Average Error: 33.8 → 9.1
Time: 21.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.970010565552108757188050455448622102575 \cdot 10^{58}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.241327699195247001117817134883838709979 \cdot 10^{-253}:\\ \;\;\;\;\frac{\left(b \cdot b - b \cdot b\right) + 4 \cdot \left(c \cdot a\right)}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b} \cdot \frac{1}{a \cdot 2}\\ \mathbf{elif}\;b \le 3.628799960716311990444092539387346352569 \cdot 10^{50}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]
\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 -1.970010565552108757188050455448622102575 \cdot 10^{58}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -1.241327699195247001117817134883838709979 \cdot 10^{-253}:\\
\;\;\;\;\frac{\left(b \cdot b - b \cdot b\right) + 4 \cdot \left(c \cdot a\right)}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b} \cdot \frac{1}{a \cdot 2}\\

\mathbf{elif}\;b \le 3.628799960716311990444092539387346352569 \cdot 10^{50}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\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 r4134371 = b;
        double r4134372 = -r4134371;
        double r4134373 = r4134371 * r4134371;
        double r4134374 = 4.0;
        double r4134375 = a;
        double r4134376 = c;
        double r4134377 = r4134375 * r4134376;
        double r4134378 = r4134374 * r4134377;
        double r4134379 = r4134373 - r4134378;
        double r4134380 = sqrt(r4134379);
        double r4134381 = r4134372 - r4134380;
        double r4134382 = 2.0;
        double r4134383 = r4134382 * r4134375;
        double r4134384 = r4134381 / r4134383;
        return r4134384;
}
↓
double f(double a, double b, double c) {
        double r4134385 = b;
        double r4134386 = -1.9700105655521088e+58;
        bool r4134387 = r4134385 <= r4134386;
        double r4134388 = -1.0;
        double r4134389 = c;
        double r4134390 = r4134389 / r4134385;
        double r4134391 = r4134388 * r4134390;
        double r4134392 = -1.241327699195247e-253;
        bool r4134393 = r4134385 <= r4134392;
        double r4134394 = r4134385 * r4134385;
        double r4134395 = r4134394 - r4134394;
        double r4134396 = 4.0;
        double r4134397 = a;
        double r4134398 = r4134389 * r4134397;
        double r4134399 = r4134396 * r4134398;
        double r4134400 = r4134395 + r4134399;
        double r4134401 = r4134394 - r4134399;
        double r4134402 = sqrt(r4134401);
        double r4134403 = r4134402 - r4134385;
        double r4134404 = r4134400 / r4134403;
        double r4134405 = 1.0;
        double r4134406 = 2.0;
        double r4134407 = r4134397 * r4134406;
        double r4134408 = r4134405 / r4134407;
        double r4134409 = r4134404 * r4134408;
        double r4134410 = 3.628799960716312e+50;
        bool r4134411 = r4134385 <= r4134410;
        double r4134412 = -r4134385;
        double r4134413 = r4134412 - r4134402;
        double r4134414 = r4134413 / r4134407;
        double r4134415 = r4134385 / r4134397;
        double r4134416 = r4134390 - r4134415;
        double r4134417 = 1.0;
        double r4134418 = r4134416 * r4134417;
        double r4134419 = r4134411 ? r4134414 : r4134418;
        double r4134420 = r4134393 ? r4134409 : r4134419;
        double r4134421 = r4134387 ? r4134391 : r4134420;
        return r4134421;
}

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
Target20.7
Herbie9.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.9700105655521088e+58

    1. Initial program 57.5

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

    2. Taylor expanded around -inf 3.4

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

    if -1.9700105655521088e+58 < b < -1.241327699195247e-253

    1. Initial program 31.7

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

    3. Applied div-inv31.7

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

    5. Applied flip--31.8

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

    6. Simplified17.2

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

    7. Simplified17.2

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

    if -1.241327699195247e-253 < b < 3.628799960716312e+50

    1. Initial program 10.1

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

    3. Applied div-inv10.2

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

    5. Applied un-div-inv10.1

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

    if 3.628799960716312e+50 < b

    1. Initial program 38.2

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

    2. Taylor expanded around inf 6.1

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

    3. Simplified6.1

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.970010565552108757188050455448622102575 \cdot 10^{58}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.241327699195247001117817134883838709979 \cdot 10^{-253}:\\ \;\;\;\;\frac{\left(b \cdot b - b \cdot b\right) + 4 \cdot \left(c \cdot a\right)}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b} \cdot \frac{1}{a \cdot 2}\\ \mathbf{elif}\;b \le 3.628799960716311990444092539387346352569 \cdot 10^{50}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +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)))