Average Error: 33.4 → 10.1
Time: 20.9s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.840085388791461 \cdot 10^{-68}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.5949594684703287 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\left(-b\right) - \sqrt{\sqrt[3]{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \left(\sqrt[3]{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \sqrt[3]{b \cdot b - \left(c \cdot a\right) \cdot 4}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -2.840085388791461 \cdot 10^{-68}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 1.5949594684703287 \cdot 10^{+126}:\\
\;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\left(-b\right) - \sqrt{\sqrt[3]{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \left(\sqrt[3]{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \sqrt[3]{b \cdot b - \left(c \cdot a\right) \cdot 4}\right)}\right)\\

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

\end{array}
double f(double a, double b, double c) {
        double r2868105 = b;
        double r2868106 = -r2868105;
        double r2868107 = r2868105 * r2868105;
        double r2868108 = 4.0;
        double r2868109 = a;
        double r2868110 = c;
        double r2868111 = r2868109 * r2868110;
        double r2868112 = r2868108 * r2868111;
        double r2868113 = r2868107 - r2868112;
        double r2868114 = sqrt(r2868113);
        double r2868115 = r2868106 - r2868114;
        double r2868116 = 2.0;
        double r2868117 = r2868116 * r2868109;
        double r2868118 = r2868115 / r2868117;
        return r2868118;
}
↓
double f(double a, double b, double c) {
        double r2868119 = b;
        double r2868120 = -2.840085388791461e-68;
        bool r2868121 = r2868119 <= r2868120;
        double r2868122 = c;
        double r2868123 = r2868122 / r2868119;
        double r2868124 = -r2868123;
        double r2868125 = 1.5949594684703287e+126;
        bool r2868126 = r2868119 <= r2868125;
        double r2868127 = 0.5;
        double r2868128 = a;
        double r2868129 = r2868127 / r2868128;
        double r2868130 = -r2868119;
        double r2868131 = r2868119 * r2868119;
        double r2868132 = r2868122 * r2868128;
        double r2868133 = 4.0;
        double r2868134 = r2868132 * r2868133;
        double r2868135 = r2868131 - r2868134;
        double r2868136 = cbrt(r2868135);
        double r2868137 = r2868136 * r2868136;
        double r2868138 = r2868136 * r2868137;
        double r2868139 = sqrt(r2868138);
        double r2868140 = r2868130 - r2868139;
        double r2868141 = r2868129 * r2868140;
        double r2868142 = r2868119 / r2868128;
        double r2868143 = r2868123 - r2868142;
        double r2868144 = r2868126 ? r2868141 : r2868143;
        double r2868145 = r2868121 ? r2868124 : r2868144;
        return r2868145;
}

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

Original33.4
Target20.2
Herbie10.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.840085388791461e-68

    1. Initial program 52.7

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

    3. Applied div-inv52.7

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

    4. Simplified52.7

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

    5. Taylor expanded around -inf 9.0

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

    6. Simplified9.0

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

    if -2.840085388791461e-68 < b < 1.5949594684703287e+126

    1. Initial program 12.5

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

    3. Applied div-inv12.7

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

    4. Simplified12.7

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

    6. Applied add-cube-cbrt13.0

      \[\leadsto \left(\left(-b\right) - \sqrt{\color{blue}{\left(\sqrt[3]{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \sqrt[3]{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\right) \cdot \frac{\frac{1}{2}}{a}\]

    if 1.5949594684703287e+126 < b

    1. Initial program 50.8

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

    2. Taylor expanded around inf 3.1

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.840085388791461 \cdot 10^{-68}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.5949594684703287 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\left(-b\right) - \sqrt{\sqrt[3]{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \left(\sqrt[3]{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \sqrt[3]{b \cdot b - \left(c \cdot a\right) \cdot 4}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :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)))