Average Error: 34.4 → 10.8
Time: 19.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.617913947565299992326164335754974391576 \cdot 10^{116}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 2.898348930695269343280527497904161468201 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{\sqrt{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]
\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 -5.617913947565299992326164335754974391576 \cdot 10^{116}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\mathbf{elif}\;b \le 2.898348930695269343280527497904161468201 \cdot 10^{-35}:\\
\;\;\;\;\frac{\frac{\sqrt{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}} - b}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r4015807 = b;
        double r4015808 = -r4015807;
        double r4015809 = r4015807 * r4015807;
        double r4015810 = 4.0;
        double r4015811 = a;
        double r4015812 = c;
        double r4015813 = r4015811 * r4015812;
        double r4015814 = r4015810 * r4015813;
        double r4015815 = r4015809 - r4015814;
        double r4015816 = sqrt(r4015815);
        double r4015817 = r4015808 + r4015816;
        double r4015818 = 2.0;
        double r4015819 = r4015818 * r4015811;
        double r4015820 = r4015817 / r4015819;
        return r4015820;
}
↓
double f(double a, double b, double c) {
        double r4015821 = b;
        double r4015822 = -5.6179139475653e+116;
        bool r4015823 = r4015821 <= r4015822;
        double r4015824 = c;
        double r4015825 = r4015824 / r4015821;
        double r4015826 = a;
        double r4015827 = r4015821 / r4015826;
        double r4015828 = r4015825 - r4015827;
        double r4015829 = 1.0;
        double r4015830 = r4015828 * r4015829;
        double r4015831 = 2.8983489306952693e-35;
        bool r4015832 = r4015821 <= r4015831;
        double r4015833 = r4015821 * r4015821;
        double r4015834 = 4.0;
        double r4015835 = r4015826 * r4015834;
        double r4015836 = r4015835 * r4015824;
        double r4015837 = r4015833 - r4015836;
        double r4015838 = sqrt(r4015837);
        double r4015839 = sqrt(r4015838);
        double r4015840 = r4015839 * r4015839;
        double r4015841 = r4015840 - r4015821;
        double r4015842 = 2.0;
        double r4015843 = r4015841 / r4015842;
        double r4015844 = r4015843 / r4015826;
        double r4015845 = -1.0;
        double r4015846 = r4015825 * r4015845;
        double r4015847 = r4015832 ? r4015844 : r4015846;
        double r4015848 = r4015823 ? r4015830 : r4015847;
        return r4015848;
}

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

Original34.4
Target21.5
Herbie10.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -5.6179139475653e+116

    1. Initial program 52.0

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

    2. Simplified52.0

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

    3. Taylor expanded around -inf 3.7

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

    4. Simplified3.7

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

    if -5.6179139475653e+116 < b < 2.8983489306952693e-35

    1. Initial program 15.0

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

    2. Simplified15.1

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

    4. Applied add-sqr-sqrt15.1

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

    5. Applied sqrt-prod15.3

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

    if 2.8983489306952693e-35 < b

    1. Initial program 54.4

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

    2. Simplified54.4

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

    3. Taylor expanded around inf 7.3

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

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.617913947565299992326164335754974391576 \cdot 10^{116}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 2.898348930695269343280527497904161468201 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{\sqrt{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :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)))