Average Error: 34.5 → 10.6
Time: 9.0s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.78285893492843261 \cdot 10^{-126}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.6627135292415903 \cdot 10^{111}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]
\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 -4.78285893492843261 \cdot 10^{-126}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 3.6627135292415903 \cdot 10^{111}:\\
\;\;\;\;\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 r83657 = b;
        double r83658 = -r83657;
        double r83659 = r83657 * r83657;
        double r83660 = 4.0;
        double r83661 = a;
        double r83662 = c;
        double r83663 = r83661 * r83662;
        double r83664 = r83660 * r83663;
        double r83665 = r83659 - r83664;
        double r83666 = sqrt(r83665);
        double r83667 = r83658 - r83666;
        double r83668 = 2.0;
        double r83669 = r83668 * r83661;
        double r83670 = r83667 / r83669;
        return r83670;
}
↓
double f(double a, double b, double c) {
        double r83671 = b;
        double r83672 = -4.7828589349284326e-126;
        bool r83673 = r83671 <= r83672;
        double r83674 = -1.0;
        double r83675 = c;
        double r83676 = r83675 / r83671;
        double r83677 = r83674 * r83676;
        double r83678 = 3.6627135292415903e+111;
        bool r83679 = r83671 <= r83678;
        double r83680 = -r83671;
        double r83681 = r83671 * r83671;
        double r83682 = 4.0;
        double r83683 = a;
        double r83684 = r83683 * r83675;
        double r83685 = r83682 * r83684;
        double r83686 = r83681 - r83685;
        double r83687 = sqrt(r83686);
        double r83688 = r83680 - r83687;
        double r83689 = 1.0;
        double r83690 = 2.0;
        double r83691 = r83690 * r83683;
        double r83692 = r83689 / r83691;
        double r83693 = r83688 * r83692;
        double r83694 = r83671 / r83683;
        double r83695 = r83674 * r83694;
        double r83696 = r83679 ? r83693 : r83695;
        double r83697 = r83673 ? r83677 : r83696;
        return r83697;
}

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

Original34.5
Target20.6
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.7828589349284326e-126

    1. Initial program 51.3

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

    2. Taylor expanded around -inf 11.3

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

    if -4.7828589349284326e-126 < b < 3.6627135292415903e+111

    1. Initial program 12.0

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

    3. Applied div-inv12.1

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

    if 3.6627135292415903e+111 < b

    1. Initial program 49.7

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

    3. Applied div-inv49.8

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

    5. Applied associate-*r/49.7

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

    6. Simplified49.7

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

    8. Applied flip--63.3

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

    9. Simplified62.3

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

    10. Simplified62.3

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

    11. Taylor expanded around 0 3.6

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.78285893492843261 \cdot 10^{-126}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.6627135292415903 \cdot 10^{111}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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)))