Average Error: 34.8 → 10.3
Time: 5.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.3732770006881601 \cdot 10^{-89}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 9.18109192604693914 \cdot 10^{128}:\\ \;\;\;\;1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -2.3732770006881601 \cdot 10^{-89}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 9.18109192604693914 \cdot 10^{128}:\\
\;\;\;\;1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r115113 = b;
        double r115114 = -r115113;
        double r115115 = r115113 * r115113;
        double r115116 = 4.0;
        double r115117 = a;
        double r115118 = c;
        double r115119 = r115117 * r115118;
        double r115120 = r115116 * r115119;
        double r115121 = r115115 - r115120;
        double r115122 = sqrt(r115121);
        double r115123 = r115114 - r115122;
        double r115124 = 2.0;
        double r115125 = r115124 * r115117;
        double r115126 = r115123 / r115125;
        return r115126;
}
↓
double f(double a, double b, double c) {
        double r115127 = b;
        double r115128 = -2.37327700068816e-89;
        bool r115129 = r115127 <= r115128;
        double r115130 = -1.0;
        double r115131 = c;
        double r115132 = r115131 / r115127;
        double r115133 = r115130 * r115132;
        double r115134 = 9.181091926046939e+128;
        bool r115135 = r115127 <= r115134;
        double r115136 = 1.0;
        double r115137 = -r115127;
        double r115138 = r115127 * r115127;
        double r115139 = 4.0;
        double r115140 = a;
        double r115141 = r115140 * r115131;
        double r115142 = r115139 * r115141;
        double r115143 = r115138 - r115142;
        double r115144 = sqrt(r115143);
        double r115145 = r115137 - r115144;
        double r115146 = 2.0;
        double r115147 = r115146 * r115140;
        double r115148 = r115145 / r115147;
        double r115149 = r115136 * r115148;
        double r115150 = 1.0;
        double r115151 = r115127 / r115140;
        double r115152 = r115132 - r115151;
        double r115153 = r115150 * r115152;
        double r115154 = r115135 ? r115149 : r115153;
        double r115155 = r115129 ? r115133 : r115154;
        return r115155;
}

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

Original34.8
Target21.2
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 < -2.37327700068816e-89

    1. Initial program 52.5

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

    2. Taylor expanded around -inf 10.0

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

    if -2.37327700068816e-89 < b < 9.181091926046939e+128

    1. Initial program 12.7

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

    3. Applied div-inv12.9

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

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

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

    6. Applied associate-*l*12.9

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

    7. Simplified12.7

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

    if 9.181091926046939e+128 < b

    1. Initial program 55.7

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

    2. Taylor expanded around inf 3.0

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

    3. Simplified3.0

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.3732770006881601 \cdot 10^{-89}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 9.18109192604693914 \cdot 10^{128}:\\ \;\;\;\;1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :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)))