Average Error: 34.3 → 7.1
Time: 36.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.05669002671933381232315467688999002364 \cdot 10^{153}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.411004807395853104361669776563711353544 \cdot 10^{-303}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le 0.173897874048477174557802982235443778336:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]
\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 -6.05669002671933381232315467688999002364 \cdot 10^{153}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 3.411004807395853104361669776563711353544 \cdot 10^{-303}:\\
\;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\

\mathbf{elif}\;b \le 0.173897874048477174557802982235443778336:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r4348525 = b;
        double r4348526 = -r4348525;
        double r4348527 = r4348525 * r4348525;
        double r4348528 = 4.0;
        double r4348529 = a;
        double r4348530 = c;
        double r4348531 = r4348529 * r4348530;
        double r4348532 = r4348528 * r4348531;
        double r4348533 = r4348527 - r4348532;
        double r4348534 = sqrt(r4348533);
        double r4348535 = r4348526 - r4348534;
        double r4348536 = 2.0;
        double r4348537 = r4348536 * r4348529;
        double r4348538 = r4348535 / r4348537;
        return r4348538;
}
↓
double f(double a, double b, double c) {
        double r4348539 = b;
        double r4348540 = -6.056690026719334e+153;
        bool r4348541 = r4348539 <= r4348540;
        double r4348542 = -1.0;
        double r4348543 = c;
        double r4348544 = r4348543 / r4348539;
        double r4348545 = r4348542 * r4348544;
        double r4348546 = 3.411004807395853e-303;
        bool r4348547 = r4348539 <= r4348546;
        double r4348548 = 2.0;
        double r4348549 = r4348543 * r4348548;
        double r4348550 = -r4348539;
        double r4348551 = r4348539 * r4348539;
        double r4348552 = 4.0;
        double r4348553 = a;
        double r4348554 = r4348553 * r4348543;
        double r4348555 = r4348552 * r4348554;
        double r4348556 = r4348551 - r4348555;
        double r4348557 = sqrt(r4348556);
        double r4348558 = r4348550 + r4348557;
        double r4348559 = r4348549 / r4348558;
        double r4348560 = 0.17389787404847717;
        bool r4348561 = r4348539 <= r4348560;
        double r4348562 = r4348550 - r4348557;
        double r4348563 = r4348553 * r4348548;
        double r4348564 = r4348562 / r4348563;
        double r4348565 = r4348539 / r4348553;
        double r4348566 = r4348544 - r4348565;
        double r4348567 = 1.0;
        double r4348568 = r4348566 * r4348567;
        double r4348569 = r4348561 ? r4348564 : r4348568;
        double r4348570 = r4348547 ? r4348559 : r4348569;
        double r4348571 = r4348541 ? r4348545 : r4348570;
        return r4348571;
}

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

Original34.3
Target21.3
Herbie7.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -6.056690026719334e+153

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 1.1

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

    if -6.056690026719334e+153 < b < 3.411004807395853e-303

    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--34.1

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

    6. Applied associate-*l/34.1

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

    7. Simplified13.2

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

    8. Taylor expanded around 0 7.6

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

    if 3.411004807395853e-303 < b < 0.17389787404847717

    1. Initial program 11.3

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

    if 0.17389787404847717 < b

    1. Initial program 31.2

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

    2. Taylor expanded around inf 7.3

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

    3. Simplified7.3

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

  4. Final simplification7.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.05669002671933381232315467688999002364 \cdot 10^{153}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.411004807395853104361669776563711353544 \cdot 10^{-303}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le 0.173897874048477174557802982235443778336:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))