Average Error: 34.2 → 10.0
Time: 4.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.2389466313579672 \cdot 10^{127}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.6670468245058271 \cdot 10^{-85}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -5.2389466313579672 \cdot 10^{127}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.6670468245058271 \cdot 10^{-85}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r85084 = b;
        double r85085 = -r85084;
        double r85086 = r85084 * r85084;
        double r85087 = 4.0;
        double r85088 = a;
        double r85089 = r85087 * r85088;
        double r85090 = c;
        double r85091 = r85089 * r85090;
        double r85092 = r85086 - r85091;
        double r85093 = sqrt(r85092);
        double r85094 = r85085 + r85093;
        double r85095 = 2.0;
        double r85096 = r85095 * r85088;
        double r85097 = r85094 / r85096;
        return r85097;
}
↓
double f(double a, double b, double c) {
        double r85098 = b;
        double r85099 = -5.238946631357967e+127;
        bool r85100 = r85098 <= r85099;
        double r85101 = 1.0;
        double r85102 = c;
        double r85103 = r85102 / r85098;
        double r85104 = a;
        double r85105 = r85098 / r85104;
        double r85106 = r85103 - r85105;
        double r85107 = r85101 * r85106;
        double r85108 = 1.667046824505827e-85;
        bool r85109 = r85098 <= r85108;
        double r85110 = 1.0;
        double r85111 = 2.0;
        double r85112 = r85111 * r85104;
        double r85113 = -r85098;
        double r85114 = r85098 * r85098;
        double r85115 = 4.0;
        double r85116 = r85115 * r85104;
        double r85117 = r85116 * r85102;
        double r85118 = r85114 - r85117;
        double r85119 = sqrt(r85118);
        double r85120 = r85113 + r85119;
        double r85121 = r85112 / r85120;
        double r85122 = r85110 / r85121;
        double r85123 = -1.0;
        double r85124 = r85123 * r85103;
        double r85125 = r85109 ? r85122 : r85124;
        double r85126 = r85100 ? r85107 : r85125;
        return r85126;
}

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
Target21.6
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -5.238946631357967e+127

    1. Initial program 54.2

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

    2. Taylor expanded around -inf 3.3

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

    3. Simplified3.3

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

    if -5.238946631357967e+127 < b < 1.667046824505827e-85

    1. Initial program 12.2

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

    3. Applied clear-num12.3

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

    if 1.667046824505827e-85 < b

    1. Initial program 52.8

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

    2. Taylor expanded around inf 9.7

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.2389466313579672 \cdot 10^{127}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.6670468245058271 \cdot 10^{-85}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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)))