Average Error: 34.3 → 10.3
Time: 5.3s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.125553485370055 \cdot 10^{-113}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 6.51740022507215 \cdot 10^{112}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -2.125553485370055 \cdot 10^{-113}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 6.51740022507215 \cdot 10^{112}:\\
\;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{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 r92512 = b;
        double r92513 = -r92512;
        double r92514 = r92512 * r92512;
        double r92515 = 4.0;
        double r92516 = a;
        double r92517 = c;
        double r92518 = r92516 * r92517;
        double r92519 = r92515 * r92518;
        double r92520 = r92514 - r92519;
        double r92521 = sqrt(r92520);
        double r92522 = r92513 - r92521;
        double r92523 = 2.0;
        double r92524 = r92523 * r92516;
        double r92525 = r92522 / r92524;
        return r92525;
}
↓
double f(double a, double b, double c) {
        double r92526 = b;
        double r92527 = -2.125553485370055e-113;
        bool r92528 = r92526 <= r92527;
        double r92529 = -1.0;
        double r92530 = c;
        double r92531 = r92530 / r92526;
        double r92532 = r92529 * r92531;
        double r92533 = 6.51740022507215e+112;
        bool r92534 = r92526 <= r92533;
        double r92535 = -r92526;
        double r92536 = r92526 * r92526;
        double r92537 = 4.0;
        double r92538 = a;
        double r92539 = r92538 * r92530;
        double r92540 = r92537 * r92539;
        double r92541 = r92536 - r92540;
        double r92542 = sqrt(r92541);
        double r92543 = r92535 - r92542;
        double r92544 = 1.0;
        double r92545 = 2.0;
        double r92546 = r92545 * r92538;
        double r92547 = r92544 / r92546;
        double r92548 = r92543 * r92547;
        double r92549 = 1.0;
        double r92550 = r92526 / r92538;
        double r92551 = r92531 - r92550;
        double r92552 = r92549 * r92551;
        double r92553 = r92534 ? r92548 : r92552;
        double r92554 = r92528 ? r92532 : r92553;
        return r92554;
}

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

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

Derivation

  1. Split input into 3 regimes

  2. if b < -2.125553485370055e-113

    1. Initial program 51.3

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

    2. Taylor expanded around -inf 10.8

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

    if -2.125553485370055e-113 < b < 6.51740022507215e+112

    1. Initial program 12.2

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

    3. Applied div-inv12.3

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

    if 6.51740022507215e+112 < b

    1. Initial program 49.8

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

    2. Taylor expanded around inf 2.8

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

    3. Simplified2.8

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.125553485370055 \cdot 10^{-113}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 6.51740022507215 \cdot 10^{112}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +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)))