Average Error: 34.1 → 10.0
Time: 16.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.11981154530853106611761327467786604265 \cdot 10^{143}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 4.718890261991468628346768591871377778707 \cdot 10^{-106}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.11981154530853106611761327467786604265 \cdot 10^{143}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 4.718890261991468628346768591871377778707 \cdot 10^{-106}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r52130 = b;
        double r52131 = -r52130;
        double r52132 = r52130 * r52130;
        double r52133 = 4.0;
        double r52134 = a;
        double r52135 = c;
        double r52136 = r52134 * r52135;
        double r52137 = r52133 * r52136;
        double r52138 = r52132 - r52137;
        double r52139 = sqrt(r52138);
        double r52140 = r52131 + r52139;
        double r52141 = 2.0;
        double r52142 = r52141 * r52134;
        double r52143 = r52140 / r52142;
        return r52143;
}
↓
double f(double a, double b, double c) {
        double r52144 = b;
        double r52145 = -1.119811545308531e+143;
        bool r52146 = r52144 <= r52145;
        double r52147 = 1.0;
        double r52148 = c;
        double r52149 = r52148 / r52144;
        double r52150 = a;
        double r52151 = r52144 / r52150;
        double r52152 = r52149 - r52151;
        double r52153 = r52147 * r52152;
        double r52154 = 4.718890261991469e-106;
        bool r52155 = r52144 <= r52154;
        double r52156 = r52144 * r52144;
        double r52157 = 4.0;
        double r52158 = r52150 * r52148;
        double r52159 = r52157 * r52158;
        double r52160 = r52156 - r52159;
        double r52161 = sqrt(r52160);
        double r52162 = r52161 - r52144;
        double r52163 = 2.0;
        double r52164 = r52163 * r52150;
        double r52165 = r52162 / r52164;
        double r52166 = -1.0;
        double r52167 = r52166 * r52149;
        double r52168 = r52155 ? r52165 : r52167;
        double r52169 = r52146 ? r52153 : r52168;
        return r52169;
}

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

Original34.1
Target21.0
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.119811545308531e+143

    1. Initial program 59.0

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

    2. Simplified59.0

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

    3. Taylor expanded around -inf 2.4

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

    4. Simplified2.4

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

    if -1.119811545308531e+143 < b < 4.718890261991469e-106

    1. Initial program 11.1

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

    2. Simplified11.1

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

    4. Applied div-inv11.2

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

    6. Applied pow111.2

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

    7. Applied pow111.2

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

    8. Applied pow-prod-down11.2

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

    9. Simplified11.1

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

    if 4.718890261991469e-106 < b

    1. Initial program 52.4

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

    2. Simplified52.4

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

    3. Taylor expanded around inf 10.9

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.11981154530853106611761327467786604265 \cdot 10^{143}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 4.718890261991468628346768591871377778707 \cdot 10^{-106}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :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)))