Average Error: 33.7 → 10.6
Time: 17.0s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.363255598823911 \cdot 10^{-15}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -6.936587154412951 \cdot 10^{-28}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{2 \cdot a}\\ \mathbf{elif}\;b \le -2.3344326820285623 \cdot 10^{-123}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.6691257204922504 \cdot 10^{+85}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -7.363255598823911 \cdot 10^{-15}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le -6.936587154412951 \cdot 10^{-28}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{2 \cdot a}\\

\mathbf{elif}\;b \le -2.3344326820285623 \cdot 10^{-123}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 1.6691257204922504 \cdot 10^{+85}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3761476 = b;
        double r3761477 = -r3761476;
        double r3761478 = r3761476 * r3761476;
        double r3761479 = 4.0;
        double r3761480 = a;
        double r3761481 = c;
        double r3761482 = r3761480 * r3761481;
        double r3761483 = r3761479 * r3761482;
        double r3761484 = r3761478 - r3761483;
        double r3761485 = sqrt(r3761484);
        double r3761486 = r3761477 - r3761485;
        double r3761487 = 2.0;
        double r3761488 = r3761487 * r3761480;
        double r3761489 = r3761486 / r3761488;
        return r3761489;
}
↓
double f(double a, double b, double c) {
        double r3761490 = b;
        double r3761491 = -7.363255598823911e-15;
        bool r3761492 = r3761490 <= r3761491;
        double r3761493 = c;
        double r3761494 = r3761493 / r3761490;
        double r3761495 = -r3761494;
        double r3761496 = -6.936587154412951e-28;
        bool r3761497 = r3761490 <= r3761496;
        double r3761498 = -r3761490;
        double r3761499 = 2.0;
        double r3761500 = a;
        double r3761501 = r3761499 * r3761500;
        double r3761502 = r3761498 / r3761501;
        double r3761503 = r3761490 * r3761490;
        double r3761504 = r3761500 * r3761493;
        double r3761505 = 4.0;
        double r3761506 = r3761504 * r3761505;
        double r3761507 = r3761503 - r3761506;
        double r3761508 = sqrt(r3761507);
        double r3761509 = r3761508 / r3761501;
        double r3761510 = r3761502 - r3761509;
        double r3761511 = -2.3344326820285623e-123;
        bool r3761512 = r3761490 <= r3761511;
        double r3761513 = 1.6691257204922504e+85;
        bool r3761514 = r3761490 <= r3761513;
        double r3761515 = r3761490 / r3761500;
        double r3761516 = r3761494 - r3761515;
        double r3761517 = r3761514 ? r3761510 : r3761516;
        double r3761518 = r3761512 ? r3761495 : r3761517;
        double r3761519 = r3761497 ? r3761510 : r3761518;
        double r3761520 = r3761492 ? r3761495 : r3761519;
        return r3761520;
}

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
Target21.0
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -7.363255598823911e-15 or -6.936587154412951e-28 < b < -2.3344326820285623e-123

    1. Initial program 50.9

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

    2. Taylor expanded around -inf 10.6

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

    3. Simplified10.6

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

    if -7.363255598823911e-15 < b < -6.936587154412951e-28 or -2.3344326820285623e-123 < b < 1.6691257204922504e+85

    1. Initial program 13.4

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

    3. Applied div-sub13.4

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

    if 1.6691257204922504e+85 < b

    1. Initial program 42.9

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

    2. Taylor expanded around inf 3.7

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.363255598823911 \cdot 10^{-15}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -6.936587154412951 \cdot 10^{-28}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{2 \cdot a}\\ \mathbf{elif}\;b \le -2.3344326820285623 \cdot 10^{-123}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.6691257204922504 \cdot 10^{+85}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(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)))