Average Error: 34.5 → 8.8
Time: 19.8s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.102895015780532348136946077262401346805 \cdot 10^{69}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -8.767251655423633534328588307438915014497 \cdot 10^{-253}:\\ \;\;\;\;\frac{\frac{\left(4 \cdot a\right) \cdot c}{2 \cdot a}}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 2.125563079851438727208684227808951636731 \cdot 10^{135}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -3.102895015780532348136946077262401346805 \cdot 10^{69}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -8.767251655423633534328588307438915014497 \cdot 10^{-253}:\\
\;\;\;\;\frac{\frac{\left(4 \cdot a\right) \cdot c}{2 \cdot a}}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\

\mathbf{elif}\;b \le 2.125563079851438727208684227808951636731 \cdot 10^{135}:\\
\;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r74345 = b;
        double r74346 = -r74345;
        double r74347 = r74345 * r74345;
        double r74348 = 4.0;
        double r74349 = a;
        double r74350 = c;
        double r74351 = r74349 * r74350;
        double r74352 = r74348 * r74351;
        double r74353 = r74347 - r74352;
        double r74354 = sqrt(r74353);
        double r74355 = r74346 - r74354;
        double r74356 = 2.0;
        double r74357 = r74356 * r74349;
        double r74358 = r74355 / r74357;
        return r74358;
}
↓
double f(double a, double b, double c) {
        double r74359 = b;
        double r74360 = -3.1028950157805323e+69;
        bool r74361 = r74359 <= r74360;
        double r74362 = -1.0;
        double r74363 = c;
        double r74364 = r74363 / r74359;
        double r74365 = r74362 * r74364;
        double r74366 = -8.767251655423634e-253;
        bool r74367 = r74359 <= r74366;
        double r74368 = 4.0;
        double r74369 = a;
        double r74370 = r74368 * r74369;
        double r74371 = r74370 * r74363;
        double r74372 = 2.0;
        double r74373 = r74372 * r74369;
        double r74374 = r74371 / r74373;
        double r74375 = 2.0;
        double r74376 = pow(r74359, r74375);
        double r74377 = r74369 * r74363;
        double r74378 = r74368 * r74377;
        double r74379 = r74376 - r74378;
        double r74380 = sqrt(r74379);
        double r74381 = r74380 - r74359;
        double r74382 = r74374 / r74381;
        double r74383 = 2.1255630798514387e+135;
        bool r74384 = r74359 <= r74383;
        double r74385 = -r74359;
        double r74386 = r74359 * r74359;
        double r74387 = r74386 - r74378;
        double r74388 = sqrt(r74387);
        double r74389 = r74385 - r74388;
        double r74390 = 1.0;
        double r74391 = r74390 / r74373;
        double r74392 = r74389 * r74391;
        double r74393 = 1.0;
        double r74394 = r74359 / r74369;
        double r74395 = r74364 - r74394;
        double r74396 = r74393 * r74395;
        double r74397 = r74384 ? r74392 : r74396;
        double r74398 = r74367 ? r74382 : r74397;
        double r74399 = r74361 ? r74365 : r74398;
        return r74399;
}

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
Target21.3
Herbie8.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.1028950157805323e+69

    1. Initial program 58.6

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

    2. Taylor expanded around -inf 3.1

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

    if -3.1028950157805323e+69 < b < -8.767251655423634e-253

    1. Initial program 33.1

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

    3. Applied flip--33.1

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

    4. Simplified16.9

      \[\leadsto \frac{\frac{\color{blue}{0 + c \cdot \left(4 \cdot a\right)}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]

    5. Simplified16.9

      \[\leadsto \frac{\frac{0 + c \cdot \left(4 \cdot a\right)}{\color{blue}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}}}{2 \cdot a}\]
    6. Using strategy rm

    7. Applied div-inv16.9

      \[\leadsto \color{blue}{\frac{0 + c \cdot \left(4 \cdot a\right)}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b} \cdot \frac{1}{2 \cdot a}}\]
    8. Using strategy rm

    9. Applied *-un-lft-identity16.9

      \[\leadsto \frac{0 + c \cdot \left(4 \cdot a\right)}{\color{blue}{1 \cdot \left(\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b\right)}} \cdot \frac{1}{2 \cdot a}\]

    10. Applied *-un-lft-identity16.9

      \[\leadsto \frac{\color{blue}{1 \cdot \left(0 + c \cdot \left(4 \cdot a\right)\right)}}{1 \cdot \left(\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b\right)} \cdot \frac{1}{2 \cdot a}\]

    11. Applied times-frac16.9

      \[\leadsto \color{blue}{\left(\frac{1}{1} \cdot \frac{0 + c \cdot \left(4 \cdot a\right)}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\right)} \cdot \frac{1}{2 \cdot a}\]

    12. Applied associate-*l*16.9

      \[\leadsto \color{blue}{\frac{1}{1} \cdot \left(\frac{0 + c \cdot \left(4 \cdot a\right)}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b} \cdot \frac{1}{2 \cdot a}\right)}\]

    13. Simplified16.3

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

    if -8.767251655423634e-253 < b < 2.1255630798514387e+135

    1. Initial program 9.6

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

    3. Applied div-inv9.8

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

    if 2.1255630798514387e+135 < b

    1. Initial program 58.2

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

    2. Taylor expanded around inf 3.0

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

    3. Simplified3.0

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

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.102895015780532348136946077262401346805 \cdot 10^{69}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -8.767251655423633534328588307438915014497 \cdot 10^{-253}:\\ \;\;\;\;\frac{\frac{\left(4 \cdot a\right) \cdot c}{2 \cdot a}}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 2.125563079851438727208684227808951636731 \cdot 10^{135}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +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)))