Average Error: 34.2 → 11.9
Time: 14.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.547666603636537260513437138645901028344 \cdot 10^{50}:\\ \;\;\;\;\left(\frac{\frac{c}{\sqrt[3]{b}}}{\sqrt[3]{b} \cdot \sqrt[3]{b}} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.547666603636537260513437138645901028344 \cdot 10^{50}:\\
\;\;\;\;\left(\frac{\frac{c}{\sqrt[3]{b}}}{\sqrt[3]{b} \cdot \sqrt[3]{b}} - \frac{b}{a}\right) \cdot 1\\

\mathbf{elif}\;b \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r50730 = b;
        double r50731 = -r50730;
        double r50732 = r50730 * r50730;
        double r50733 = 4.0;
        double r50734 = a;
        double r50735 = r50733 * r50734;
        double r50736 = c;
        double r50737 = r50735 * r50736;
        double r50738 = r50732 - r50737;
        double r50739 = sqrt(r50738);
        double r50740 = r50731 + r50739;
        double r50741 = 2.0;
        double r50742 = r50741 * r50734;
        double r50743 = r50740 / r50742;
        return r50743;
}
↓
double f(double a, double b, double c) {
        double r50744 = b;
        double r50745 = -1.5476666036365373e+50;
        bool r50746 = r50744 <= r50745;
        double r50747 = c;
        double r50748 = cbrt(r50744);
        double r50749 = r50747 / r50748;
        double r50750 = r50748 * r50748;
        double r50751 = r50749 / r50750;
        double r50752 = a;
        double r50753 = r50744 / r50752;
        double r50754 = r50751 - r50753;
        double r50755 = 1.0;
        double r50756 = r50754 * r50755;
        double r50757 = 7.455592343308264e-170;
        bool r50758 = r50744 <= r50757;
        double r50759 = 1.0;
        double r50760 = 2.0;
        double r50761 = r50760 * r50752;
        double r50762 = r50744 * r50744;
        double r50763 = 4.0;
        double r50764 = r50763 * r50752;
        double r50765 = r50764 * r50747;
        double r50766 = r50762 - r50765;
        double r50767 = sqrt(r50766);
        double r50768 = r50767 - r50744;
        double r50769 = r50761 / r50768;
        double r50770 = r50759 / r50769;
        double r50771 = -1.0;
        double r50772 = r50747 / r50744;
        double r50773 = r50771 * r50772;
        double r50774 = r50758 ? r50770 : r50773;
        double r50775 = r50746 ? r50756 : r50774;
        return r50775;
}

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

Original34.2
Target20.8
Herbie11.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.5476666036365373e+50

    1. Initial program 37.8

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

    2. Simplified37.8

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

    3. Taylor expanded around -inf 5.8

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

    4. Simplified5.8

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

    6. Applied add-cube-cbrt7.0

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

    7. Applied add-cube-cbrt7.0

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

    8. Applied add-sqr-sqrt35.2

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

    9. Applied times-frac35.2

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

    10. Applied prod-diff35.2

      \[\leadsto 1 \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{\sqrt{c}}{\sqrt[3]{b} \cdot \sqrt[3]{b}}, \frac{\sqrt{c}}{\sqrt[3]{b}}, -\sqrt[3]{\frac{b}{a}} \cdot \left(\sqrt[3]{\frac{b}{a}} \cdot \sqrt[3]{\frac{b}{a}}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\frac{b}{a}}, \sqrt[3]{\frac{b}{a}} \cdot \sqrt[3]{\frac{b}{a}}, \sqrt[3]{\frac{b}{a}} \cdot \left(\sqrt[3]{\frac{b}{a}} \cdot \sqrt[3]{\frac{b}{a}}\right)\right)\right)}\]

    11. Simplified5.8

      \[\leadsto 1 \cdot \left(\color{blue}{\left(\frac{\frac{c}{\sqrt[3]{b}}}{\sqrt[3]{b} \cdot \sqrt[3]{b}} - \frac{b}{a}\right)} + \mathsf{fma}\left(-\sqrt[3]{\frac{b}{a}}, \sqrt[3]{\frac{b}{a}} \cdot \sqrt[3]{\frac{b}{a}}, \sqrt[3]{\frac{b}{a}} \cdot \left(\sqrt[3]{\frac{b}{a}} \cdot \sqrt[3]{\frac{b}{a}}\right)\right)\right)\]

    12. Simplified5.8

      \[\leadsto 1 \cdot \left(\left(\frac{\frac{c}{\sqrt[3]{b}}}{\sqrt[3]{b} \cdot \sqrt[3]{b}} - \frac{b}{a}\right) + \color{blue}{0}\right)\]

    if -1.5476666036365373e+50 < b < 7.455592343308264e-170

    1. Initial program 12.4

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

    2. Simplified12.4

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

    4. Applied clear-num12.5

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

    if 7.455592343308264e-170 < b

    1. Initial program 48.9

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

    2. Simplified48.9

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

    3. Taylor expanded around inf 14.1

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification11.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.547666603636537260513437138645901028344 \cdot 10^{50}:\\ \;\;\;\;\left(\frac{\frac{c}{\sqrt[3]{b}}}{\sqrt[3]{b} \cdot \sqrt[3]{b}} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- 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)))