Average Error: 33.3 → 6.6
Time: 40.8s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.411206454162785 \cdot 10^{+120}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 8.142093116881289 \cdot 10^{-248}:\\ \;\;\;\;\frac{4}{\sqrt{c \cdot \left(-4 \cdot a\right) + b \cdot b} - b} \cdot \left(\frac{1}{2} \cdot c\right)\\ \mathbf{elif}\;b \le 5.419916601733116 \cdot 10^{+77}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -3.411206454162785 \cdot 10^{+120}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 8.142093116881289 \cdot 10^{-248}:\\
\;\;\;\;\frac{4}{\sqrt{c \cdot \left(-4 \cdot a\right) + b \cdot b} - b} \cdot \left(\frac{1}{2} \cdot c\right)\\

\mathbf{elif}\;b \le 5.419916601733116 \cdot 10^{+77}:\\
\;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right) \cdot \frac{1}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r9171760 = b;
        double r9171761 = -r9171760;
        double r9171762 = r9171760 * r9171760;
        double r9171763 = 4.0;
        double r9171764 = a;
        double r9171765 = c;
        double r9171766 = r9171764 * r9171765;
        double r9171767 = r9171763 * r9171766;
        double r9171768 = r9171762 - r9171767;
        double r9171769 = sqrt(r9171768);
        double r9171770 = r9171761 - r9171769;
        double r9171771 = 2.0;
        double r9171772 = r9171771 * r9171764;
        double r9171773 = r9171770 / r9171772;
        return r9171773;
}
↓
double f(double a, double b, double c) {
        double r9171774 = b;
        double r9171775 = -3.411206454162785e+120;
        bool r9171776 = r9171774 <= r9171775;
        double r9171777 = c;
        double r9171778 = r9171777 / r9171774;
        double r9171779 = -r9171778;
        double r9171780 = 8.142093116881289e-248;
        bool r9171781 = r9171774 <= r9171780;
        double r9171782 = 4.0;
        double r9171783 = -4.0;
        double r9171784 = a;
        double r9171785 = r9171783 * r9171784;
        double r9171786 = r9171777 * r9171785;
        double r9171787 = r9171774 * r9171774;
        double r9171788 = r9171786 + r9171787;
        double r9171789 = sqrt(r9171788);
        double r9171790 = r9171789 - r9171774;
        double r9171791 = r9171782 / r9171790;
        double r9171792 = 0.5;
        double r9171793 = r9171792 * r9171777;
        double r9171794 = r9171791 * r9171793;
        double r9171795 = 5.419916601733116e+77;
        bool r9171796 = r9171774 <= r9171795;
        double r9171797 = -r9171774;
        double r9171798 = r9171784 * r9171777;
        double r9171799 = r9171798 * r9171782;
        double r9171800 = r9171787 - r9171799;
        double r9171801 = sqrt(r9171800);
        double r9171802 = r9171797 - r9171801;
        double r9171803 = 1.0;
        double r9171804 = 2.0;
        double r9171805 = r9171804 * r9171784;
        double r9171806 = r9171803 / r9171805;
        double r9171807 = r9171802 * r9171806;
        double r9171808 = r9171774 / r9171784;
        double r9171809 = r9171778 - r9171808;
        double r9171810 = r9171796 ? r9171807 : r9171809;
        double r9171811 = r9171781 ? r9171794 : r9171810;
        double r9171812 = r9171776 ? r9171779 : r9171811;
        return r9171812;
}

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

Original33.3
Target20.6
Herbie6.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.411206454162785e+120

    1. Initial program 59.4

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

    3. Applied div-inv59.4

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

    4. Taylor expanded around -inf 1.9

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

    5. Simplified1.9

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

    if -3.411206454162785e+120 < b < 8.142093116881289e-248

    1. Initial program 30.9

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

    3. Applied flip--31.1

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

    4. Applied associate-/l/36.0

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

    5. Simplified20.1

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

    7. Applied times-frac14.3

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

    8. Simplified8.9

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

    9. Simplified9.0

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

    if 8.142093116881289e-248 < b < 5.419916601733116e+77

    1. Initial program 8.5

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

    3. Applied div-inv8.7

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

    if 5.419916601733116e+77 < b

    1. Initial program 40.5

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

    3. Applied div-inv40.6

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

    4. Taylor expanded around inf 4.7

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.411206454162785 \cdot 10^{+120}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 8.142093116881289 \cdot 10^{-248}:\\ \;\;\;\;\frac{4}{\sqrt{c \cdot \left(-4 \cdot a\right) + b \cdot b} - b} \cdot \left(\frac{1}{2} \cdot c\right)\\ \mathbf{elif}\;b \le 5.419916601733116 \cdot 10^{+77}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

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