Average Error: 33.4 → 10.1
Time: 22.0s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.840085388791461 \cdot 10^{-68}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.5949594684703287 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\left(-b\right) - \sqrt{\sqrt[3]{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \left(\sqrt[3]{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \sqrt[3]{b \cdot b - \left(c \cdot a\right) \cdot 4}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -2.840085388791461 \cdot 10^{-68}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 1.5949594684703287 \cdot 10^{+126}:\\
\;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\left(-b\right) - \sqrt{\sqrt[3]{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \left(\sqrt[3]{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \sqrt[3]{b \cdot b - \left(c \cdot a\right) \cdot 4}\right)}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r2730673 = b;
        double r2730674 = -r2730673;
        double r2730675 = r2730673 * r2730673;
        double r2730676 = 4.0;
        double r2730677 = a;
        double r2730678 = c;
        double r2730679 = r2730677 * r2730678;
        double r2730680 = r2730676 * r2730679;
        double r2730681 = r2730675 - r2730680;
        double r2730682 = sqrt(r2730681);
        double r2730683 = r2730674 - r2730682;
        double r2730684 = 2.0;
        double r2730685 = r2730684 * r2730677;
        double r2730686 = r2730683 / r2730685;
        return r2730686;
}
↓
double f(double a, double b, double c) {
        double r2730687 = b;
        double r2730688 = -2.840085388791461e-68;
        bool r2730689 = r2730687 <= r2730688;
        double r2730690 = c;
        double r2730691 = r2730690 / r2730687;
        double r2730692 = -r2730691;
        double r2730693 = 1.5949594684703287e+126;
        bool r2730694 = r2730687 <= r2730693;
        double r2730695 = 0.5;
        double r2730696 = a;
        double r2730697 = r2730695 / r2730696;
        double r2730698 = -r2730687;
        double r2730699 = r2730687 * r2730687;
        double r2730700 = r2730690 * r2730696;
        double r2730701 = 4.0;
        double r2730702 = r2730700 * r2730701;
        double r2730703 = r2730699 - r2730702;
        double r2730704 = cbrt(r2730703);
        double r2730705 = r2730704 * r2730704;
        double r2730706 = r2730704 * r2730705;
        double r2730707 = sqrt(r2730706);
        double r2730708 = r2730698 - r2730707;
        double r2730709 = r2730697 * r2730708;
        double r2730710 = r2730687 / r2730696;
        double r2730711 = r2730691 - r2730710;
        double r2730712 = r2730694 ? r2730709 : r2730711;
        double r2730713 = r2730689 ? r2730692 : r2730712;
        return r2730713;
}

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

Original33.4
Target20.2
Herbie10.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.840085388791461e-68

    1. Initial program 52.7

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

    2. Taylor expanded around -inf 9.0

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

    3. Simplified9.0

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

    if -2.840085388791461e-68 < b < 1.5949594684703287e+126

    1. Initial program 12.5

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

    3. Applied div-inv12.7

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

    4. Simplified12.7

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

    6. Applied add-cube-cbrt13.0

      \[\leadsto \left(\left(-b\right) - \sqrt{\color{blue}{\left(\sqrt[3]{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \sqrt[3]{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\right) \cdot \frac{\frac{1}{2}}{a}\]

    if 1.5949594684703287e+126 < b

    1. Initial program 50.8

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

    3. Applied div-inv50.8

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

    4. Simplified50.8

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

    5. Taylor expanded around inf 3.1

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.840085388791461 \cdot 10^{-68}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.5949594684703287 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\left(-b\right) - \sqrt{\sqrt[3]{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \left(\sqrt[3]{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \sqrt[3]{b \cdot b - \left(c \cdot a\right) \cdot 4}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :herbie-target
  (if (< b 0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))