Average Error: 34.7 → 10.1
Time: 6.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.244774291407710824026233990502584030865 \cdot 10^{109}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 6.485606601696406255086078549712143397431 \cdot 10^{-71}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -1.244774291407710824026233990502584030865 \cdot 10^{109}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 6.485606601696406255086078549712143397431 \cdot 10^{-71}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r71003 = b;
        double r71004 = -r71003;
        double r71005 = r71003 * r71003;
        double r71006 = 4.0;
        double r71007 = a;
        double r71008 = r71006 * r71007;
        double r71009 = c;
        double r71010 = r71008 * r71009;
        double r71011 = r71005 - r71010;
        double r71012 = sqrt(r71011);
        double r71013 = r71004 + r71012;
        double r71014 = 2.0;
        double r71015 = r71014 * r71007;
        double r71016 = r71013 / r71015;
        return r71016;
}
↓
double f(double a, double b, double c) {
        double r71017 = b;
        double r71018 = -1.2447742914077108e+109;
        bool r71019 = r71017 <= r71018;
        double r71020 = 1.0;
        double r71021 = c;
        double r71022 = r71021 / r71017;
        double r71023 = a;
        double r71024 = r71017 / r71023;
        double r71025 = r71022 - r71024;
        double r71026 = r71020 * r71025;
        double r71027 = 6.485606601696406e-71;
        bool r71028 = r71017 <= r71027;
        double r71029 = -r71017;
        double r71030 = r71017 * r71017;
        double r71031 = 4.0;
        double r71032 = r71031 * r71023;
        double r71033 = r71032 * r71021;
        double r71034 = r71030 - r71033;
        double r71035 = sqrt(r71034);
        double r71036 = r71029 + r71035;
        double r71037 = 1.0;
        double r71038 = 2.0;
        double r71039 = r71038 * r71023;
        double r71040 = r71037 / r71039;
        double r71041 = r71036 * r71040;
        double r71042 = -1.0;
        double r71043 = r71042 * r71022;
        double r71044 = r71028 ? r71041 : r71043;
        double r71045 = r71019 ? r71026 : r71044;
        return r71045;
}

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

Original34.7
Target21.5
Herbie10.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.2447742914077108e+109

    1. Initial program 49.3

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

    2. Taylor expanded around -inf 4.0

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

    3. Simplified4.0

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

    if -1.2447742914077108e+109 < b < 6.485606601696406e-71

    1. Initial program 13.5

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

    3. Applied div-inv13.6

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

    if 6.485606601696406e-71 < b

    1. Initial program 53.3

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

    2. Taylor expanded around inf 8.4

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.244774291407710824026233990502584030865 \cdot 10^{109}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 6.485606601696406255086078549712143397431 \cdot 10^{-71}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

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