Average Error: 34.5 → 10.2
Time: 4.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.706781135059311758856471716413486308072 \cdot 10^{-92}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 5.722235152988638272816037483919181313619 \cdot 10^{98}:\\ \;\;\;\;{\left(\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -4.706781135059311758856471716413486308072 \cdot 10^{-92}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 5.722235152988638272816037483919181313619 \cdot 10^{98}:\\
\;\;\;\;{\left(\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right)}^{1}\\

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

\end{array}
double f(double a, double b, double c) {
        double r68249 = b;
        double r68250 = -r68249;
        double r68251 = r68249 * r68249;
        double r68252 = 4.0;
        double r68253 = a;
        double r68254 = c;
        double r68255 = r68253 * r68254;
        double r68256 = r68252 * r68255;
        double r68257 = r68251 - r68256;
        double r68258 = sqrt(r68257);
        double r68259 = r68250 - r68258;
        double r68260 = 2.0;
        double r68261 = r68260 * r68253;
        double r68262 = r68259 / r68261;
        return r68262;
}
↓
double f(double a, double b, double c) {
        double r68263 = b;
        double r68264 = -4.706781135059312e-92;
        bool r68265 = r68263 <= r68264;
        double r68266 = -1.0;
        double r68267 = c;
        double r68268 = r68267 / r68263;
        double r68269 = r68266 * r68268;
        double r68270 = 5.722235152988638e+98;
        bool r68271 = r68263 <= r68270;
        double r68272 = -r68263;
        double r68273 = r68263 * r68263;
        double r68274 = 4.0;
        double r68275 = a;
        double r68276 = r68275 * r68267;
        double r68277 = r68274 * r68276;
        double r68278 = r68273 - r68277;
        double r68279 = sqrt(r68278);
        double r68280 = r68272 - r68279;
        double r68281 = 2.0;
        double r68282 = r68281 * r68275;
        double r68283 = r68280 / r68282;
        double r68284 = 1.0;
        double r68285 = pow(r68283, r68284);
        double r68286 = 1.0;
        double r68287 = r68263 / r68275;
        double r68288 = r68268 - r68287;
        double r68289 = r68286 * r68288;
        double r68290 = r68271 ? r68285 : r68289;
        double r68291 = r68265 ? r68269 : r68290;
        return r68291;
}

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

Original34.5
Target21.5
Herbie10.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.706781135059312e-92

    1. Initial program 52.4

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

    2. Taylor expanded around -inf 10.3

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

    if -4.706781135059312e-92 < b < 5.722235152988638e+98

    1. Initial program 12.7

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

    3. Applied div-inv12.8

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

    5. Applied pow112.8

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

    6. Applied pow112.8

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

    7. Applied pow-prod-down12.8

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

    8. Simplified12.7

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

    if 5.722235152988638e+98 < b

    1. Initial program 47.2

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

    2. Taylor expanded around inf 3.6

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

    3. Simplified3.6

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.706781135059311758856471716413486308072 \cdot 10^{-92}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 5.722235152988638272816037483919181313619 \cdot 10^{98}:\\ \;\;\;\;{\left(\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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)))