Average Error: 33.7 → 6.9
Time: 5.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.5975400610846271 \cdot 10^{115}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -1.06237398994986779 \cdot 10^{-305}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{elif}\;b \le 3.29571176074688 \cdot 10^{130}:\\ \;\;\;\;\frac{1}{\frac{0.5}{c} \cdot \left(-b\right) + \frac{0.5}{c} \cdot \left(-\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -9.5975400610846271 \cdot 10^{115}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -1.06237398994986779 \cdot 10^{-305}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{elif}\;b \le 3.29571176074688 \cdot 10^{130}:\\
\;\;\;\;\frac{1}{\frac{0.5}{c} \cdot \left(-b\right) + \frac{0.5}{c} \cdot \left(-\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r67524 = b;
        double r67525 = -r67524;
        double r67526 = r67524 * r67524;
        double r67527 = 4.0;
        double r67528 = a;
        double r67529 = r67527 * r67528;
        double r67530 = c;
        double r67531 = r67529 * r67530;
        double r67532 = r67526 - r67531;
        double r67533 = sqrt(r67532);
        double r67534 = r67525 + r67533;
        double r67535 = 2.0;
        double r67536 = r67535 * r67528;
        double r67537 = r67534 / r67536;
        return r67537;
}
↓
double f(double a, double b, double c) {
        double r67538 = b;
        double r67539 = -9.597540061084627e+115;
        bool r67540 = r67538 <= r67539;
        double r67541 = 1.0;
        double r67542 = c;
        double r67543 = r67542 / r67538;
        double r67544 = a;
        double r67545 = r67538 / r67544;
        double r67546 = r67543 - r67545;
        double r67547 = r67541 * r67546;
        double r67548 = -1.0623739899498678e-305;
        bool r67549 = r67538 <= r67548;
        double r67550 = -r67538;
        double r67551 = r67538 * r67538;
        double r67552 = 4.0;
        double r67553 = r67552 * r67544;
        double r67554 = r67553 * r67542;
        double r67555 = r67551 - r67554;
        double r67556 = sqrt(r67555);
        double r67557 = r67550 + r67556;
        double r67558 = 2.0;
        double r67559 = r67558 * r67544;
        double r67560 = r67557 / r67559;
        double r67561 = 3.29571176074688e+130;
        bool r67562 = r67538 <= r67561;
        double r67563 = 1.0;
        double r67564 = 0.5;
        double r67565 = r67564 / r67542;
        double r67566 = r67565 * r67550;
        double r67567 = -r67556;
        double r67568 = r67565 * r67567;
        double r67569 = r67566 + r67568;
        double r67570 = r67563 / r67569;
        double r67571 = -1.0;
        double r67572 = r67571 * r67543;
        double r67573 = r67562 ? r67570 : r67572;
        double r67574 = r67549 ? r67560 : r67573;
        double r67575 = r67540 ? r67547 : r67574;
        return r67575;
}

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

Original33.7
Target20.9
Herbie6.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -9.597540061084627e+115

    1. Initial program 48.7

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

    2. Taylor expanded around -inf 3.3

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

    3. Simplified3.3

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

    if -9.597540061084627e+115 < b < -1.0623739899498678e-305

    1. Initial program 8.7

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

    if -1.0623739899498678e-305 < b < 3.29571176074688e+130

    1. Initial program 34.4

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

    3. Applied flip-+34.4

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

    4. Simplified16.9

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

    6. Applied clear-num17.1

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

    7. Simplified16.0

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

    8. Taylor expanded around 0 9.6

      \[\leadsto \frac{1}{\color{blue}{\frac{0.5}{c}} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\]
    9. Using strategy rm

    10. Applied sub-neg9.6

      \[\leadsto \frac{1}{\frac{0.5}{c} \cdot \color{blue}{\left(\left(-b\right) + \left(-\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)\right)}}\]

    11. Applied distribute-lft-in9.6

      \[\leadsto \frac{1}{\color{blue}{\frac{0.5}{c} \cdot \left(-b\right) + \frac{0.5}{c} \cdot \left(-\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}\]

    if 3.29571176074688e+130 < b

    1. Initial program 61.2

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

    2. Taylor expanded around inf 2.1

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

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.5975400610846271 \cdot 10^{115}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -1.06237398994986779 \cdot 10^{-305}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{elif}\;b \le 3.29571176074688 \cdot 10^{130}:\\ \;\;\;\;\frac{1}{\frac{0.5}{c} \cdot \left(-b\right) + \frac{0.5}{c} \cdot \left(-\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))