Average Error: 34.7 → 10.2
Time: 33.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.270528699455007486596308100489334356636 \cdot 10^{152}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 1.744031351412432972171902712116585209201 \cdot 10^{-142}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]
\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 -1.270528699455007486596308100489334356636 \cdot 10^{152}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\mathbf{elif}\;b \le 1.744031351412432972171902712116585209201 \cdot 10^{-142}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} - b}{a \cdot 2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r5097496 = b;
        double r5097497 = -r5097496;
        double r5097498 = r5097496 * r5097496;
        double r5097499 = 4.0;
        double r5097500 = a;
        double r5097501 = r5097499 * r5097500;
        double r5097502 = c;
        double r5097503 = r5097501 * r5097502;
        double r5097504 = r5097498 - r5097503;
        double r5097505 = sqrt(r5097504);
        double r5097506 = r5097497 + r5097505;
        double r5097507 = 2.0;
        double r5097508 = r5097507 * r5097500;
        double r5097509 = r5097506 / r5097508;
        return r5097509;
}
↓
double f(double a, double b, double c) {
        double r5097510 = b;
        double r5097511 = -1.2705286994550075e+152;
        bool r5097512 = r5097510 <= r5097511;
        double r5097513 = c;
        double r5097514 = r5097513 / r5097510;
        double r5097515 = a;
        double r5097516 = r5097510 / r5097515;
        double r5097517 = r5097514 - r5097516;
        double r5097518 = 1.0;
        double r5097519 = r5097517 * r5097518;
        double r5097520 = 1.744031351412433e-142;
        bool r5097521 = r5097510 <= r5097520;
        double r5097522 = r5097510 * r5097510;
        double r5097523 = 4.0;
        double r5097524 = r5097513 * r5097523;
        double r5097525 = r5097524 * r5097515;
        double r5097526 = r5097522 - r5097525;
        double r5097527 = sqrt(r5097526);
        double r5097528 = r5097527 - r5097510;
        double r5097529 = 2.0;
        double r5097530 = r5097515 * r5097529;
        double r5097531 = r5097528 / r5097530;
        double r5097532 = -1.0;
        double r5097533 = r5097514 * r5097532;
        double r5097534 = r5097521 ? r5097531 : r5097533;
        double r5097535 = r5097512 ? r5097519 : r5097534;
        return r5097535;
}

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

Original34.7
Target21.1
Herbie10.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.2705286994550075e+152

    1. Initial program 62.9

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

    3. Applied div-inv62.9

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

    5. Applied associate-*r/62.9

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

    6. Simplified62.9

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

    7. Taylor expanded around -inf 1.7

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

    8. Simplified1.7

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

    if -1.2705286994550075e+152 < b < 1.744031351412433e-142

    1. Initial program 10.4

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

    3. Applied div-inv10.5

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

    5. Applied associate-*r/10.4

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

    6. Simplified10.4

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

    if 1.744031351412433e-142 < b

    1. Initial program 50.7

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

    2. Taylor expanded around inf 12.1

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.270528699455007486596308100489334356636 \cdot 10^{152}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 1.744031351412432972171902712116585209201 \cdot 10^{-142}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (a b c)
  :name "The quadratic formula (r1)"

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

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