Average Error: 20.1 → 6.9
Time: 18.8s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.25165686884117225057308430661709452775 \cdot 10^{152}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\left(\frac{2}{\left(b \cdot b - b \cdot b\right) + \left(a \cdot 4\right) \cdot c} \cdot c\right) \cdot \left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \le 1.113819743837194612716812540397097008684 \cdot 10^{86}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}} + \left(-b\right)}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - \frac{a}{b} \cdot \left(2 \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)}{a \cdot 2}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \le -3.25165686884117225057308430661709452775 \cdot 10^{152}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\left(\frac{2}{\left(b \cdot b - b \cdot b\right) + \left(a \cdot 4\right) \cdot c} \cdot c\right) \cdot \left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)\right)\\

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

\end{array}\\

\mathbf{elif}\;b \le 1.113819743837194612716812540397097008684 \cdot 10^{86}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}} + \left(-b\right)}{a \cdot 2}\\

\end{array}\\

\mathbf{elif}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - \frac{a}{b} \cdot \left(2 \cdot c\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)}{a \cdot 2}\\

\end{array}
double f(double a, double b, double c) {
        double r1266550 = b;
        double r1266551 = 0.0;
        bool r1266552 = r1266550 >= r1266551;
        double r1266553 = 2.0;
        double r1266554 = c;
        double r1266555 = r1266553 * r1266554;
        double r1266556 = -r1266550;
        double r1266557 = r1266550 * r1266550;
        double r1266558 = 4.0;
        double r1266559 = a;
        double r1266560 = r1266558 * r1266559;
        double r1266561 = r1266560 * r1266554;
        double r1266562 = r1266557 - r1266561;
        double r1266563 = sqrt(r1266562);
        double r1266564 = r1266556 - r1266563;
        double r1266565 = r1266555 / r1266564;
        double r1266566 = r1266556 + r1266563;
        double r1266567 = r1266553 * r1266559;
        double r1266568 = r1266566 / r1266567;
        double r1266569 = r1266552 ? r1266565 : r1266568;
        return r1266569;
}
↓
double f(double a, double b, double c) {
        double r1266570 = b;
        double r1266571 = -3.2516568688411723e+152;
        bool r1266572 = r1266570 <= r1266571;
        double r1266573 = 0.0;
        bool r1266574 = r1266570 >= r1266573;
        double r1266575 = 2.0;
        double r1266576 = r1266570 * r1266570;
        double r1266577 = r1266576 - r1266576;
        double r1266578 = a;
        double r1266579 = 4.0;
        double r1266580 = r1266578 * r1266579;
        double r1266581 = c;
        double r1266582 = r1266580 * r1266581;
        double r1266583 = r1266577 + r1266582;
        double r1266584 = r1266575 / r1266583;
        double r1266585 = r1266584 * r1266581;
        double r1266586 = r1266576 - r1266582;
        double r1266587 = sqrt(r1266586);
        double r1266588 = -r1266570;
        double r1266589 = r1266587 + r1266588;
        double r1266590 = r1266585 * r1266589;
        double r1266591 = 1.0;
        double r1266592 = r1266581 / r1266570;
        double r1266593 = r1266570 / r1266578;
        double r1266594 = r1266592 - r1266593;
        double r1266595 = r1266591 * r1266594;
        double r1266596 = r1266574 ? r1266590 : r1266595;
        double r1266597 = 1.1138197438371946e+86;
        bool r1266598 = r1266570 <= r1266597;
        double r1266599 = r1266575 * r1266581;
        double r1266600 = r1266588 - r1266587;
        double r1266601 = r1266599 / r1266600;
        double r1266602 = sqrt(r1266587);
        double r1266603 = r1266602 * r1266602;
        double r1266604 = r1266603 + r1266588;
        double r1266605 = r1266578 * r1266575;
        double r1266606 = r1266604 / r1266605;
        double r1266607 = r1266574 ? r1266601 : r1266606;
        double r1266608 = r1266578 / r1266570;
        double r1266609 = r1266608 * r1266599;
        double r1266610 = r1266570 - r1266609;
        double r1266611 = r1266588 - r1266610;
        double r1266612 = r1266599 / r1266611;
        double r1266613 = r1266589 / r1266605;
        double r1266614 = r1266574 ? r1266612 : r1266613;
        double r1266615 = r1266598 ? r1266607 : r1266614;
        double r1266616 = r1266572 ? r1266596 : r1266615;
        return r1266616;
}

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

Derivation

  1. Split input into 3 regimes

  2. if b < -3.2516568688411723e+152

    1. Initial program 63.5

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

    2. Taylor expanded around -inf 11.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(2 \cdot \frac{a \cdot c}{b} - b\right)}{2 \cdot a}\\ \end{array}\]

    3. Simplified2.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(\left(2 \cdot c\right) \cdot \frac{a}{b} - b\right)}{2 \cdot a}\\ \end{array}\]

    4. Taylor expanded around 0 2.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{c}{b} - 1 \cdot \frac{b}{a}\\ \end{array}\]

    5. Simplified2.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]
    6. Using strategy rm

    7. Applied flip--2.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]

    8. Applied associate-/r/2.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]

    9. Simplified2.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\color{blue}{\left(\frac{2}{\left(b \cdot b - b \cdot b\right) + \left(4 \cdot a\right) \cdot c} \cdot c\right)} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]

    if -3.2516568688411723e+152 < b < 1.1138197438371946e+86

    1. Initial program 9.2

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt9.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]

    4. Applied sqrt-prod9.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]

    if 1.1138197438371946e+86 < b

    1. Initial program 28.2

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

    2. Taylor expanded around inf 6.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\left(b - 2 \cdot \frac{a \cdot c}{b}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    3. Simplified2.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\left(b - \left(2 \cdot c\right) \cdot \frac{a}{b}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.25165686884117225057308430661709452775 \cdot 10^{152}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\left(\frac{2}{\left(b \cdot b - b \cdot b\right) + \left(a \cdot 4\right) \cdot c} \cdot c\right) \cdot \left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\\ \mathbf{elif}\;b \le 1.113819743837194612716812540397097008684 \cdot 10^{86}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}} + \left(-b\right)}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b - \frac{a}{b} \cdot \left(2 \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)}{a \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))