Average Error: 33.5 → 10.5
Time: 25.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -227369802444031.66:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 2.0617732603635578 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \frac{1}{a} + b \cdot \frac{-1}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]
\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 -227369802444031.66:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 2.0617732603635578 \cdot 10^{-61}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \frac{1}{a} + b \cdot \frac{-1}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r3490358 = b;
        double r3490359 = -r3490358;
        double r3490360 = r3490358 * r3490358;
        double r3490361 = 4.0;
        double r3490362 = a;
        double r3490363 = r3490361 * r3490362;
        double r3490364 = c;
        double r3490365 = r3490363 * r3490364;
        double r3490366 = r3490360 - r3490365;
        double r3490367 = sqrt(r3490366);
        double r3490368 = r3490359 + r3490367;
        double r3490369 = 2.0;
        double r3490370 = r3490369 * r3490362;
        double r3490371 = r3490368 / r3490370;
        return r3490371;
}
↓
double f(double a, double b, double c) {
        double r3490372 = b;
        double r3490373 = -227369802444031.66;
        bool r3490374 = r3490372 <= r3490373;
        double r3490375 = c;
        double r3490376 = r3490375 / r3490372;
        double r3490377 = a;
        double r3490378 = r3490372 / r3490377;
        double r3490379 = r3490376 - r3490378;
        double r3490380 = 2.0;
        double r3490381 = r3490379 * r3490380;
        double r3490382 = r3490381 / r3490380;
        double r3490383 = 2.0617732603635578e-61;
        bool r3490384 = r3490372 <= r3490383;
        double r3490385 = r3490372 * r3490372;
        double r3490386 = r3490375 * r3490377;
        double r3490387 = 4.0;
        double r3490388 = r3490386 * r3490387;
        double r3490389 = r3490385 - r3490388;
        double r3490390 = sqrt(r3490389);
        double r3490391 = 1.0;
        double r3490392 = r3490391 / r3490377;
        double r3490393 = r3490390 * r3490392;
        double r3490394 = -1.0;
        double r3490395 = r3490394 / r3490377;
        double r3490396 = r3490372 * r3490395;
        double r3490397 = r3490393 + r3490396;
        double r3490398 = r3490397 / r3490380;
        double r3490399 = -2.0;
        double r3490400 = r3490376 * r3490399;
        double r3490401 = r3490400 / r3490380;
        double r3490402 = r3490384 ? r3490398 : r3490401;
        double r3490403 = r3490374 ? r3490382 : r3490402;
        return r3490403;
}

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

Original33.5
Target20.6
Herbie10.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -227369802444031.66

    1. Initial program 32.9

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

    2. Simplified32.9

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

    4. Applied div-inv33.1

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

    5. Taylor expanded around -inf 6.8

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]

    6. Simplified6.8

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

    if -227369802444031.66 < b < 2.0617732603635578e-61

    1. Initial program 14.9

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

    2. Simplified15.0

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

    4. Applied div-inv15.0

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

    6. Applied *-commutative15.0

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

    8. Applied sub-neg15.0

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

    9. Applied distribute-lft-in15.0

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

    if 2.0617732603635578e-61 < b

    1. Initial program 52.8

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

    2. Simplified52.8

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

    3. Taylor expanded around inf 8.3

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -227369802444031.66:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 2.0617732603635578 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \frac{1}{a} + b \cdot \frac{-1}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 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)))