Average Error: 34.4 → 10.6
Time: 14.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.668263498157484 \cdot 10^{-23}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.938039658404583 \cdot 10^{94}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -7.668263498157484 \cdot 10^{-23}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 2.938039658404583 \cdot 10^{94}:\\
\;\;\;\;\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 r48345 = b;
        double r48346 = -r48345;
        double r48347 = r48345 * r48345;
        double r48348 = 4.0;
        double r48349 = a;
        double r48350 = c;
        double r48351 = r48349 * r48350;
        double r48352 = r48348 * r48351;
        double r48353 = r48347 - r48352;
        double r48354 = sqrt(r48353);
        double r48355 = r48346 - r48354;
        double r48356 = 2.0;
        double r48357 = r48356 * r48349;
        double r48358 = r48355 / r48357;
        return r48358;
}
↓
double f(double a, double b, double c) {
        double r48359 = b;
        double r48360 = -7.668263498157484e-23;
        bool r48361 = r48359 <= r48360;
        double r48362 = -1.0;
        double r48363 = c;
        double r48364 = r48363 / r48359;
        double r48365 = r48362 * r48364;
        double r48366 = 2.9380396584045826e+94;
        bool r48367 = r48359 <= r48366;
        double r48368 = -r48359;
        double r48369 = r48359 * r48359;
        double r48370 = 4.0;
        double r48371 = a;
        double r48372 = r48371 * r48363;
        double r48373 = r48370 * r48372;
        double r48374 = r48369 - r48373;
        double r48375 = sqrt(r48374);
        double r48376 = r48368 - r48375;
        double r48377 = 1.0;
        double r48378 = 2.0;
        double r48379 = r48378 * r48371;
        double r48380 = r48377 / r48379;
        double r48381 = r48376 * r48380;
        double r48382 = 1.0;
        double r48383 = r48359 / r48371;
        double r48384 = r48364 - r48383;
        double r48385 = r48382 * r48384;
        double r48386 = r48367 ? r48381 : r48385;
        double r48387 = r48361 ? r48365 : r48386;
        return r48387;
}

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
Target21.0
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -7.668263498157484e-23

    1. Initial program 55.0

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

    2. Taylor expanded around -inf 6.4

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

    if -7.668263498157484e-23 < b < 2.9380396584045826e+94

    1. Initial program 15.6

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

    3. Applied div-inv15.7

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

    if 2.9380396584045826e+94 < b

    1. Initial program 46.7

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

    2. Taylor expanded around inf 3.9

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

    3. Simplified3.9

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.668263498157484 \cdot 10^{-23}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.938039658404583 \cdot 10^{94}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))