Average Error: 34.4 → 14.1
Time: 18.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.390658213785421360285622940547871300736 \cdot 10^{101}:\\ \;\;\;\;\frac{\left(\frac{a}{\frac{b}{c}} - b\right) \cdot 1}{a}\\ \mathbf{elif}\;b \le 4.330541687749954965862284767620099540245 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{\frac{2}{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot c}{b} \cdot -1}{a}\\ \end{array}\]
\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 -1.390658213785421360285622940547871300736 \cdot 10^{101}:\\
\;\;\;\;\frac{\left(\frac{a}{\frac{b}{c}} - b\right) \cdot 1}{a}\\

\mathbf{elif}\;b \le 4.330541687749954965862284767620099540245 \cdot 10^{-17}:\\
\;\;\;\;\frac{\frac{1}{\frac{2}{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r4147499 = b;
        double r4147500 = -r4147499;
        double r4147501 = r4147499 * r4147499;
        double r4147502 = 4.0;
        double r4147503 = a;
        double r4147504 = c;
        double r4147505 = r4147503 * r4147504;
        double r4147506 = r4147502 * r4147505;
        double r4147507 = r4147501 - r4147506;
        double r4147508 = sqrt(r4147507);
        double r4147509 = r4147500 + r4147508;
        double r4147510 = 2.0;
        double r4147511 = r4147510 * r4147503;
        double r4147512 = r4147509 / r4147511;
        return r4147512;
}
↓
double f(double a, double b, double c) {
        double r4147513 = b;
        double r4147514 = -1.3906582137854214e+101;
        bool r4147515 = r4147513 <= r4147514;
        double r4147516 = a;
        double r4147517 = c;
        double r4147518 = r4147513 / r4147517;
        double r4147519 = r4147516 / r4147518;
        double r4147520 = r4147519 - r4147513;
        double r4147521 = 1.0;
        double r4147522 = r4147520 * r4147521;
        double r4147523 = r4147522 / r4147516;
        double r4147524 = 4.330541687749955e-17;
        bool r4147525 = r4147513 <= r4147524;
        double r4147526 = 1.0;
        double r4147527 = 2.0;
        double r4147528 = r4147513 * r4147513;
        double r4147529 = 4.0;
        double r4147530 = r4147529 * r4147516;
        double r4147531 = r4147517 * r4147530;
        double r4147532 = r4147528 - r4147531;
        double r4147533 = sqrt(r4147532);
        double r4147534 = r4147533 - r4147513;
        double r4147535 = r4147527 / r4147534;
        double r4147536 = r4147526 / r4147535;
        double r4147537 = r4147536 / r4147516;
        double r4147538 = r4147516 * r4147517;
        double r4147539 = r4147538 / r4147513;
        double r4147540 = -1.0;
        double r4147541 = r4147539 * r4147540;
        double r4147542 = r4147541 / r4147516;
        double r4147543 = r4147525 ? r4147537 : r4147542;
        double r4147544 = r4147515 ? r4147523 : r4147543;
        return r4147544;
}

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

Original34.4
Target21.3
Herbie14.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.3906582137854214e+101

    1. Initial program 48.2

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

    2. Simplified48.2

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

    3. Taylor expanded around -inf 10.3

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

    4. Simplified3.5

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

    if -1.3906582137854214e+101 < b < 4.330541687749955e-17

    1. Initial program 15.2

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

    2. Simplified15.2

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

    4. Applied clear-num15.3

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

    if 4.330541687749955e-17 < b

    1. Initial program 55.6

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

    2. Simplified55.6

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

    3. Taylor expanded around inf 17.3

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

  4. Final simplification14.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.390658213785421360285622940547871300736 \cdot 10^{101}:\\ \;\;\;\;\frac{\left(\frac{a}{\frac{b}{c}} - b\right) \cdot 1}{a}\\ \mathbf{elif}\;b \le 4.330541687749954965862284767620099540245 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{\frac{2}{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot c}{b} \cdot -1}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))