Average Error: 33.8 → 10.3
Time: 24.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.569494919068124572690421335939486791404 \cdot 10^{-64}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.8653816703769607550753035783606354728 \cdot 10^{117}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]
\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 -2.569494919068124572690421335939486791404 \cdot 10^{-64}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 2.8653816703769607550753035783606354728 \cdot 10^{117}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{b}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r3268479 = b;
        double r3268480 = -r3268479;
        double r3268481 = r3268479 * r3268479;
        double r3268482 = 4.0;
        double r3268483 = a;
        double r3268484 = c;
        double r3268485 = r3268483 * r3268484;
        double r3268486 = r3268482 * r3268485;
        double r3268487 = r3268481 - r3268486;
        double r3268488 = sqrt(r3268487);
        double r3268489 = r3268480 - r3268488;
        double r3268490 = 2.0;
        double r3268491 = r3268490 * r3268483;
        double r3268492 = r3268489 / r3268491;
        return r3268492;
}
↓
double f(double a, double b, double c) {
        double r3268493 = b;
        double r3268494 = -2.5694949190681246e-64;
        bool r3268495 = r3268493 <= r3268494;
        double r3268496 = -1.0;
        double r3268497 = c;
        double r3268498 = r3268497 / r3268493;
        double r3268499 = r3268496 * r3268498;
        double r3268500 = 2.865381670376961e+117;
        bool r3268501 = r3268493 <= r3268500;
        double r3268502 = 1.0;
        double r3268503 = 2.0;
        double r3268504 = a;
        double r3268505 = r3268503 * r3268504;
        double r3268506 = -r3268493;
        double r3268507 = r3268493 * r3268493;
        double r3268508 = 4.0;
        double r3268509 = r3268504 * r3268508;
        double r3268510 = r3268509 * r3268497;
        double r3268511 = r3268507 - r3268510;
        double r3268512 = sqrt(r3268511);
        double r3268513 = r3268506 - r3268512;
        double r3268514 = r3268505 / r3268513;
        double r3268515 = r3268502 / r3268514;
        double r3268516 = r3268493 / r3268504;
        double r3268517 = r3268496 * r3268516;
        double r3268518 = r3268501 ? r3268515 : r3268517;
        double r3268519 = r3268495 ? r3268499 : r3268518;
        return r3268519;
}

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

Original33.8
Target20.9
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.5694949190681246e-64

    1. Initial program 53.0

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

    2. Taylor expanded around -inf 9.1

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

    if -2.5694949190681246e-64 < b < 2.865381670376961e+117

    1. Initial program 13.1

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

    2. Taylor expanded around 0 13.1

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

    3. Simplified13.1

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

    5. Applied clear-num13.2

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

    if 2.865381670376961e+117 < b

    1. Initial program 52.1

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

    2. Taylor expanded around 0 52.1

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

    3. Simplified52.1

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

    5. Applied clear-num52.2

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

    6. Taylor expanded around 0 3.1

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.569494919068124572690421335939486791404 \cdot 10^{-64}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.8653816703769607550753035783606354728 \cdot 10^{117}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(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)))