Average Error: 33.6 → 10.5
Time: 18.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.1144981103869975 \cdot 10^{+131}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 4.5810084990875205 \cdot 10^{-68}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\frac{1}{2}}{\sqrt{2}}}}{\frac{a}{\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{\sqrt{2}}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -2.1144981103869975 \cdot 10^{+131}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 4.5810084990875205 \cdot 10^{-68}:\\
\;\;\;\;\frac{\sqrt[3]{\frac{\frac{1}{2}}{\sqrt{2}}}}{\frac{a}{\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{\sqrt{2}}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3424100 = b;
        double r3424101 = -r3424100;
        double r3424102 = r3424100 * r3424100;
        double r3424103 = 4.0;
        double r3424104 = a;
        double r3424105 = c;
        double r3424106 = r3424104 * r3424105;
        double r3424107 = r3424103 * r3424106;
        double r3424108 = r3424102 - r3424107;
        double r3424109 = sqrt(r3424108);
        double r3424110 = r3424101 + r3424109;
        double r3424111 = 2.0;
        double r3424112 = r3424111 * r3424104;
        double r3424113 = r3424110 / r3424112;
        return r3424113;
}
↓
double f(double a, double b, double c) {
        double r3424114 = b;
        double r3424115 = -2.1144981103869975e+131;
        bool r3424116 = r3424114 <= r3424115;
        double r3424117 = c;
        double r3424118 = r3424117 / r3424114;
        double r3424119 = a;
        double r3424120 = r3424114 / r3424119;
        double r3424121 = r3424118 - r3424120;
        double r3424122 = 4.5810084990875205e-68;
        bool r3424123 = r3424114 <= r3424122;
        double r3424124 = 0.5;
        double r3424125 = 2.0;
        double r3424126 = sqrt(r3424125);
        double r3424127 = r3424124 / r3424126;
        double r3424128 = cbrt(r3424127);
        double r3424129 = r3424114 * r3424114;
        double r3424130 = 4.0;
        double r3424131 = r3424130 * r3424119;
        double r3424132 = r3424117 * r3424131;
        double r3424133 = r3424129 - r3424132;
        double r3424134 = sqrt(r3424133);
        double r3424135 = r3424134 - r3424114;
        double r3424136 = r3424135 / r3424126;
        double r3424137 = r3424119 / r3424136;
        double r3424138 = r3424128 / r3424137;
        double r3424139 = -r3424118;
        double r3424140 = r3424123 ? r3424138 : r3424139;
        double r3424141 = r3424116 ? r3424121 : r3424140;
        return r3424141;
}

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
Target21.0
Herbie10.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.1144981103869975e+131

    1. Initial program 53.8

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

    2. Simplified53.8

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

    3. Taylor expanded around -inf 2.6

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

    if -2.1144981103869975e+131 < b < 4.5810084990875205e-68

    1. Initial program 13.3

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

    2. Simplified13.3

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

    4. Applied add-sqr-sqrt14.1

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

    5. Applied *-un-lft-identity14.1

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

    6. Applied times-frac14.0

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

    7. Applied associate-/l*13.9

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

    9. Applied add-cbrt-cube14.4

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

    10. Applied add-cbrt-cube14.4

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

    11. Applied cbrt-undiv13.9

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

    12. Simplified13.5

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

    if 4.5810084990875205e-68 < b

    1. Initial program 51.9

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

    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 9.3

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

    4. Simplified9.3

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.1144981103869975 \cdot 10^{+131}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 4.5810084990875205 \cdot 10^{-68}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\frac{1}{2}}{\sqrt{2}}}}{\frac{a}{\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{\sqrt{2}}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :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)))