Average Error: 34.2 → 10.5
Time: 5.5s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.4270058556435274 \cdot 10^{-117}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.49922826628406174 \cdot 10^{84}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{1}{\frac{a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -4.4270058556435274 \cdot 10^{-117}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 2.49922826628406174 \cdot 10^{84}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{1}{\frac{a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r58174 = b;
        double r58175 = -r58174;
        double r58176 = r58174 * r58174;
        double r58177 = 4.0;
        double r58178 = a;
        double r58179 = c;
        double r58180 = r58178 * r58179;
        double r58181 = r58177 * r58180;
        double r58182 = r58176 - r58181;
        double r58183 = sqrt(r58182);
        double r58184 = r58175 - r58183;
        double r58185 = 2.0;
        double r58186 = r58185 * r58178;
        double r58187 = r58184 / r58186;
        return r58187;
}
↓
double f(double a, double b, double c) {
        double r58188 = b;
        double r58189 = -4.4270058556435274e-117;
        bool r58190 = r58188 <= r58189;
        double r58191 = -1.0;
        double r58192 = c;
        double r58193 = r58192 / r58188;
        double r58194 = r58191 * r58193;
        double r58195 = 2.4992282662840617e+84;
        bool r58196 = r58188 <= r58195;
        double r58197 = 1.0;
        double r58198 = 2.0;
        double r58199 = r58197 / r58198;
        double r58200 = a;
        double r58201 = -r58188;
        double r58202 = r58188 * r58188;
        double r58203 = 4.0;
        double r58204 = r58200 * r58192;
        double r58205 = r58203 * r58204;
        double r58206 = r58202 - r58205;
        double r58207 = sqrt(r58206);
        double r58208 = r58201 - r58207;
        double r58209 = r58200 / r58208;
        double r58210 = r58197 / r58209;
        double r58211 = r58199 * r58210;
        double r58212 = 1.0;
        double r58213 = r58188 / r58200;
        double r58214 = r58193 - r58213;
        double r58215 = r58212 * r58214;
        double r58216 = r58196 ? r58211 : r58215;
        double r58217 = r58190 ? r58194 : r58216;
        return r58217;
}

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

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

Derivation

  1. Split input into 3 regimes

  2. if b < -4.4270058556435274e-117

    1. Initial program 51.5

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

    2. Taylor expanded around -inf 11.1

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

    if -4.4270058556435274e-117 < b < 2.4992282662840617e+84

    1. Initial program 12.4

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

    3. Applied clear-num12.5

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

    5. Applied *-un-lft-identity12.5

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

    6. Applied times-frac12.5

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

    7. Applied add-cube-cbrt12.5

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

    8. Applied times-frac12.5

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

    9. Simplified12.5

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

    10. Simplified12.4

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

    12. Applied clear-num12.5

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

    if 2.4992282662840617e+84 < b

    1. Initial program 43.2

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

    2. Taylor expanded around inf 4.1

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

    3. Simplified4.1

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.4270058556435274 \cdot 10^{-117}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.49922826628406174 \cdot 10^{84}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{1}{\frac{a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ 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)))