Average Error: 34.0 → 10.5
Time: 17.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.900769547116861 \cdot 10^{+46}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.6528810740721013 \cdot 10^{-142}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -2.900769547116861 \cdot 10^{+46}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 1.6528810740721013 \cdot 10^{-142}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\right) \cdot \frac{\frac{1}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3455544 = b;
        double r3455545 = -r3455544;
        double r3455546 = r3455544 * r3455544;
        double r3455547 = 4.0;
        double r3455548 = a;
        double r3455549 = r3455547 * r3455548;
        double r3455550 = c;
        double r3455551 = r3455549 * r3455550;
        double r3455552 = r3455546 - r3455551;
        double r3455553 = sqrt(r3455552);
        double r3455554 = r3455545 + r3455553;
        double r3455555 = 2.0;
        double r3455556 = r3455555 * r3455548;
        double r3455557 = r3455554 / r3455556;
        return r3455557;
}
↓
double f(double a, double b, double c) {
        double r3455558 = b;
        double r3455559 = -2.900769547116861e+46;
        bool r3455560 = r3455558 <= r3455559;
        double r3455561 = c;
        double r3455562 = r3455561 / r3455558;
        double r3455563 = a;
        double r3455564 = r3455558 / r3455563;
        double r3455565 = r3455562 - r3455564;
        double r3455566 = 1.6528810740721013e-142;
        bool r3455567 = r3455558 <= r3455566;
        double r3455568 = -r3455558;
        double r3455569 = r3455558 * r3455558;
        double r3455570 = 4.0;
        double r3455571 = r3455570 * r3455563;
        double r3455572 = r3455561 * r3455571;
        double r3455573 = r3455569 - r3455572;
        double r3455574 = sqrt(r3455573);
        double r3455575 = r3455568 + r3455574;
        double r3455576 = 0.5;
        double r3455577 = r3455576 / r3455563;
        double r3455578 = r3455575 * r3455577;
        double r3455579 = -r3455562;
        double r3455580 = r3455567 ? r3455578 : r3455579;
        double r3455581 = r3455560 ? r3455565 : r3455580;
        return r3455581;
}

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

Original34.0
Target20.7
Herbie10.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.900769547116861e+46

    1. Initial program 35.9

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

    2. Taylor expanded around -inf 5.3

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

    if -2.900769547116861e+46 < b < 1.6528810740721013e-142

    1. Initial program 11.5

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

    3. Applied div-inv11.7

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

    4. Simplified11.7

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

    if 1.6528810740721013e-142 < b

    1. Initial program 50.1

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

    2. Taylor expanded around inf 12.0

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

    3. Simplified12.0

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.900769547116861 \cdot 10^{+46}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.6528810740721013 \cdot 10^{-142}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 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)))