Average Error: 33.1 → 8.5
Time: 51.3s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.9319920094724875 \cdot 10^{+55}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -6.516529709605321 \cdot 10^{-197}:\\ \;\;\;\;-\frac{\frac{\frac{a \cdot c}{\frac{-1}{2}}}{a}}{\left(-b\right) + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}\\ \mathbf{elif}\;b \le 2.0734884367796945 \cdot 10^{+89}:\\ \;\;\;\;\frac{-b}{a \cdot 2} - \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -1.9319920094724875 \cdot 10^{+55}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le -6.516529709605321 \cdot 10^{-197}:\\
\;\;\;\;-\frac{\frac{\frac{a \cdot c}{\frac{-1}{2}}}{a}}{\left(-b\right) + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}\\

\mathbf{elif}\;b \le 2.0734884367796945 \cdot 10^{+89}:\\
\;\;\;\;\frac{-b}{a \cdot 2} - \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r4192491 = b;
        double r4192492 = -r4192491;
        double r4192493 = r4192491 * r4192491;
        double r4192494 = 4.0;
        double r4192495 = a;
        double r4192496 = c;
        double r4192497 = r4192495 * r4192496;
        double r4192498 = r4192494 * r4192497;
        double r4192499 = r4192493 - r4192498;
        double r4192500 = sqrt(r4192499);
        double r4192501 = r4192492 - r4192500;
        double r4192502 = 2.0;
        double r4192503 = r4192502 * r4192495;
        double r4192504 = r4192501 / r4192503;
        return r4192504;
}
↓
double f(double a, double b, double c) {
        double r4192505 = b;
        double r4192506 = -1.9319920094724875e+55;
        bool r4192507 = r4192505 <= r4192506;
        double r4192508 = c;
        double r4192509 = r4192508 / r4192505;
        double r4192510 = -r4192509;
        double r4192511 = -6.516529709605321e-197;
        bool r4192512 = r4192505 <= r4192511;
        double r4192513 = a;
        double r4192514 = r4192513 * r4192508;
        double r4192515 = -0.5;
        double r4192516 = r4192514 / r4192515;
        double r4192517 = r4192516 / r4192513;
        double r4192518 = -r4192505;
        double r4192519 = r4192505 * r4192505;
        double r4192520 = 4.0;
        double r4192521 = r4192514 * r4192520;
        double r4192522 = r4192519 - r4192521;
        double r4192523 = sqrt(r4192522);
        double r4192524 = r4192518 + r4192523;
        double r4192525 = r4192517 / r4192524;
        double r4192526 = -r4192525;
        double r4192527 = 2.0734884367796945e+89;
        bool r4192528 = r4192505 <= r4192527;
        double r4192529 = 2.0;
        double r4192530 = r4192513 * r4192529;
        double r4192531 = r4192518 / r4192530;
        double r4192532 = r4192523 / r4192530;
        double r4192533 = r4192531 - r4192532;
        double r4192534 = r4192505 / r4192513;
        double r4192535 = r4192509 - r4192534;
        double r4192536 = r4192528 ? r4192533 : r4192535;
        double r4192537 = r4192512 ? r4192526 : r4192536;
        double r4192538 = r4192507 ? r4192510 : r4192537;
        return r4192538;
}

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

Original33.1
Target20.3
Herbie8.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.9319920094724875e+55

    1. Initial program 56.5

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

    3. Applied div-inv56.5

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

    4. Taylor expanded around -inf 3.7

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

    5. Simplified3.7

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

    if -1.9319920094724875e+55 < b < -6.516529709605321e-197

    1. Initial program 34.0

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

    3. Applied div-inv34.1

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

    5. Applied flip--34.2

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

    6. Applied associate-*l/34.2

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

    7. Simplified15.9

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

    if -6.516529709605321e-197 < b < 2.0734884367796945e+89

    1. Initial program 10.3

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

    3. Applied div-sub10.3

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

    if 2.0734884367796945e+89 < b

    1. Initial program 42.0

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

    2. Taylor expanded around inf 4.1

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

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.9319920094724875 \cdot 10^{+55}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -6.516529709605321 \cdot 10^{-197}:\\ \;\;\;\;-\frac{\frac{\frac{a \cdot c}{\frac{-1}{2}}}{a}}{\left(-b\right) + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}\\ \mathbf{elif}\;b \le 2.0734884367796945 \cdot 10^{+89}:\\ \;\;\;\;\frac{-b}{a \cdot 2} - \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 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)))