Average Error: 34.2 → 6.5
Time: 29.9s
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 r4104989 = b;
        double r4104990 = -r4104989;
        double r4104991 = r4104989 * r4104989;
        double r4104992 = 4.0;
        double r4104993 = a;
        double r4104994 = c;
        double r4104995 = r4104993 * r4104994;
        double r4104996 = r4104992 * r4104995;
        double r4104997 = r4104991 - r4104996;
        double r4104998 = sqrt(r4104997);
        double r4104999 = r4104990 - r4104998;
        double r4105000 = 2.0;
        double r4105001 = r4105000 * r4104993;
        double r4105002 = r4104999 / r4105001;
        return r4105002;
}
↓
double f(double a, double b, double c) {
        double r4105003 = b;
        double r4105004 = -1.6507168034351889e+100;
        bool r4105005 = r4105003 <= r4105004;
        double r4105006 = -1.0;
        double r4105007 = c;
        double r4105008 = r4105007 / r4105003;
        double r4105009 = r4105006 * r4105008;
        double r4105010 = 1.5861830220858986e-203;
        bool r4105011 = r4105003 <= r4105010;
        double r4105012 = 2.0;
        double r4105013 = r4105012 * r4105007;
        double r4105014 = r4105003 * r4105003;
        double r4105015 = a;
        double r4105016 = r4105007 * r4105015;
        double r4105017 = 4.0;
        double r4105018 = r4105016 * r4105017;
        double r4105019 = r4105014 - r4105018;
        double r4105020 = sqrt(r4105019);
        double r4105021 = -r4105003;
        double r4105022 = r4105020 + r4105021;
        double r4105023 = r4105013 / r4105022;
        double r4105024 = 6.994525771494005e+142;
        bool r4105025 = r4105003 <= r4105024;
        double r4105026 = -r4105017;
        double r4105027 = r4105016 * r4105026;
        double r4105028 = fma(r4105003, r4105003, r4105027);
        double r4105029 = sqrt(r4105028);
        double r4105030 = r4105021 - r4105029;
        double r4105031 = r4105012 * r4105015;
        double r4105032 = r4105030 / r4105031;
        double r4105033 = 1.0;
        double r4105034 = r4105003 / r4105015;
        double r4105035 = r4105008 - r4105034;
        double r4105036 = r4105033 * r4105035;
        double r4105037 = r4105025 ? r4105032 : r4105036;
        double r4105038 = r4105011 ? r4105023 : r4105037;
        double r4105039 = r4105005 ? r4105009 : r4105038;
        return r4105039;
}

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 "The quadratic formula (r2)"

  :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)))