Average Error: 34.1 → 10.5
Time: 4.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.643947230437265585428917170074785083411 \cdot 10^{-71}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.498338218964205825262884582884276173268 \cdot 10^{54}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\frac{1}{a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}\right)\\ \end{array}\]
\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 -5.643947230437265585428917170074785083411 \cdot 10^{-71}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 1.498338218964205825262884582884276173268 \cdot 10^{54}:\\
\;\;\;\;\frac{1}{2} \cdot \left(\frac{1}{a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{2} \cdot \left(2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r79453 = b;
        double r79454 = -r79453;
        double r79455 = r79453 * r79453;
        double r79456 = 4.0;
        double r79457 = a;
        double r79458 = c;
        double r79459 = r79457 * r79458;
        double r79460 = r79456 * r79459;
        double r79461 = r79455 - r79460;
        double r79462 = sqrt(r79461);
        double r79463 = r79454 - r79462;
        double r79464 = 2.0;
        double r79465 = r79464 * r79457;
        double r79466 = r79463 / r79465;
        return r79466;
}
↓
double f(double a, double b, double c) {
        double r79467 = b;
        double r79468 = -5.6439472304372656e-71;
        bool r79469 = r79467 <= r79468;
        double r79470 = -1.0;
        double r79471 = c;
        double r79472 = r79471 / r79467;
        double r79473 = r79470 * r79472;
        double r79474 = 1.4983382189642058e+54;
        bool r79475 = r79467 <= r79474;
        double r79476 = 1.0;
        double r79477 = 2.0;
        double r79478 = r79476 / r79477;
        double r79479 = a;
        double r79480 = r79476 / r79479;
        double r79481 = -r79467;
        double r79482 = r79467 * r79467;
        double r79483 = 4.0;
        double r79484 = r79479 * r79471;
        double r79485 = r79483 * r79484;
        double r79486 = r79482 - r79485;
        double r79487 = sqrt(r79486);
        double r79488 = r79481 - r79487;
        double r79489 = r79480 * r79488;
        double r79490 = r79478 * r79489;
        double r79491 = r79477 * r79472;
        double r79492 = 2.0;
        double r79493 = r79467 / r79479;
        double r79494 = r79492 * r79493;
        double r79495 = r79491 - r79494;
        double r79496 = r79478 * r79495;
        double r79497 = r79475 ? r79490 : r79496;
        double r79498 = r79469 ? r79473 : r79497;
        return r79498;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original34.1
Target20.8
Herbie10.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -5.6439472304372656e-71

    1. Initial program 53.3

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

    2. Taylor expanded around -inf 9.2

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

    if -5.6439472304372656e-71 < b < 1.4983382189642058e+54

    1. Initial program 14.3

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

    3. Applied *-un-lft-identity14.3

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

    4. Applied times-frac14.2

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

    6. Applied clear-num14.4

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

    8. Applied div-inv14.4

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

    9. Applied add-cube-cbrt14.4

      \[\leadsto \frac{1}{2} \cdot \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{a \cdot \frac{1}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

    10. Applied times-frac14.4

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{a} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\right)}\]

    11. Simplified14.4

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\frac{1}{a}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\right)\]

    12. Simplified14.4

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

    if 1.4983382189642058e+54 < b

    1. Initial program 37.9

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

    3. Applied *-un-lft-identity37.9

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

    4. Applied times-frac38.0

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

    5. Taylor expanded around inf 5.1

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.643947230437265585428917170074785083411 \cdot 10^{-71}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.498338218964205825262884582884276173268 \cdot 10^{54}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\frac{1}{a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +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)))