Average Error: 34.5 → 10.2
Time: 8.1s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.3645547041066157 \cdot 10^{-80}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 4.1199128263687574 \cdot 10^{46}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -8.3645547041066157 \cdot 10^{-80}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 4.1199128263687574 \cdot 10^{46}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r109932 = b;
        double r109933 = -r109932;
        double r109934 = r109932 * r109932;
        double r109935 = 4.0;
        double r109936 = a;
        double r109937 = c;
        double r109938 = r109936 * r109937;
        double r109939 = r109935 * r109938;
        double r109940 = r109934 - r109939;
        double r109941 = sqrt(r109940);
        double r109942 = r109933 - r109941;
        double r109943 = 2.0;
        double r109944 = r109943 * r109936;
        double r109945 = r109942 / r109944;
        return r109945;
}
↓
double f(double a, double b, double c) {
        double r109946 = b;
        double r109947 = -8.364554704106616e-80;
        bool r109948 = r109946 <= r109947;
        double r109949 = -1.0;
        double r109950 = c;
        double r109951 = r109950 / r109946;
        double r109952 = r109949 * r109951;
        double r109953 = 4.1199128263687574e+46;
        bool r109954 = r109946 <= r109953;
        double r109955 = 1.0;
        double r109956 = 2.0;
        double r109957 = a;
        double r109958 = r109956 * r109957;
        double r109959 = -r109946;
        double r109960 = r109946 * r109946;
        double r109961 = 4.0;
        double r109962 = r109957 * r109950;
        double r109963 = r109961 * r109962;
        double r109964 = r109960 - r109963;
        double r109965 = sqrt(r109964);
        double r109966 = r109959 - r109965;
        double r109967 = r109958 / r109966;
        double r109968 = r109955 / r109967;
        double r109969 = 1.0;
        double r109970 = r109946 / r109957;
        double r109971 = r109951 - r109970;
        double r109972 = r109969 * r109971;
        double r109973 = r109954 ? r109968 : r109972;
        double r109974 = r109948 ? r109952 : r109973;
        return r109974;
}

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

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

Derivation

  1. Split input into 3 regimes

  2. if b < -8.364554704106616e-80

    1. Initial program 53.8

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

    2. Taylor expanded around -inf 9.1

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

    if -8.364554704106616e-80 < b < 4.1199128263687574e+46

    1. Initial program 13.8

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

    3. Applied clear-num13.9

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

    if 4.1199128263687574e+46 < b

    1. Initial program 36.8

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

    2. Taylor expanded around inf 5.2

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

    3. Simplified5.2

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.3645547041066157 \cdot 10^{-80}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 4.1199128263687574 \cdot 10^{46}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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)))