Average Error: 34.1 → 9.5
Time: 17.3s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.356959927988237168348139414849710212524 \cdot 10^{-56}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.087668654677018032633364446323411964642 \cdot 10^{130}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -4.356959927988237168348139414849710212524 \cdot 10^{-56}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 3.087668654677018032633364446323411964642 \cdot 10^{130}:\\
\;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{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 r5648631 = b;
        double r5648632 = -r5648631;
        double r5648633 = r5648631 * r5648631;
        double r5648634 = 4.0;
        double r5648635 = a;
        double r5648636 = c;
        double r5648637 = r5648635 * r5648636;
        double r5648638 = r5648634 * r5648637;
        double r5648639 = r5648633 - r5648638;
        double r5648640 = sqrt(r5648639);
        double r5648641 = r5648632 - r5648640;
        double r5648642 = 2.0;
        double r5648643 = r5648642 * r5648635;
        double r5648644 = r5648641 / r5648643;
        return r5648644;
}
↓
double f(double a, double b, double c) {
        double r5648645 = b;
        double r5648646 = -4.356959927988237e-56;
        bool r5648647 = r5648645 <= r5648646;
        double r5648648 = -1.0;
        double r5648649 = c;
        double r5648650 = r5648649 / r5648645;
        double r5648651 = r5648648 * r5648650;
        double r5648652 = 3.087668654677018e+130;
        bool r5648653 = r5648645 <= r5648652;
        double r5648654 = -r5648645;
        double r5648655 = r5648645 * r5648645;
        double r5648656 = 4.0;
        double r5648657 = a;
        double r5648658 = r5648657 * r5648649;
        double r5648659 = r5648656 * r5648658;
        double r5648660 = r5648655 - r5648659;
        double r5648661 = sqrt(r5648660);
        double r5648662 = r5648654 - r5648661;
        double r5648663 = 1.0;
        double r5648664 = 2.0;
        double r5648665 = r5648664 * r5648657;
        double r5648666 = r5648663 / r5648665;
        double r5648667 = r5648662 * r5648666;
        double r5648668 = 1.0;
        double r5648669 = r5648645 / r5648657;
        double r5648670 = r5648650 - r5648669;
        double r5648671 = r5648668 * r5648670;
        double r5648672 = r5648653 ? r5648667 : r5648671;
        double r5648673 = r5648647 ? r5648651 : r5648672;
        return r5648673;
}

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

Original34.1
Target21.1
Herbie9.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.356959927988237e-56

    1. Initial program 54.0

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

    2. Taylor expanded around -inf 7.7

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

    if -4.356959927988237e-56 < b < 3.087668654677018e+130

    1. Initial program 12.6

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

    3. Applied div-inv12.8

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

    if 3.087668654677018e+130 < b

    1. Initial program 56.2

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

    2. Taylor expanded around inf 2.4

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

    3. Simplified2.4

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

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.356959927988237168348139414849710212524 \cdot 10^{-56}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.087668654677018032633364446323411964642 \cdot 10^{130}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))