Average Error: 34.6 → 9.9
Time: 16.9s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.884773494239208074673838159500017127083 \cdot 10^{102}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.296584755784870148043824818440281520371 \cdot 10^{-151}:\\ \;\;\;\;\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b} \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 0.01064842317658122247681085070780682144687:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -4.884773494239208074673838159500017127083 \cdot 10^{102}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -1.296584755784870148043824818440281520371 \cdot 10^{-151}:\\
\;\;\;\;\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b} \cdot \frac{1}{2 \cdot a}\\

\mathbf{elif}\;b \le 0.01064842317658122247681085070780682144687:\\
\;\;\;\;\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 r72577 = b;
        double r72578 = -r72577;
        double r72579 = r72577 * r72577;
        double r72580 = 4.0;
        double r72581 = a;
        double r72582 = c;
        double r72583 = r72581 * r72582;
        double r72584 = r72580 * r72583;
        double r72585 = r72579 - r72584;
        double r72586 = sqrt(r72585);
        double r72587 = r72578 - r72586;
        double r72588 = 2.0;
        double r72589 = r72588 * r72581;
        double r72590 = r72587 / r72589;
        return r72590;
}
↓
double f(double a, double b, double c) {
        double r72591 = b;
        double r72592 = -4.884773494239208e+102;
        bool r72593 = r72591 <= r72592;
        double r72594 = -1.0;
        double r72595 = c;
        double r72596 = r72595 / r72591;
        double r72597 = r72594 * r72596;
        double r72598 = -1.2965847557848701e-151;
        bool r72599 = r72591 <= r72598;
        double r72600 = 4.0;
        double r72601 = a;
        double r72602 = r72601 * r72595;
        double r72603 = r72600 * r72602;
        double r72604 = r72591 * r72591;
        double r72605 = r72604 - r72603;
        double r72606 = sqrt(r72605);
        double r72607 = r72606 - r72591;
        double r72608 = r72603 / r72607;
        double r72609 = 1.0;
        double r72610 = 2.0;
        double r72611 = r72610 * r72601;
        double r72612 = r72609 / r72611;
        double r72613 = r72608 * r72612;
        double r72614 = 0.010648423176581222;
        bool r72615 = r72591 <= r72614;
        double r72616 = -r72591;
        double r72617 = r72616 - r72606;
        double r72618 = r72617 * r72612;
        double r72619 = 1.0;
        double r72620 = r72591 / r72601;
        double r72621 = r72596 - r72620;
        double r72622 = r72619 * r72621;
        double r72623 = r72615 ? r72618 : r72622;
        double r72624 = r72599 ? r72613 : r72623;
        double r72625 = r72593 ? r72597 : r72624;
        return r72625;
}

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

Original34.6
Target21.1
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -4.884773494239208e+102

    1. Initial program 59.9

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

    2. Taylor expanded around -inf 2.4

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

    if -4.884773494239208e+102 < b < -1.2965847557848701e-151

    1. Initial program 38.6

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

    3. Applied flip--38.7

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

    4. Simplified15.7

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

    5. Simplified15.7

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

    7. Applied div-inv15.8

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

    if -1.2965847557848701e-151 < b < 0.010648423176581222

    1. Initial program 12.9

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

    3. Applied div-inv13.0

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

    if 0.010648423176581222 < b

    1. Initial program 32.9

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

    2. Taylor expanded around inf 8.1

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

    3. Simplified8.1

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.884773494239208074673838159500017127083 \cdot 10^{102}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.296584755784870148043824818440281520371 \cdot 10^{-151}:\\ \;\;\;\;\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b} \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 0.01064842317658122247681085070780682144687:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 
(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)))