Average Error: 33.6 → 10.4
Time: 20.9s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.136683434005781 \cdot 10^{-32}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.927598127340643 \cdot 10^{+124}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -3.136683434005781 \cdot 10^{-32}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 2.927598127340643 \cdot 10^{+124}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r3085333 = b;
        double r3085334 = -r3085333;
        double r3085335 = r3085333 * r3085333;
        double r3085336 = 4.0;
        double r3085337 = a;
        double r3085338 = c;
        double r3085339 = r3085337 * r3085338;
        double r3085340 = r3085336 * r3085339;
        double r3085341 = r3085335 - r3085340;
        double r3085342 = sqrt(r3085341);
        double r3085343 = r3085334 - r3085342;
        double r3085344 = 2.0;
        double r3085345 = r3085344 * r3085337;
        double r3085346 = r3085343 / r3085345;
        return r3085346;
}
↓
double f(double a, double b, double c) {
        double r3085347 = b;
        double r3085348 = -3.136683434005781e-32;
        bool r3085349 = r3085347 <= r3085348;
        double r3085350 = c;
        double r3085351 = r3085350 / r3085347;
        double r3085352 = -r3085351;
        double r3085353 = 2.927598127340643e+124;
        bool r3085354 = r3085347 <= r3085353;
        double r3085355 = -r3085347;
        double r3085356 = -4.0;
        double r3085357 = a;
        double r3085358 = r3085357 * r3085350;
        double r3085359 = r3085347 * r3085347;
        double r3085360 = fma(r3085356, r3085358, r3085359);
        double r3085361 = sqrt(r3085360);
        double r3085362 = r3085355 - r3085361;
        double r3085363 = 2.0;
        double r3085364 = r3085363 * r3085357;
        double r3085365 = r3085362 / r3085364;
        double r3085366 = r3085347 / r3085357;
        double r3085367 = r3085351 - r3085366;
        double r3085368 = r3085354 ? r3085365 : r3085367;
        double r3085369 = r3085349 ? r3085352 : r3085368;
        return r3085369;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.6
Target20.8
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.136683434005781e-32

    1. Initial program 53.4

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

    2. Taylor expanded around -inf 7.3

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

    3. Simplified7.3

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

    if -3.136683434005781e-32 < b < 2.927598127340643e+124

    1. Initial program 14.7

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

    2. Taylor expanded around 0 14.7

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

    3. Simplified14.7

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

    if 2.927598127340643e+124 < b

    1. Initial program 50.6

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

    2. Taylor expanded around inf 2.9

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.136683434005781 \cdot 10^{-32}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.927598127340643 \cdot 10^{+124}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 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)))