Average Error: 32.9 → 10.3
Time: 15.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.088000531423294 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 9.354082991670835 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{a}}{2} - \frac{\frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -9.088000531423294 \cdot 10^{+152}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 9.354082991670835 \cdot 10^{-125}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{a}}{2} - \frac{\frac{b}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2984582 = b;
        double r2984583 = -r2984582;
        double r2984584 = r2984582 * r2984582;
        double r2984585 = 4.0;
        double r2984586 = a;
        double r2984587 = r2984585 * r2984586;
        double r2984588 = c;
        double r2984589 = r2984587 * r2984588;
        double r2984590 = r2984584 - r2984589;
        double r2984591 = sqrt(r2984590);
        double r2984592 = r2984583 + r2984591;
        double r2984593 = 2.0;
        double r2984594 = r2984593 * r2984586;
        double r2984595 = r2984592 / r2984594;
        return r2984595;
}
↓
double f(double a, double b, double c) {
        double r2984596 = b;
        double r2984597 = -9.088000531423294e+152;
        bool r2984598 = r2984596 <= r2984597;
        double r2984599 = c;
        double r2984600 = r2984599 / r2984596;
        double r2984601 = a;
        double r2984602 = r2984596 / r2984601;
        double r2984603 = r2984600 - r2984602;
        double r2984604 = 9.354082991670835e-125;
        bool r2984605 = r2984596 <= r2984604;
        double r2984606 = r2984596 * r2984596;
        double r2984607 = r2984599 * r2984601;
        double r2984608 = 4.0;
        double r2984609 = r2984607 * r2984608;
        double r2984610 = r2984606 - r2984609;
        double r2984611 = sqrt(r2984610);
        double r2984612 = r2984611 / r2984601;
        double r2984613 = 2.0;
        double r2984614 = r2984612 / r2984613;
        double r2984615 = r2984602 / r2984613;
        double r2984616 = r2984614 - r2984615;
        double r2984617 = -r2984600;
        double r2984618 = r2984605 ? r2984616 : r2984617;
        double r2984619 = r2984598 ? r2984603 : r2984618;
        return r2984619;
}

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

Original32.9
Target20.3
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -9.088000531423294e+152

    1. Initial program 60.4

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

    2. Simplified60.4

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

    4. Applied *-un-lft-identity60.4

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

    5. Applied div-inv60.4

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

    6. Applied times-frac60.4

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

    7. Simplified60.4

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

    8. Taylor expanded around -inf 1.5

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

    if -9.088000531423294e+152 < b < 9.354082991670835e-125

    1. Initial program 10.9

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

    2. Simplified10.9

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

    4. Applied div-sub10.9

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

    5. Applied div-sub10.9

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

    if 9.354082991670835e-125 < b

    1. Initial program 49.8

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

    2. Simplified49.8

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

    4. Applied *-un-lft-identity49.8

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

    5. Applied div-inv49.8

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

    6. Applied times-frac49.8

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

    7. Simplified49.8

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

    8. Taylor expanded around inf 11.9

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

    9. Simplified11.9

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.088000531423294 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 9.354082991670835 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{a}}{2} - \frac{\frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 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)))