Average Error: 34.2 → 9.8
Time: 24.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.446364120144488225689247638815792820209 \cdot 10^{-54}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 6.060593179890297485125936384251379457639 \cdot 10^{143}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]
\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 -9.446364120144488225689247638815792820209 \cdot 10^{-54}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 6.060593179890297485125936384251379457639 \cdot 10^{143}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r4571559 = b;
        double r4571560 = -r4571559;
        double r4571561 = r4571559 * r4571559;
        double r4571562 = 4.0;
        double r4571563 = a;
        double r4571564 = c;
        double r4571565 = r4571563 * r4571564;
        double r4571566 = r4571562 * r4571565;
        double r4571567 = r4571561 - r4571566;
        double r4571568 = sqrt(r4571567);
        double r4571569 = r4571560 - r4571568;
        double r4571570 = 2.0;
        double r4571571 = r4571570 * r4571563;
        double r4571572 = r4571569 / r4571571;
        return r4571572;
}
↓
double f(double a, double b, double c) {
        double r4571573 = b;
        double r4571574 = -9.446364120144488e-54;
        bool r4571575 = r4571573 <= r4571574;
        double r4571576 = -1.0;
        double r4571577 = c;
        double r4571578 = r4571577 / r4571573;
        double r4571579 = r4571576 * r4571578;
        double r4571580 = 6.0605931798902975e+143;
        bool r4571581 = r4571573 <= r4571580;
        double r4571582 = -r4571573;
        double r4571583 = r4571573 * r4571573;
        double r4571584 = 4.0;
        double r4571585 = a;
        double r4571586 = r4571584 * r4571585;
        double r4571587 = r4571586 * r4571577;
        double r4571588 = r4571583 - r4571587;
        double r4571589 = sqrt(r4571588);
        double r4571590 = r4571582 - r4571589;
        double r4571591 = 2.0;
        double r4571592 = r4571585 * r4571591;
        double r4571593 = r4571590 / r4571592;
        double r4571594 = r4571573 / r4571585;
        double r4571595 = r4571578 - r4571594;
        double r4571596 = 1.0;
        double r4571597 = r4571595 * r4571596;
        double r4571598 = r4571581 ? r4571593 : r4571597;
        double r4571599 = r4571575 ? r4571579 : r4571598;
        return r4571599;
}

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

Original34.2
Target21.1
Herbie9.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -9.446364120144488e-54

    1. Initial program 54.2

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

    2. Taylor expanded around -inf 7.7

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

    if -9.446364120144488e-54 < b < 6.0605931798902975e+143

    1. Initial program 13.1

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

    2. Taylor expanded around 0 13.1

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

    3. Simplified13.1

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

    if 6.0605931798902975e+143 < b

    1. Initial program 59.9

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

    2. Taylor expanded around inf 2.1

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

    3. Simplified2.1

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.446364120144488225689247638815792820209 \cdot 10^{-54}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 6.060593179890297485125936384251379457639 \cdot 10^{143}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

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

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