Average Error: 33.6 → 9.7
Time: 24.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.694684309811035 \cdot 10^{+121}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 4.6659701943749105 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]
\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 -4.694684309811035 \cdot 10^{+121}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 4.6659701943749105 \cdot 10^{-84}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r5287753 = b;
        double r5287754 = -r5287753;
        double r5287755 = r5287753 * r5287753;
        double r5287756 = 4.0;
        double r5287757 = a;
        double r5287758 = r5287756 * r5287757;
        double r5287759 = c;
        double r5287760 = r5287758 * r5287759;
        double r5287761 = r5287755 - r5287760;
        double r5287762 = sqrt(r5287761);
        double r5287763 = r5287754 + r5287762;
        double r5287764 = 2.0;
        double r5287765 = r5287764 * r5287757;
        double r5287766 = r5287763 / r5287765;
        return r5287766;
}
↓
double f(double a, double b, double c) {
        double r5287767 = b;
        double r5287768 = -4.694684309811035e+121;
        bool r5287769 = r5287767 <= r5287768;
        double r5287770 = c;
        double r5287771 = r5287770 / r5287767;
        double r5287772 = a;
        double r5287773 = r5287767 / r5287772;
        double r5287774 = r5287771 - r5287773;
        double r5287775 = 2.0;
        double r5287776 = r5287774 * r5287775;
        double r5287777 = r5287776 / r5287775;
        double r5287778 = 4.6659701943749105e-84;
        bool r5287779 = r5287767 <= r5287778;
        double r5287780 = 1.0;
        double r5287781 = r5287780 / r5287772;
        double r5287782 = r5287767 * r5287767;
        double r5287783 = 4.0;
        double r5287784 = r5287770 * r5287772;
        double r5287785 = r5287783 * r5287784;
        double r5287786 = r5287782 - r5287785;
        double r5287787 = sqrt(r5287786);
        double r5287788 = r5287787 - r5287767;
        double r5287789 = r5287781 * r5287788;
        double r5287790 = r5287789 / r5287775;
        double r5287791 = -2.0;
        double r5287792 = r5287771 * r5287791;
        double r5287793 = r5287792 / r5287775;
        double r5287794 = r5287779 ? r5287790 : r5287793;
        double r5287795 = r5287769 ? r5287777 : r5287794;
        return r5287795;
}

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

Original33.6
Target20.4
Herbie9.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.694684309811035e+121

    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 div-inv49.9

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

    5. Taylor expanded around -inf 2.6

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]

    6. Simplified2.6

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

    if -4.694684309811035e+121 < b < 4.6659701943749105e-84

    1. Initial program 12.2

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

    2. Simplified12.2

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

    4. Applied div-inv12.3

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

    if 4.6659701943749105e-84 < b

    1. Initial program 52.2

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

    2. Simplified52.2

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

    3. Taylor expanded around inf 9.3

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.694684309811035 \cdot 10^{+121}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 4.6659701943749105 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019144 
(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)))