Average Error: 33.2 → 10.0
Time: 17.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.397994825724217 \cdot 10^{+150}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.2158870426682226 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -7.397994825724217 \cdot 10^{+150}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 1.2158870426682226 \cdot 10^{-82}:\\
\;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}} - \frac{b}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r5091674 = b;
        double r5091675 = -r5091674;
        double r5091676 = r5091674 * r5091674;
        double r5091677 = 4.0;
        double r5091678 = a;
        double r5091679 = r5091677 * r5091678;
        double r5091680 = c;
        double r5091681 = r5091679 * r5091680;
        double r5091682 = r5091676 - r5091681;
        double r5091683 = sqrt(r5091682);
        double r5091684 = r5091675 + r5091683;
        double r5091685 = 2.0;
        double r5091686 = r5091685 * r5091678;
        double r5091687 = r5091684 / r5091686;
        return r5091687;
}
↓
double f(double a, double b, double c) {
        double r5091688 = b;
        double r5091689 = -7.397994825724217e+150;
        bool r5091690 = r5091688 <= r5091689;
        double r5091691 = c;
        double r5091692 = r5091691 / r5091688;
        double r5091693 = a;
        double r5091694 = r5091688 / r5091693;
        double r5091695 = r5091692 - r5091694;
        double r5091696 = 2.0;
        double r5091697 = r5091695 * r5091696;
        double r5091698 = r5091697 / r5091696;
        double r5091699 = 1.2158870426682226e-82;
        bool r5091700 = r5091688 <= r5091699;
        double r5091701 = 1.0;
        double r5091702 = r5091688 * r5091688;
        double r5091703 = r5091693 * r5091691;
        double r5091704 = 4.0;
        double r5091705 = r5091703 * r5091704;
        double r5091706 = r5091702 - r5091705;
        double r5091707 = sqrt(r5091706);
        double r5091708 = r5091693 / r5091707;
        double r5091709 = r5091701 / r5091708;
        double r5091710 = r5091709 - r5091694;
        double r5091711 = r5091710 / r5091696;
        double r5091712 = -2.0;
        double r5091713 = r5091712 * r5091692;
        double r5091714 = r5091713 / r5091696;
        double r5091715 = r5091700 ? r5091711 : r5091714;
        double r5091716 = r5091690 ? r5091698 : r5091715;
        return r5091716;
}

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

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

Derivation

  1. Split input into 3 regimes

  2. if b < -7.397994825724217e+150

    1. Initial program 59.1

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

    2. Simplified59.1

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

    3. Taylor expanded around -inf 2.2

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

    4. Simplified2.2

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

    if -7.397994825724217e+150 < b < 1.2158870426682226e-82

    1. Initial program 11.8

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

    2. Simplified11.7

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

    4. Applied div-sub11.7

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

    6. Applied clear-num11.8

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

    if 1.2158870426682226e-82 < b

    1. Initial program 52.3

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

    2. Simplified52.3

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

    3. Taylor expanded around inf 9.9

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.397994825724217 \cdot 10^{+150}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.2158870426682226 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(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)))