Average Error: 34.0 → 10.6
Time: 17.3s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.76351621427461392823177571966952670558 \cdot 10^{80}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -4.275041523480463755960237531055658758378 \cdot 10^{-132}:\\ \;\;\;\;\frac{\left(a \cdot c\right) \cdot 4}{2} \cdot \frac{\frac{1}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{a}\\ \mathbf{elif}\;b \le 2098867031.934578418731689453125:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -1.76351621427461392823177571966952670558 \cdot 10^{80}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -4.275041523480463755960237531055658758378 \cdot 10^{-132}:\\
\;\;\;\;\frac{\left(a \cdot c\right) \cdot 4}{2} \cdot \frac{\frac{1}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{a}\\

\mathbf{elif}\;b \le 2098867031.934578418731689453125:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{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 r80562 = b;
        double r80563 = -r80562;
        double r80564 = r80562 * r80562;
        double r80565 = 4.0;
        double r80566 = a;
        double r80567 = c;
        double r80568 = r80566 * r80567;
        double r80569 = r80565 * r80568;
        double r80570 = r80564 - r80569;
        double r80571 = sqrt(r80570);
        double r80572 = r80563 - r80571;
        double r80573 = 2.0;
        double r80574 = r80573 * r80566;
        double r80575 = r80572 / r80574;
        return r80575;
}
↓
double f(double a, double b, double c) {
        double r80576 = b;
        double r80577 = -1.763516214274614e+80;
        bool r80578 = r80576 <= r80577;
        double r80579 = -1.0;
        double r80580 = c;
        double r80581 = r80580 / r80576;
        double r80582 = r80579 * r80581;
        double r80583 = -4.275041523480464e-132;
        bool r80584 = r80576 <= r80583;
        double r80585 = a;
        double r80586 = r80585 * r80580;
        double r80587 = 4.0;
        double r80588 = r80586 * r80587;
        double r80589 = 2.0;
        double r80590 = r80588 / r80589;
        double r80591 = 1.0;
        double r80592 = r80576 * r80576;
        double r80593 = r80587 * r80586;
        double r80594 = r80592 - r80593;
        double r80595 = sqrt(r80594);
        double r80596 = r80595 - r80576;
        double r80597 = r80591 / r80596;
        double r80598 = r80597 / r80585;
        double r80599 = r80590 * r80598;
        double r80600 = 2098867031.9345784;
        bool r80601 = r80576 <= r80600;
        double r80602 = -r80576;
        double r80603 = r80602 - r80595;
        double r80604 = r80589 * r80585;
        double r80605 = r80603 / r80604;
        double r80606 = 1.0;
        double r80607 = r80576 / r80585;
        double r80608 = r80581 - r80607;
        double r80609 = r80606 * r80608;
        double r80610 = r80601 ? r80605 : r80609;
        double r80611 = r80584 ? r80599 : r80610;
        double r80612 = r80578 ? r80582 : r80611;
        return r80612;
}

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

Original34.0
Target20.9
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.763516214274614e+80

    1. Initial program 58.0

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

    2. Taylor expanded around -inf 2.8

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

    if -1.763516214274614e+80 < b < -4.275041523480464e-132

    1. Initial program 39.7

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

    3. Applied flip--39.7

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

    4. Simplified15.4

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

    5. Simplified15.4

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

    7. Applied div-inv15.4

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

    8. Applied times-frac18.9

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

    9. Simplified18.9

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

    if -4.275041523480464e-132 < b < 2098867031.9345784

    1. Initial program 14.4

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

    if 2098867031.9345784 < b

    1. Initial program 32.3

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

    2. Taylor expanded around inf 7.0

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

    3. Simplified7.0

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.76351621427461392823177571966952670558 \cdot 10^{80}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -4.275041523480463755960237531055658758378 \cdot 10^{-132}:\\ \;\;\;\;\frac{\left(a \cdot c\right) \cdot 4}{2} \cdot \frac{\frac{1}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{a}\\ \mathbf{elif}\;b \le 2098867031.934578418731689453125:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 
(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)))