Average Error: 32.7 → 10.0
Time: 29.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.3975762435547 \cdot 10^{+118}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.3115303715225787 \cdot 10^{-131}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -7.3975762435547 \cdot 10^{+118}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 1.3115303715225787 \cdot 10^{-131}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r4319559 = b;
        double r4319560 = -r4319559;
        double r4319561 = r4319559 * r4319559;
        double r4319562 = 4.0;
        double r4319563 = a;
        double r4319564 = r4319562 * r4319563;
        double r4319565 = c;
        double r4319566 = r4319564 * r4319565;
        double r4319567 = r4319561 - r4319566;
        double r4319568 = sqrt(r4319567);
        double r4319569 = r4319560 + r4319568;
        double r4319570 = 2.0;
        double r4319571 = r4319570 * r4319563;
        double r4319572 = r4319569 / r4319571;
        return r4319572;
}
↓
double f(double a, double b, double c) {
        double r4319573 = b;
        double r4319574 = -7.3975762435547e+118;
        bool r4319575 = r4319573 <= r4319574;
        double r4319576 = c;
        double r4319577 = r4319576 / r4319573;
        double r4319578 = a;
        double r4319579 = r4319573 / r4319578;
        double r4319580 = r4319577 - r4319579;
        double r4319581 = 1.3115303715225787e-131;
        bool r4319582 = r4319573 <= r4319581;
        double r4319583 = r4319573 * r4319573;
        double r4319584 = 4.0;
        double r4319585 = r4319578 * r4319584;
        double r4319586 = r4319585 * r4319576;
        double r4319587 = r4319583 - r4319586;
        double r4319588 = sqrt(r4319587);
        double r4319589 = r4319588 - r4319573;
        double r4319590 = 2.0;
        double r4319591 = r4319589 / r4319590;
        double r4319592 = r4319591 / r4319578;
        double r4319593 = -r4319577;
        double r4319594 = r4319582 ? r4319592 : r4319593;
        double r4319595 = r4319575 ? r4319580 : r4319594;
        return r4319595;
}

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

Original32.7
Target20.0
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -7.3975762435547e+118

    1. Initial program 49.0

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

    2. Simplified49.0

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

    3. Taylor expanded around -inf 49.0

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

    4. Simplified49.0

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

    5. Taylor expanded around -inf 3.0

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

    if -7.3975762435547e+118 < b < 1.3115303715225787e-131

    1. Initial program 10.7

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

    2. Simplified10.7

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

    3. Taylor expanded around -inf 10.7

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

    4. Simplified10.7

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

    if 1.3115303715225787e-131 < b

    1. Initial program 50.3

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

    2. Simplified50.3

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

    3. Taylor expanded around -inf 50.3

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

    4. Simplified50.3

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

    5. Taylor expanded around inf 11.7

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

    6. Simplified11.7

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.3975762435547 \cdot 10^{+118}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.3115303715225787 \cdot 10^{-131}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(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)))