Average Error: 33.6 → 10.3
Time: 8.0s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.16908657181932359 \cdot 10^{-104}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \le 1.3316184968738608 \cdot 10^{61}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{b}{a}\right)\\ \end{array}\]
\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 -4.16908657181932359 \cdot 10^{-104}:\\
\;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{c}{b}\right)\\

\mathbf{elif}\;b \le 1.3316184968738608 \cdot 10^{61}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r90140 = b;
        double r90141 = -r90140;
        double r90142 = r90140 * r90140;
        double r90143 = 4.0;
        double r90144 = a;
        double r90145 = c;
        double r90146 = r90144 * r90145;
        double r90147 = r90143 * r90146;
        double r90148 = r90142 - r90147;
        double r90149 = sqrt(r90148);
        double r90150 = r90141 - r90149;
        double r90151 = 2.0;
        double r90152 = r90151 * r90144;
        double r90153 = r90150 / r90152;
        return r90153;
}
↓
double f(double a, double b, double c) {
        double r90154 = b;
        double r90155 = -4.1690865718193236e-104;
        bool r90156 = r90154 <= r90155;
        double r90157 = 1.0;
        double r90158 = 2.0;
        double r90159 = r90157 / r90158;
        double r90160 = -2.0;
        double r90161 = c;
        double r90162 = r90161 / r90154;
        double r90163 = r90160 * r90162;
        double r90164 = r90159 * r90163;
        double r90165 = 1.3316184968738608e+61;
        bool r90166 = r90154 <= r90165;
        double r90167 = -r90154;
        double r90168 = r90154 * r90154;
        double r90169 = 4.0;
        double r90170 = a;
        double r90171 = r90170 * r90161;
        double r90172 = r90169 * r90171;
        double r90173 = r90168 - r90172;
        double r90174 = sqrt(r90173);
        double r90175 = r90167 - r90174;
        double r90176 = r90175 / r90170;
        double r90177 = r90159 * r90176;
        double r90178 = -2.0;
        double r90179 = r90154 / r90170;
        double r90180 = r90178 * r90179;
        double r90181 = r90159 * r90180;
        double r90182 = r90166 ? r90177 : r90181;
        double r90183 = r90156 ? r90164 : r90182;
        return r90183;
}

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

Original33.6
Target20.6
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.1690865718193236e-104

    1. Initial program 51.5

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

    3. Applied *-un-lft-identity51.5

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

    4. Applied times-frac51.5

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

    5. Taylor expanded around -inf 11.0

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

    if -4.1690865718193236e-104 < b < 1.3316184968738608e+61

    1. Initial program 12.3

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

    3. Applied *-un-lft-identity12.3

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

    4. Applied times-frac12.3

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

    6. Applied clear-num12.4

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

    8. Applied *-un-lft-identity12.4

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

    9. Applied *-un-lft-identity12.4

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

    10. Applied times-frac12.4

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

    11. Applied add-cube-cbrt12.4

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

    12. Applied times-frac12.4

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

    13. Simplified12.4

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

    14. Simplified12.3

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

    if 1.3316184968738608e+61 < b

    1. Initial program 39.6

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

    3. Applied *-un-lft-identity39.6

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

    4. Applied times-frac39.5

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

    6. Applied clear-num39.6

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

    7. Taylor expanded around 0 4.6

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(-2 \cdot \frac{b}{a}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.16908657181932359 \cdot 10^{-104}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \le 1.3316184968738608 \cdot 10^{61}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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)))