Average Error: 34.0 → 11.7
Time: 5.4s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.92145859080318459 \cdot 10^{30}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.35303540148363475 \cdot 10^{-114}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le -3.70403707285546 \cdot 10^{-127}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.3866645776898725 \cdot 10^{85}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]
\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 -5.92145859080318459 \cdot 10^{30}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -1.35303540148363475 \cdot 10^{-114}:\\
\;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\

\mathbf{elif}\;b \le -3.70403707285546 \cdot 10^{-127}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 2.3866645776898725 \cdot 10^{85}:\\
\;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r62564 = b;
        double r62565 = -r62564;
        double r62566 = r62564 * r62564;
        double r62567 = 4.0;
        double r62568 = a;
        double r62569 = c;
        double r62570 = r62568 * r62569;
        double r62571 = r62567 * r62570;
        double r62572 = r62566 - r62571;
        double r62573 = sqrt(r62572);
        double r62574 = r62565 - r62573;
        double r62575 = 2.0;
        double r62576 = r62575 * r62568;
        double r62577 = r62574 / r62576;
        return r62577;
}
↓
double f(double a, double b, double c) {
        double r62578 = b;
        double r62579 = -5.921458590803185e+30;
        bool r62580 = r62578 <= r62579;
        double r62581 = -1.0;
        double r62582 = c;
        double r62583 = r62582 / r62578;
        double r62584 = r62581 * r62583;
        double r62585 = -1.3530354014836348e-114;
        bool r62586 = r62578 <= r62585;
        double r62587 = -r62578;
        double r62588 = r62578 * r62578;
        double r62589 = 4.0;
        double r62590 = a;
        double r62591 = r62590 * r62582;
        double r62592 = r62589 * r62591;
        double r62593 = r62588 - r62592;
        double r62594 = sqrt(r62593);
        double r62595 = r62587 - r62594;
        double r62596 = 1.0;
        double r62597 = 2.0;
        double r62598 = r62597 * r62590;
        double r62599 = r62596 / r62598;
        double r62600 = r62595 * r62599;
        double r62601 = -3.704037072855455e-127;
        bool r62602 = r62578 <= r62601;
        double r62603 = 2.3866645776898725e+85;
        bool r62604 = r62578 <= r62603;
        double r62605 = r62578 / r62590;
        double r62606 = r62581 * r62605;
        double r62607 = r62604 ? r62600 : r62606;
        double r62608 = r62602 ? r62584 : r62607;
        double r62609 = r62586 ? r62600 : r62608;
        double r62610 = r62580 ? r62584 : r62609;
        return r62610;
}

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
Target21.2
Herbie11.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -5.921458590803185e+30 or -1.3530354014836348e-114 < b < -3.704037072855455e-127

    1. Initial program 56.0

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

    2. Taylor expanded around -inf 6.0

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

    if -5.921458590803185e+30 < b < -1.3530354014836348e-114 or -3.704037072855455e-127 < b < 2.3866645776898725e+85

    1. Initial program 17.6

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

    3. Applied div-inv17.7

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

    if 2.3866645776898725e+85 < b

    1. Initial program 43.7

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

    3. Applied clear-num43.8

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

    4. Taylor expanded around 0 3.8

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

  4. Final simplification11.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.92145859080318459 \cdot 10^{30}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.35303540148363475 \cdot 10^{-114}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le -3.70403707285546 \cdot 10^{-127}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.3866645776898725 \cdot 10^{85}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ 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)))