Average Error: 34.2 → 6.5
Time: 1.1m
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.650716803435188882222807352588915693143 \cdot 10^{100}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.586183022085898567784328170731092391431 \cdot 10^{-203}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + \left(-b\right)}\\ \mathbf{elif}\;b \le 6.994525771494005489896684232081336893823 \cdot 10^{142}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot \left(-4\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r4206588 = b;
        double r4206589 = -r4206588;
        double r4206590 = r4206588 * r4206588;
        double r4206591 = 4.0;
        double r4206592 = a;
        double r4206593 = c;
        double r4206594 = r4206592 * r4206593;
        double r4206595 = r4206591 * r4206594;
        double r4206596 = r4206590 - r4206595;
        double r4206597 = sqrt(r4206596);
        double r4206598 = r4206589 - r4206597;
        double r4206599 = 2.0;
        double r4206600 = r4206599 * r4206592;
        double r4206601 = r4206598 / r4206600;
        return r4206601;
}
↓
double f(double a, double b, double c) {
        double r4206602 = b;
        double r4206603 = -1.6507168034351889e+100;
        bool r4206604 = r4206602 <= r4206603;
        double r4206605 = -1.0;
        double r4206606 = c;
        double r4206607 = r4206606 / r4206602;
        double r4206608 = r4206605 * r4206607;
        double r4206609 = 1.5861830220858986e-203;
        bool r4206610 = r4206602 <= r4206609;
        double r4206611 = 2.0;
        double r4206612 = r4206611 * r4206606;
        double r4206613 = r4206602 * r4206602;
        double r4206614 = a;
        double r4206615 = r4206606 * r4206614;
        double r4206616 = 4.0;
        double r4206617 = r4206615 * r4206616;
        double r4206618 = r4206613 - r4206617;
        double r4206619 = sqrt(r4206618);
        double r4206620 = -r4206602;
        double r4206621 = r4206619 + r4206620;
        double r4206622 = r4206612 / r4206621;
        double r4206623 = 6.994525771494005e+142;
        bool r4206624 = r4206602 <= r4206623;
        double r4206625 = -r4206616;
        double r4206626 = r4206615 * r4206625;
        double r4206627 = fma(r4206602, r4206602, r4206626);
        double r4206628 = sqrt(r4206627);
        double r4206629 = r4206620 - r4206628;
        double r4206630 = r4206611 * r4206614;
        double r4206631 = r4206629 / r4206630;
        double r4206632 = 1.0;
        double r4206633 = r4206602 / r4206614;
        double r4206634 = r4206607 - r4206633;
        double r4206635 = r4206632 * r4206634;
        double r4206636 = r4206624 ? r4206631 : r4206635;
        double r4206637 = r4206610 ? r4206622 : r4206636;
        double r4206638 = r4206604 ? r4206608 : r4206637;
        return r4206638;
}

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. 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 4 regimes into one program.

  4. Final simplification6.5

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

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)))