Average Error: 34.4 → 10.3
Time: 5.8s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.4278904486834676 \cdot 10^{-42}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.8046284917653458 \cdot 10^{91}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -5.4278904486834676 \cdot 10^{-42}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 2.8046284917653458 \cdot 10^{91}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r92354 = b;
        double r92355 = -r92354;
        double r92356 = r92354 * r92354;
        double r92357 = 4.0;
        double r92358 = a;
        double r92359 = c;
        double r92360 = r92358 * r92359;
        double r92361 = r92357 * r92360;
        double r92362 = r92356 - r92361;
        double r92363 = sqrt(r92362);
        double r92364 = r92355 - r92363;
        double r92365 = 2.0;
        double r92366 = r92365 * r92358;
        double r92367 = r92364 / r92366;
        return r92367;
}
↓
double f(double a, double b, double c) {
        double r92368 = b;
        double r92369 = -5.4278904486834676e-42;
        bool r92370 = r92368 <= r92369;
        double r92371 = -1.0;
        double r92372 = c;
        double r92373 = r92372 / r92368;
        double r92374 = r92371 * r92373;
        double r92375 = 2.8046284917653458e+91;
        bool r92376 = r92368 <= r92375;
        double r92377 = 1.0;
        double r92378 = 2.0;
        double r92379 = a;
        double r92380 = r92378 * r92379;
        double r92381 = -r92368;
        double r92382 = r92368 * r92368;
        double r92383 = 4.0;
        double r92384 = r92379 * r92372;
        double r92385 = r92383 * r92384;
        double r92386 = r92382 - r92385;
        double r92387 = sqrt(r92386);
        double r92388 = r92381 - r92387;
        double r92389 = r92380 / r92388;
        double r92390 = r92377 / r92389;
        double r92391 = 1.0;
        double r92392 = r92368 / r92379;
        double r92393 = r92373 - r92392;
        double r92394 = r92391 * r92393;
        double r92395 = r92376 ? r92390 : r92394;
        double r92396 = r92370 ? r92374 : r92395;
        return r92396;
}

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

Original34.4
Target20.8
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 < -5.4278904486834676e-42

    1. Initial program 54.7

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

    2. Taylor expanded around -inf 7.1

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

    if -5.4278904486834676e-42 < b < 2.8046284917653458e+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 clear-num14.8

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

    if 2.8046284917653458e+91 < b

    1. Initial program 45.4

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

    2. Taylor expanded around inf 4.1

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

    3. Simplified4.1

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.4278904486834676 \cdot 10^{-42}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.8046284917653458 \cdot 10^{91}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +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)))