Average Error: 33.7 → 10.7
Time: 19.5s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.363255598823911 \cdot 10^{-15}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -6.936587154412951 \cdot 10^{-28}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{1}{2 \cdot a} \cdot \sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c}\\ \mathbf{elif}\;b \le -2.3344326820285623 \cdot 10^{-123}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.6691257204922504 \cdot 10^{+85}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{1}{2 \cdot a} \cdot \sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -7.363255598823911 \cdot 10^{-15}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le -6.936587154412951 \cdot 10^{-28}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{1}{2 \cdot a} \cdot \sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c}\\

\mathbf{elif}\;b \le -2.3344326820285623 \cdot 10^{-123}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 1.6691257204922504 \cdot 10^{+85}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{1}{2 \cdot a} \cdot \sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3053636 = b;
        double r3053637 = -r3053636;
        double r3053638 = r3053636 * r3053636;
        double r3053639 = 4.0;
        double r3053640 = a;
        double r3053641 = c;
        double r3053642 = r3053640 * r3053641;
        double r3053643 = r3053639 * r3053642;
        double r3053644 = r3053638 - r3053643;
        double r3053645 = sqrt(r3053644);
        double r3053646 = r3053637 - r3053645;
        double r3053647 = 2.0;
        double r3053648 = r3053647 * r3053640;
        double r3053649 = r3053646 / r3053648;
        return r3053649;
}
↓
double f(double a, double b, double c) {
        double r3053650 = b;
        double r3053651 = -7.363255598823911e-15;
        bool r3053652 = r3053650 <= r3053651;
        double r3053653 = c;
        double r3053654 = r3053653 / r3053650;
        double r3053655 = -r3053654;
        double r3053656 = -6.936587154412951e-28;
        bool r3053657 = r3053650 <= r3053656;
        double r3053658 = -r3053650;
        double r3053659 = 2.0;
        double r3053660 = a;
        double r3053661 = r3053659 * r3053660;
        double r3053662 = r3053658 / r3053661;
        double r3053663 = 1.0;
        double r3053664 = r3053663 / r3053661;
        double r3053665 = r3053650 * r3053650;
        double r3053666 = -4.0;
        double r3053667 = r3053666 * r3053660;
        double r3053668 = r3053667 * r3053653;
        double r3053669 = r3053665 + r3053668;
        double r3053670 = sqrt(r3053669);
        double r3053671 = r3053664 * r3053670;
        double r3053672 = r3053662 - r3053671;
        double r3053673 = -2.3344326820285623e-123;
        bool r3053674 = r3053650 <= r3053673;
        double r3053675 = 1.6691257204922504e+85;
        bool r3053676 = r3053650 <= r3053675;
        double r3053677 = r3053650 / r3053660;
        double r3053678 = r3053654 - r3053677;
        double r3053679 = r3053676 ? r3053672 : r3053678;
        double r3053680 = r3053674 ? r3053655 : r3053679;
        double r3053681 = r3053657 ? r3053672 : r3053680;
        double r3053682 = r3053652 ? r3053655 : r3053681;
        return r3053682;
}

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

Original33.7
Target21.0
Herbie10.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -7.363255598823911e-15 or -6.936587154412951e-28 < b < -2.3344326820285623e-123

    1. Initial program 50.9

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

    2. Taylor expanded around -inf 10.6

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

    3. Simplified10.6

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

    if -7.363255598823911e-15 < b < -6.936587154412951e-28 or -2.3344326820285623e-123 < b < 1.6691257204922504e+85

    1. Initial program 13.4

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

    3. Applied sub-neg13.4

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

    4. Simplified13.4

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

    6. Applied div-sub13.4

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

    8. Applied div-inv13.5

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

    if 1.6691257204922504e+85 < b

    1. Initial program 42.9

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

    2. Taylor expanded around inf 3.7

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

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.363255598823911 \cdot 10^{-15}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -6.936587154412951 \cdot 10^{-28}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{1}{2 \cdot a} \cdot \sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c}\\ \mathbf{elif}\;b \le -2.3344326820285623 \cdot 10^{-123}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.6691257204922504 \cdot 10^{+85}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{1}{2 \cdot a} \cdot \sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :herbie-target
  (if (< b 0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))