Average Error: 34.2 → 6.5
Time: 31.4s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.650716803435188882222807352588915693143 \cdot 10^{100}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.586183022085898567784328170731092391431 \cdot 10^{-203}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + \left(-b\right)}\\ \mathbf{elif}\;b \le 6.994525771494005489896684232081336893823 \cdot 10^{142}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot \left(-4\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]
\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 -1.650716803435188882222807352588915693143 \cdot 10^{100}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 1.586183022085898567784328170731092391431 \cdot 10^{-203}:\\
\;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + \left(-b\right)}\\

\mathbf{elif}\;b \le 6.994525771494005489896684232081336893823 \cdot 10^{142}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot \left(-4\right)\right)}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3491567 = b;
        double r3491568 = -r3491567;
        double r3491569 = r3491567 * r3491567;
        double r3491570 = 4.0;
        double r3491571 = a;
        double r3491572 = c;
        double r3491573 = r3491571 * r3491572;
        double r3491574 = r3491570 * r3491573;
        double r3491575 = r3491569 - r3491574;
        double r3491576 = sqrt(r3491575);
        double r3491577 = r3491568 - r3491576;
        double r3491578 = 2.0;
        double r3491579 = r3491578 * r3491571;
        double r3491580 = r3491577 / r3491579;
        return r3491580;
}
↓
double f(double a, double b, double c) {
        double r3491581 = b;
        double r3491582 = -1.6507168034351889e+100;
        bool r3491583 = r3491581 <= r3491582;
        double r3491584 = -1.0;
        double r3491585 = c;
        double r3491586 = r3491585 / r3491581;
        double r3491587 = r3491584 * r3491586;
        double r3491588 = 1.5861830220858986e-203;
        bool r3491589 = r3491581 <= r3491588;
        double r3491590 = 2.0;
        double r3491591 = r3491590 * r3491585;
        double r3491592 = r3491581 * r3491581;
        double r3491593 = a;
        double r3491594 = r3491585 * r3491593;
        double r3491595 = 4.0;
        double r3491596 = r3491594 * r3491595;
        double r3491597 = r3491592 - r3491596;
        double r3491598 = sqrt(r3491597);
        double r3491599 = -r3491581;
        double r3491600 = r3491598 + r3491599;
        double r3491601 = r3491591 / r3491600;
        double r3491602 = 6.994525771494005e+142;
        bool r3491603 = r3491581 <= r3491602;
        double r3491604 = -r3491595;
        double r3491605 = r3491594 * r3491604;
        double r3491606 = fma(r3491581, r3491581, r3491605);
        double r3491607 = sqrt(r3491606);
        double r3491608 = r3491599 - r3491607;
        double r3491609 = r3491590 * r3491593;
        double r3491610 = r3491608 / r3491609;
        double r3491611 = r3491581 / r3491593;
        double r3491612 = r3491586 - r3491611;
        double r3491613 = 1.0;
        double r3491614 = r3491612 * r3491613;
        double r3491615 = r3491603 ? r3491610 : r3491614;
        double r3491616 = r3491589 ? r3491601 : r3491615;
        double r3491617 = r3491583 ? r3491587 : r3491616;
        return r3491617;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.2
Target21.1
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.6507168034351889e+100

    1. Initial program 59.6

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

    2. Taylor expanded around -inf 2.8

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

    if -1.6507168034351889e+100 < b < 1.5861830220858986e-203

    1. Initial program 30.0

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

    3. Applied div-inv30.1

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

    5. Applied flip--30.2

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

    6. Applied associate-*l/30.2

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

    7. Simplified15.6

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

    8. Taylor expanded around 0 9.8

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

    if 1.5861830220858986e-203 < b < 6.994525771494005e+142

    1. Initial program 7.3

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

    3. Applied fma-neg7.3

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

    if 6.994525771494005e+142 < 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. Using strategy rm

    3. Applied fma-neg59.9

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

    4. Taylor expanded around inf 2.1

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

    5. Simplified2.1

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.650716803435188882222807352588915693143 \cdot 10^{100}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.586183022085898567784328170731092391431 \cdot 10^{-203}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + \left(-b\right)}\\ \mathbf{elif}\;b \le 6.994525771494005489896684232081336893823 \cdot 10^{142}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot \left(-4\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :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)))