Average Error: 34.2 → 10.3
Time: 4.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.429521559957367003973909183894614803551 \cdot 10^{-36}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 7.895438119410103188352046975315827374151 \cdot 10^{91}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \left(0.5 \cdot \frac{b}{a} - 1 \cdot \frac{c}{b}\right)\\ \end{array}\]
\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 -3.429521559957367003973909183894614803551 \cdot 10^{-36}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 7.895438119410103188352046975315827374151 \cdot 10^{91}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \left(0.5 \cdot \frac{b}{a} - 1 \cdot \frac{c}{b}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r83363 = b;
        double r83364 = -r83363;
        double r83365 = r83363 * r83363;
        double r83366 = 4.0;
        double r83367 = a;
        double r83368 = c;
        double r83369 = r83367 * r83368;
        double r83370 = r83366 * r83369;
        double r83371 = r83365 - r83370;
        double r83372 = sqrt(r83371);
        double r83373 = r83364 - r83372;
        double r83374 = 2.0;
        double r83375 = r83374 * r83367;
        double r83376 = r83373 / r83375;
        return r83376;
}
↓
double f(double a, double b, double c) {
        double r83377 = b;
        double r83378 = -3.429521559957367e-36;
        bool r83379 = r83377 <= r83378;
        double r83380 = -1.0;
        double r83381 = c;
        double r83382 = r83381 / r83377;
        double r83383 = r83380 * r83382;
        double r83384 = 7.895438119410103e+91;
        bool r83385 = r83377 <= r83384;
        double r83386 = -r83377;
        double r83387 = 2.0;
        double r83388 = a;
        double r83389 = r83387 * r83388;
        double r83390 = r83386 / r83389;
        double r83391 = r83377 * r83377;
        double r83392 = 4.0;
        double r83393 = r83388 * r83381;
        double r83394 = r83392 * r83393;
        double r83395 = r83391 - r83394;
        double r83396 = sqrt(r83395);
        double r83397 = r83396 / r83389;
        double r83398 = r83390 - r83397;
        double r83399 = 0.5;
        double r83400 = r83377 / r83388;
        double r83401 = r83399 * r83400;
        double r83402 = 1.0;
        double r83403 = r83402 * r83382;
        double r83404 = r83401 - r83403;
        double r83405 = r83390 - r83404;
        double r83406 = r83385 ? r83398 : r83405;
        double r83407 = r83379 ? r83383 : r83406;
        return r83407;
}

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
Target21.1
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 < -3.429521559957367e-36

    1. Initial program 54.2

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

    2. Taylor expanded around -inf 7.5

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

    if -3.429521559957367e-36 < b < 7.895438119410103e+91

    1. Initial program 14.7

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

    3. Applied div-sub14.7

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

    if 7.895438119410103e+91 < b

    1. Initial program 45.8

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

    3. Applied div-sub45.8

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

    4. Taylor expanded around inf 4.2

      \[\leadsto \frac{-b}{2 \cdot a} - \color{blue}{\left(0.5 \cdot \frac{b}{a} - 1 \cdot \frac{c}{b}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.429521559957367003973909183894614803551 \cdot 10^{-36}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 7.895438119410103188352046975315827374151 \cdot 10^{91}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \left(0.5 \cdot \frac{b}{a} - 1 \cdot \frac{c}{b}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +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)))