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\right)\\ \end{array}\]
\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 -4.1515494582665793 \cdot 10^{-119}:\\
\;\;\;\;1 \cdot \left(-1 \cdot \frac{c}{b}\right)\\

\mathbf{elif}\;b \le 4.2504918589151378 \cdot 10^{117}:\\
\;\;\;\;1 \cdot \frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r91460 = b;
        double r91461 = -r91460;
        double r91462 = r91460 * r91460;
        double r91463 = 4.0;
        double r91464 = a;
        double r91465 = c;
        double r91466 = r91464 * r91465;
        double r91467 = r91463 * r91466;
        double r91468 = r91462 - r91467;
        double r91469 = sqrt(r91468);
        double r91470 = r91461 - r91469;
        double r91471 = 2.0;
        double r91472 = r91471 * r91464;
        double r91473 = r91470 / r91472;
        return r91473;
}
↓
double f(double a, double b, double c) {
        double r91474 = b;
        double r91475 = -4.1515494582665793e-119;
        bool r91476 = r91474 <= r91475;
        double r91477 = 1.0;
        double r91478 = -1.0;
        double r91479 = c;
        double r91480 = r91479 / r91474;
        double r91481 = r91478 * r91480;
        double r91482 = r91477 * r91481;
        double r91483 = 4.250491858915138e+117;
        bool r91484 = r91474 <= r91483;
        double r91485 = -r91474;
        double r91486 = 4.0;
        double r91487 = a;
        double r91488 = r91487 * r91479;
        double r91489 = r91486 * r91488;
        double r91490 = -r91489;
        double r91491 = fma(r91474, r91474, r91490);
        double r91492 = sqrt(r91491);
        double r91493 = r91485 - r91492;
        double r91494 = 2.0;
        double r91495 = r91494 * r91487;
        double r91496 = r91493 / r91495;
        double r91497 = r91477 * r91496;
        double r91498 = 1.0;
        double r91499 = r91474 / r91487;
        double r91500 = r91480 - r91499;
        double r91501 = r91498 * r91500;
        double r91502 = r91477 * r91501;
        double r91503 = r91484 ? r91497 : r91502;
        double r91504 = r91476 ? r91482 : r91503;
        return r91504;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.4
Target21.3
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.1515494582665793e-119

    1. Initial program 51.5

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

    3. Applied div-inv51.5

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

    5. Applied *-un-lft-identity51.5

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

    6. Applied associate-*l*51.5

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

    7. Simplified51.5

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

    8. Taylor expanded around -inf 11.4

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

    if -4.1515494582665793e-119 < b < 4.250491858915138e+117

    1. Initial program 11.8

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

    3. Applied div-inv11.9

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

    5. Applied *-un-lft-identity11.9

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

    6. Applied associate-*l*11.9

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

    7. Simplified11.8

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

    9. Applied fma-neg11.8

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

    if 4.250491858915138e+117 < b

    1. Initial program 52.2

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

    3. Applied div-inv52.3

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

    5. Applied *-un-lft-identity52.3

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

    6. Applied associate-*l*52.3

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

    7. Simplified52.2

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

    8. Taylor expanded around inf 2.9

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

    9. Simplified2.9

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.1515494582665793 \cdot 10^{-119}:\\ \;\;\;\;1 \cdot \left(-1 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \le 4.2504918589151378 \cdot 10^{117}:\\ \;\;\;\;1 \cdot \frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :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)))