Average Error: 34.2 → 6.8
Time: 5.1s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.40284932349203652 \cdot 10^{128}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 5.877669040907696 \cdot 10^{-167}:\\ \;\;\;\;{\left(\frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\right)}^{1}\\ \mathbf{elif}\;b \le 1.58497213944565541 \cdot 10^{84}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -2.40284932349203652 \cdot 10^{128}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 5.877669040907696 \cdot 10^{-167}:\\
\;\;\;\;{\left(\frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\right)}^{1}\\

\mathbf{elif}\;b \le 1.58497213944565541 \cdot 10^{84}:\\
\;\;\;\;\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 r106526 = b;
        double r106527 = -r106526;
        double r106528 = r106526 * r106526;
        double r106529 = 4.0;
        double r106530 = a;
        double r106531 = c;
        double r106532 = r106530 * r106531;
        double r106533 = r106529 * r106532;
        double r106534 = r106528 - r106533;
        double r106535 = sqrt(r106534);
        double r106536 = r106527 - r106535;
        double r106537 = 2.0;
        double r106538 = r106537 * r106530;
        double r106539 = r106536 / r106538;
        return r106539;
}
↓
double f(double a, double b, double c) {
        double r106540 = b;
        double r106541 = -2.4028493234920365e+128;
        bool r106542 = r106540 <= r106541;
        double r106543 = -1.0;
        double r106544 = c;
        double r106545 = r106544 / r106540;
        double r106546 = r106543 * r106545;
        double r106547 = 5.877669040907696e-167;
        bool r106548 = r106540 <= r106547;
        double r106549 = 2.0;
        double r106550 = r106549 * r106544;
        double r106551 = r106540 * r106540;
        double r106552 = 4.0;
        double r106553 = a;
        double r106554 = r106553 * r106544;
        double r106555 = r106552 * r106554;
        double r106556 = r106551 - r106555;
        double r106557 = sqrt(r106556);
        double r106558 = r106557 - r106540;
        double r106559 = r106550 / r106558;
        double r106560 = 1.0;
        double r106561 = pow(r106559, r106560);
        double r106562 = 1.5849721394456554e+84;
        bool r106563 = r106540 <= r106562;
        double r106564 = -r106540;
        double r106565 = r106564 - r106557;
        double r106566 = r106549 * r106553;
        double r106567 = r106565 / r106566;
        double r106568 = 1.0;
        double r106569 = r106540 / r106553;
        double r106570 = r106545 - r106569;
        double r106571 = r106568 * r106570;
        double r106572 = r106563 ? r106567 : r106571;
        double r106573 = r106548 ? r106561 : r106572;
        double r106574 = r106542 ? r106546 : r106573;
        return r106574;
}

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
Target21.2
Herbie6.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -2.4028493234920365e+128

    1. Initial program 61.5

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

    2. Taylor expanded around -inf 2.2

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

    if -2.4028493234920365e+128 < b < 5.877669040907696e-167

    1. Initial program 29.5

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

    3. Applied div-inv29.6

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

    5. Applied flip--29.8

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

    6. Simplified16.1

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

    7. Simplified16.1

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

    9. Applied pow116.1

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

    10. Applied pow116.1

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

    11. Applied pow-prod-down16.1

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

    12. Simplified15.0

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

    13. Taylor expanded around 0 10.0

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

    if 5.877669040907696e-167 < b < 1.5849721394456554e+84

    1. Initial program 7.0

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

    if 1.5849721394456554e+84 < b

    1. Initial program 43.1

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

    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 simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.40284932349203652 \cdot 10^{128}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 5.877669040907696 \cdot 10^{-167}:\\ \;\;\;\;{\left(\frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\right)}^{1}\\ \mathbf{elif}\;b \le 1.58497213944565541 \cdot 10^{84}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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)))