Average Error: 33.6 → 11.2
Time: 30.3s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.2890050783826923 \cdot 10^{-183}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 2.88466596167667 \cdot 10^{+141}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(2 \cdot \left(\frac{a}{b} \cdot c - b\right)\right)\\ \end{array}\]
\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 -1.2890050783826923 \cdot 10^{-183}:\\
\;\;\;\;\frac{-c}{b}\\

\mathbf{elif}\;b \le 2.88466596167667 \cdot 10^{+141}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(2 \cdot \left(\frac{a}{b} \cdot c - b\right)\right)\\

\end{array}
double f(double a, double b, double c) {
        double r2046103 = b;
        double r2046104 = -r2046103;
        double r2046105 = r2046103 * r2046103;
        double r2046106 = 4.0;
        double r2046107 = a;
        double r2046108 = c;
        double r2046109 = r2046107 * r2046108;
        double r2046110 = r2046106 * r2046109;
        double r2046111 = r2046105 - r2046110;
        double r2046112 = sqrt(r2046111);
        double r2046113 = r2046104 - r2046112;
        double r2046114 = 2.0;
        double r2046115 = r2046114 * r2046107;
        double r2046116 = r2046113 / r2046115;
        return r2046116;
}
↓
double f(double a, double b, double c) {
        double r2046117 = b;
        double r2046118 = -1.2890050783826923e-183;
        bool r2046119 = r2046117 <= r2046118;
        double r2046120 = c;
        double r2046121 = -r2046120;
        double r2046122 = r2046121 / r2046117;
        double r2046123 = 2.88466596167667e+141;
        bool r2046124 = r2046117 <= r2046123;
        double r2046125 = -r2046117;
        double r2046126 = r2046117 * r2046117;
        double r2046127 = a;
        double r2046128 = r2046120 * r2046127;
        double r2046129 = 4.0;
        double r2046130 = r2046128 * r2046129;
        double r2046131 = r2046126 - r2046130;
        double r2046132 = sqrt(r2046131);
        double r2046133 = r2046125 - r2046132;
        double r2046134 = 2.0;
        double r2046135 = r2046127 * r2046134;
        double r2046136 = r2046133 / r2046135;
        double r2046137 = 0.5;
        double r2046138 = r2046137 / r2046127;
        double r2046139 = r2046127 / r2046117;
        double r2046140 = r2046139 * r2046120;
        double r2046141 = r2046140 - r2046117;
        double r2046142 = r2046134 * r2046141;
        double r2046143 = r2046138 * r2046142;
        double r2046144 = r2046124 ? r2046136 : r2046143;
        double r2046145 = r2046119 ? r2046122 : r2046144;
        return r2046145;
}

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
Target20.5
Herbie11.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.2890050783826923e-183

    1. Initial program 48.2

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

    2. Taylor expanded around -inf 14.3

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

    3. Simplified14.3

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

    if -1.2890050783826923e-183 < b < 2.88466596167667e+141

    1. Initial program 10.0

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

    if 2.88466596167667e+141 < b

    1. Initial program 56.5

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

    3. Applied div-inv56.6

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

    4. Simplified56.5

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

    6. Applied add-cube-cbrt56.5

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

    7. Applied sqrt-prod56.5

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

    8. Taylor expanded around inf 10.1

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

    9. Simplified2.6

      \[\leadsto \color{blue}{\left(2 \cdot \left(c \cdot \frac{a}{b} - b\right)\right)} \cdot \frac{\frac{1}{2}}{a}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.2890050783826923 \cdot 10^{-183}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 2.88466596167667 \cdot 10^{+141}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(2 \cdot \left(\frac{a}{b} \cdot c - b\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(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)))