Average Error: 34.2 → 10.0
Time: 5.4s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.01157973271056712 \cdot 10^{-81}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.3176462918432122 \cdot 10^{99}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -4.01157973271056712 \cdot 10^{-81}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 1.3176462918432122 \cdot 10^{99}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{\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 r86530 = b;
        double r86531 = -r86530;
        double r86532 = r86530 * r86530;
        double r86533 = 4.0;
        double r86534 = a;
        double r86535 = c;
        double r86536 = r86534 * r86535;
        double r86537 = r86533 * r86536;
        double r86538 = r86532 - r86537;
        double r86539 = sqrt(r86538);
        double r86540 = r86531 - r86539;
        double r86541 = 2.0;
        double r86542 = r86541 * r86534;
        double r86543 = r86540 / r86542;
        return r86543;
}
↓
double f(double a, double b, double c) {
        double r86544 = b;
        double r86545 = -4.011579732710567e-81;
        bool r86546 = r86544 <= r86545;
        double r86547 = -1.0;
        double r86548 = c;
        double r86549 = r86548 / r86544;
        double r86550 = r86547 * r86549;
        double r86551 = 1.3176462918432122e+99;
        bool r86552 = r86544 <= r86551;
        double r86553 = -r86544;
        double r86554 = 2.0;
        double r86555 = a;
        double r86556 = r86554 * r86555;
        double r86557 = r86553 / r86556;
        double r86558 = r86544 * r86544;
        double r86559 = 4.0;
        double r86560 = r86555 * r86548;
        double r86561 = r86559 * r86560;
        double r86562 = r86558 - r86561;
        double r86563 = sqrt(r86562);
        double r86564 = r86563 / r86556;
        double r86565 = r86557 - r86564;
        double r86566 = 1.0;
        double r86567 = r86544 / r86555;
        double r86568 = r86549 - r86567;
        double r86569 = r86566 * r86568;
        double r86570 = r86552 ? r86565 : r86569;
        double r86571 = r86546 ? r86550 : r86570;
        return r86571;
}

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
Target21.5
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.011579732710567e-81

    1. Initial program 52.8

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

    2. Taylor expanded around -inf 9.4

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

    if -4.011579732710567e-81 < b < 1.3176462918432122e+99

    1. Initial program 12.9

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

    3. Applied div-sub12.9

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

    if 1.3176462918432122e+99 < b

    1. Initial program 46.8

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

    2. Taylor expanded around inf 3.8

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

    3. Simplified3.8

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.01157973271056712 \cdot 10^{-81}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.3176462918432122 \cdot 10^{99}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(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)))