Average Error: 34.3 → 6.4
Time: 5.1s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.1730875761889226 \cdot 10^{119}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.567927026193799 \cdot 10^{-258}:\\ \;\;\;\;1 \cdot \frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 3.5474318569254359 \cdot 10^{93}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -1.1730875761889226 \cdot 10^{119}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 1.567927026193799 \cdot 10^{-258}:\\
\;\;\;\;1 \cdot \frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\

\mathbf{elif}\;b \le 3.5474318569254359 \cdot 10^{93}:\\
\;\;\;\;\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 r89506 = b;
        double r89507 = -r89506;
        double r89508 = r89506 * r89506;
        double r89509 = 4.0;
        double r89510 = a;
        double r89511 = c;
        double r89512 = r89510 * r89511;
        double r89513 = r89509 * r89512;
        double r89514 = r89508 - r89513;
        double r89515 = sqrt(r89514);
        double r89516 = r89507 - r89515;
        double r89517 = 2.0;
        double r89518 = r89517 * r89510;
        double r89519 = r89516 / r89518;
        return r89519;
}
↓
double f(double a, double b, double c) {
        double r89520 = b;
        double r89521 = -1.1730875761889226e+119;
        bool r89522 = r89520 <= r89521;
        double r89523 = -1.0;
        double r89524 = c;
        double r89525 = r89524 / r89520;
        double r89526 = r89523 * r89525;
        double r89527 = 1.567927026193799e-258;
        bool r89528 = r89520 <= r89527;
        double r89529 = 1.0;
        double r89530 = 2.0;
        double r89531 = r89530 * r89524;
        double r89532 = r89520 * r89520;
        double r89533 = 4.0;
        double r89534 = a;
        double r89535 = r89534 * r89524;
        double r89536 = r89533 * r89535;
        double r89537 = r89532 - r89536;
        double r89538 = sqrt(r89537);
        double r89539 = r89538 - r89520;
        double r89540 = r89531 / r89539;
        double r89541 = r89529 * r89540;
        double r89542 = 3.547431856925436e+93;
        bool r89543 = r89520 <= r89542;
        double r89544 = -r89520;
        double r89545 = r89544 - r89538;
        double r89546 = r89530 * r89534;
        double r89547 = r89545 / r89546;
        double r89548 = 1.0;
        double r89549 = r89520 / r89534;
        double r89550 = r89525 - r89549;
        double r89551 = r89548 * r89550;
        double r89552 = r89543 ? r89547 : r89551;
        double r89553 = r89528 ? r89541 : r89552;
        double r89554 = r89522 ? r89526 : r89553;
        return r89554;
}

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
Target21.0
Herbie6.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.1730875761889226e+119

    1. Initial program 60.9

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

    2. Taylor expanded around -inf 1.9

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

    if -1.1730875761889226e+119 < b < 1.567927026193799e-258

    1. Initial program 32.4

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

    3. Applied div-inv32.5

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

    5. Applied flip--32.5

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

    6. Simplified15.8

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

    7. Simplified15.8

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

    9. Applied *-un-lft-identity15.8

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

    10. Applied *-un-lft-identity15.8

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

    11. Applied times-frac15.8

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

    12. Applied associate-*l*15.8

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

    13. Simplified14.7

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

    14. Taylor expanded around 0 9.0

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

    if 1.567927026193799e-258 < b < 3.547431856925436e+93

    1. Initial program 8.1

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

    3. Applied div-inv8.3

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

    5. Applied un-div-inv8.1

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

    if 3.547431856925436e+93 < b

    1. Initial program 46.1

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

    2. Taylor expanded around inf 4.1

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

    3. Simplified4.1

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.1730875761889226 \cdot 10^{119}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.567927026193799 \cdot 10^{-258}:\\ \;\;\;\;1 \cdot \frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 3.5474318569254359 \cdot 10^{93}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(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)))