Average Error: 34.2 → 8.7
Time: 5.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.16001008416394735 \cdot 10^{156}:\\ \;\;\;\;\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2 \cdot a}\\ \mathbf{elif}\;b \le -5.18636062467436046 \cdot 10^{-242}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.18992965287049363 \cdot 10^{140}:\\ \;\;\;\;\frac{\frac{c}{0.5}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.5}{c} \cdot \left(2 \cdot \frac{a \cdot c}{b} - 2 \cdot b\right)}\\ \end{array}\]
\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 -5.16001008416394735 \cdot 10^{156}:\\
\;\;\;\;\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2 \cdot a}\\

\mathbf{elif}\;b \le -5.18636062467436046 \cdot 10^{-242}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{elif}\;b \le 1.18992965287049363 \cdot 10^{140}:\\
\;\;\;\;\frac{\frac{c}{0.5}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{0.5}{c} \cdot \left(2 \cdot \frac{a \cdot c}{b} - 2 \cdot b\right)}\\

\end{array}
double f(double a, double b, double c) {
        double r178493 = b;
        double r178494 = -r178493;
        double r178495 = r178493 * r178493;
        double r178496 = 4.0;
        double r178497 = a;
        double r178498 = r178496 * r178497;
        double r178499 = c;
        double r178500 = r178498 * r178499;
        double r178501 = r178495 - r178500;
        double r178502 = sqrt(r178501);
        double r178503 = r178494 + r178502;
        double r178504 = 2.0;
        double r178505 = r178504 * r178497;
        double r178506 = r178503 / r178505;
        return r178506;
}
↓
double f(double a, double b, double c) {
        double r178507 = b;
        double r178508 = -5.160010084163947e+156;
        bool r178509 = r178507 <= r178508;
        double r178510 = 2.0;
        double r178511 = a;
        double r178512 = c;
        double r178513 = r178511 * r178512;
        double r178514 = r178513 / r178507;
        double r178515 = r178510 * r178514;
        double r178516 = 2.0;
        double r178517 = r178516 * r178507;
        double r178518 = r178515 - r178517;
        double r178519 = r178510 * r178511;
        double r178520 = r178518 / r178519;
        double r178521 = -5.1863606246743605e-242;
        bool r178522 = r178507 <= r178521;
        double r178523 = -r178507;
        double r178524 = r178507 * r178507;
        double r178525 = 4.0;
        double r178526 = r178525 * r178511;
        double r178527 = r178526 * r178512;
        double r178528 = r178524 - r178527;
        double r178529 = sqrt(r178528);
        double r178530 = r178523 + r178529;
        double r178531 = r178530 / r178519;
        double r178532 = 1.1899296528704936e+140;
        bool r178533 = r178507 <= r178532;
        double r178534 = 0.5;
        double r178535 = r178512 / r178534;
        double r178536 = r178523 - r178529;
        double r178537 = r178535 / r178536;
        double r178538 = 1.0;
        double r178539 = r178534 / r178512;
        double r178540 = r178539 * r178518;
        double r178541 = r178538 / r178540;
        double r178542 = r178533 ? r178537 : r178541;
        double r178543 = r178522 ? r178531 : r178542;
        double r178544 = r178509 ? r178520 : r178543;
        return r178544;
}

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

Original34.2
Target21.1
Herbie8.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -5.160010084163947e+156

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 10.1

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{2 \cdot a}\]

    if -5.160010084163947e+156 < b < -5.1863606246743605e-242

    1. Initial program 8.6

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

    if -5.1863606246743605e-242 < b < 1.1899296528704936e+140

    1. Initial program 32.3

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

    3. Applied flip-+32.4

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

    4. Simplified15.9

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

    6. Applied clear-num16.1

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

    7. Simplified15.1

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

    8. Taylor expanded around 0 9.7

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

    10. Applied associate-/r*9.3

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

    11. Simplified9.2

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

    if 1.1899296528704936e+140 < b

    1. Initial program 62.5

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

    3. Applied flip-+62.5

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

    4. Simplified35.1

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

    6. Applied clear-num35.1

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

    7. Simplified34.7

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

    8. Taylor expanded around 0 34.4

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

    9. Taylor expanded around inf 7.1

      \[\leadsto \frac{1}{\frac{0.5}{c} \cdot \color{blue}{\left(2 \cdot \frac{a \cdot c}{b} - 2 \cdot b\right)}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.16001008416394735 \cdot 10^{156}:\\ \;\;\;\;\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2 \cdot a}\\ \mathbf{elif}\;b \le -5.18636062467436046 \cdot 10^{-242}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.18992965287049363 \cdot 10^{140}:\\ \;\;\;\;\frac{\frac{c}{0.5}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.5}{c} \cdot \left(2 \cdot \frac{a \cdot c}{b} - 2 \cdot b\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 
(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)))