Average Error: 33.8 → 9.5
Time: 9.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.12428337420519208 \cdot 10^{57}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 6.78592592067132194 \cdot 10^{-105}:\\ \;\;\;\;\frac{1}{\frac{2}{\frac{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}{a}}}\\ \mathbf{elif}\;b \le 1.04869647254834991 \cdot 10^{-20}:\\ \;\;\;\;\frac{\left(4 \cdot a\right) \cdot c + b \cdot \left(b - b\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -3.12428337420519208 \cdot 10^{57}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 6.78592592067132194 \cdot 10^{-105}:\\
\;\;\;\;\frac{1}{\frac{2}{\frac{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}{a}}}\\

\mathbf{elif}\;b \le 1.04869647254834991 \cdot 10^{-20}:\\
\;\;\;\;\frac{\left(4 \cdot a\right) \cdot c + b \cdot \left(b - b\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r163321 = b;
        double r163322 = -r163321;
        double r163323 = r163321 * r163321;
        double r163324 = 4.0;
        double r163325 = a;
        double r163326 = r163324 * r163325;
        double r163327 = c;
        double r163328 = r163326 * r163327;
        double r163329 = r163323 - r163328;
        double r163330 = sqrt(r163329);
        double r163331 = r163322 + r163330;
        double r163332 = 2.0;
        double r163333 = r163332 * r163325;
        double r163334 = r163331 / r163333;
        return r163334;
}
↓
double f(double a, double b, double c) {
        double r163335 = b;
        double r163336 = -3.124283374205192e+57;
        bool r163337 = r163335 <= r163336;
        double r163338 = 1.0;
        double r163339 = c;
        double r163340 = r163339 / r163335;
        double r163341 = a;
        double r163342 = r163335 / r163341;
        double r163343 = r163340 - r163342;
        double r163344 = r163338 * r163343;
        double r163345 = 6.785925920671322e-105;
        bool r163346 = r163335 <= r163345;
        double r163347 = 1.0;
        double r163348 = 2.0;
        double r163349 = 2.0;
        double r163350 = pow(r163335, r163349);
        double r163351 = 4.0;
        double r163352 = r163341 * r163339;
        double r163353 = r163351 * r163352;
        double r163354 = r163350 - r163353;
        double r163355 = sqrt(r163354);
        double r163356 = r163355 - r163335;
        double r163357 = r163356 / r163341;
        double r163358 = r163348 / r163357;
        double r163359 = r163347 / r163358;
        double r163360 = 1.0486964725483499e-20;
        bool r163361 = r163335 <= r163360;
        double r163362 = r163351 * r163341;
        double r163363 = r163362 * r163339;
        double r163364 = r163335 - r163335;
        double r163365 = r163335 * r163364;
        double r163366 = r163363 + r163365;
        double r163367 = r163348 * r163341;
        double r163368 = -r163335;
        double r163369 = r163335 * r163335;
        double r163370 = r163369 - r163363;
        double r163371 = sqrt(r163370);
        double r163372 = r163368 - r163371;
        double r163373 = r163367 * r163372;
        double r163374 = r163366 / r163373;
        double r163375 = -1.0;
        double r163376 = r163375 * r163340;
        double r163377 = r163361 ? r163374 : r163376;
        double r163378 = r163346 ? r163359 : r163377;
        double r163379 = r163337 ? r163344 : r163378;
        return r163379;
}

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

Original33.8
Target20.4
Herbie9.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.124283374205192e+57

    1. Initial program 39.5

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

    2. Taylor expanded around -inf 5.4

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

    3. Simplified5.4

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

    if -3.124283374205192e+57 < b < 6.785925920671322e-105

    1. Initial program 11.8

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

    2. Taylor expanded around 0 11.8

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

    4. Applied clear-num11.9

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

    5. Simplified11.9

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

    if 6.785925920671322e-105 < b < 1.0486964725483499e-20

    1. Initial program 34.8

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

    3. Applied flip-+34.8

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

    4. Simplified17.9

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

    6. Applied div-inv18.0

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

    7. Applied associate-/l*23.7

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

    8. Simplified23.6

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

    if 1.0486964725483499e-20 < b

    1. Initial program 55.5

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

    2. Taylor expanded around inf 6.4

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

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.12428337420519208 \cdot 10^{57}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 6.78592592067132194 \cdot 10^{-105}:\\ \;\;\;\;\frac{1}{\frac{2}{\frac{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}{a}}}\\ \mathbf{elif}\;b \le 1.04869647254834991 \cdot 10^{-20}:\\ \;\;\;\;\frac{\left(4 \cdot a\right) \cdot c + b \cdot \left(b - b\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))