Average Error: 33.8 → 9.9
Time: 26.1s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.8424248185280593 \cdot 10^{-34}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 9.6230204738986831 \cdot 10^{86}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \end{array}\]
\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 -3.8424248185280593 \cdot 10^{-34}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 9.6230204738986831 \cdot 10^{86}:\\
\;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r79476 = b;
        double r79477 = -r79476;
        double r79478 = r79476 * r79476;
        double r79479 = 4.0;
        double r79480 = a;
        double r79481 = c;
        double r79482 = r79480 * r79481;
        double r79483 = r79479 * r79482;
        double r79484 = r79478 - r79483;
        double r79485 = sqrt(r79484);
        double r79486 = r79477 - r79485;
        double r79487 = 2.0;
        double r79488 = r79487 * r79480;
        double r79489 = r79486 / r79488;
        return r79489;
}
↓
double f(double a, double b, double c) {
        double r79490 = b;
        double r79491 = -3.8424248185280593e-34;
        bool r79492 = r79490 <= r79491;
        double r79493 = -1.0;
        double r79494 = c;
        double r79495 = r79494 / r79490;
        double r79496 = r79493 * r79495;
        double r79497 = 9.623020473898683e+86;
        bool r79498 = r79490 <= r79497;
        double r79499 = -r79490;
        double r79500 = r79490 * r79490;
        double r79501 = 4.0;
        double r79502 = a;
        double r79503 = r79502 * r79494;
        double r79504 = r79501 * r79503;
        double r79505 = r79500 - r79504;
        double r79506 = sqrt(r79505);
        double r79507 = r79499 - r79506;
        double r79508 = 1.0;
        double r79509 = 2.0;
        double r79510 = r79509 * r79502;
        double r79511 = r79508 / r79510;
        double r79512 = r79507 * r79511;
        double r79513 = -2.0;
        double r79514 = r79513 * r79490;
        double r79515 = r79514 / r79510;
        double r79516 = r79498 ? r79512 : r79515;
        double r79517 = r79492 ? r79496 : r79516;
        return r79517;
}

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
Target21.0
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.8424248185280593e-34

    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 7.1

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

    if -3.8424248185280593e-34 < b < 9.623020473898683e+86

    1. Initial program 13.9

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

    3. Applied div-inv14.0

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

    if 9.623020473898683e+86 < b

    1. Initial program 45.6

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

    3. Applied flip--63.0

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

    4. Simplified62.2

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

    5. Simplified62.2

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

    6. Taylor expanded around 0 4.2

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.8424248185280593 \cdot 10^{-34}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 9.6230204738986831 \cdot 10^{86}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 
(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)))