Average Error: 33.3 → 9.1
Time: 5.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.44772243792088483 \cdot 10^{-41}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -2.1406165902202071 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le 7.35146897748971308 \cdot 10^{79}:\\ \;\;\;\;\frac{\frac{-b}{2}}{a} - \frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -1.44772243792088483 \cdot 10^{-41}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -2.1406165902202071 \cdot 10^{-136}:\\
\;\;\;\;\frac{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\

\mathbf{elif}\;b \le 7.35146897748971308 \cdot 10^{79}:\\
\;\;\;\;\frac{\frac{-b}{2}}{a} - \frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r94367 = b;
        double r94368 = -r94367;
        double r94369 = r94367 * r94367;
        double r94370 = 4.0;
        double r94371 = a;
        double r94372 = c;
        double r94373 = r94371 * r94372;
        double r94374 = r94370 * r94373;
        double r94375 = r94369 - r94374;
        double r94376 = sqrt(r94375);
        double r94377 = r94368 - r94376;
        double r94378 = 2.0;
        double r94379 = r94378 * r94371;
        double r94380 = r94377 / r94379;
        return r94380;
}
↓
double f(double a, double b, double c) {
        double r94381 = b;
        double r94382 = -1.4477224379208848e-41;
        bool r94383 = r94381 <= r94382;
        double r94384 = -1.0;
        double r94385 = c;
        double r94386 = r94385 / r94381;
        double r94387 = r94384 * r94386;
        double r94388 = -2.140616590220207e-136;
        bool r94389 = r94381 <= r94388;
        double r94390 = 2.0;
        double r94391 = pow(r94381, r94390);
        double r94392 = r94391 - r94391;
        double r94393 = 4.0;
        double r94394 = a;
        double r94395 = r94394 * r94385;
        double r94396 = r94393 * r94395;
        double r94397 = r94392 + r94396;
        double r94398 = r94381 * r94381;
        double r94399 = r94398 - r94396;
        double r94400 = sqrt(r94399);
        double r94401 = r94400 - r94381;
        double r94402 = r94397 / r94401;
        double r94403 = 2.0;
        double r94404 = r94403 * r94394;
        double r94405 = r94402 / r94404;
        double r94406 = 7.351468977489713e+79;
        bool r94407 = r94381 <= r94406;
        double r94408 = -r94381;
        double r94409 = r94408 / r94403;
        double r94410 = r94409 / r94394;
        double r94411 = r94400 / r94403;
        double r94412 = r94411 / r94394;
        double r94413 = r94410 - r94412;
        double r94414 = 1.0;
        double r94415 = r94381 / r94394;
        double r94416 = r94386 - r94415;
        double r94417 = r94414 * r94416;
        double r94418 = r94407 ? r94413 : r94417;
        double r94419 = r94389 ? r94405 : r94418;
        double r94420 = r94383 ? r94387 : r94419;
        return r94420;
}

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

Original33.3
Target20.7
Herbie9.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.4477224379208848e-41

    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 7.3

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

    if -1.4477224379208848e-41 < b < -2.140616590220207e-136

    1. Initial program 28.0

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

    3. Applied flip--28.0

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

    4. Simplified15.8

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

    5. Simplified15.8

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

    if -2.140616590220207e-136 < b < 7.351468977489713e+79

    1. Initial program 11.4

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

    3. Applied associate-/r*11.4

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

    5. Applied div-sub11.4

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

    6. Applied div-sub11.4

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

    if 7.351468977489713e+79 < b

    1. Initial program 41.8

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

    2. Taylor expanded around inf 4.5

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

    3. Simplified4.5

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.44772243792088483 \cdot 10^{-41}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -2.1406165902202071 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le 7.35146897748971308 \cdot 10^{79}:\\ \;\;\;\;\frac{\frac{-b}{2}}{a} - \frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020003 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :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)))