Average Error: 34.4 → 7.0
Time: 9.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.457738542065716919858398723449020930628 \cdot 10^{153}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -2.695863739873928877277501764874065503226 \cdot 10^{-295}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}\\ \mathbf{elif}\;b \le 1.191203142513164639216663918436976252985 \cdot 10^{117}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]
\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 -1.457738542065716919858398723449020930628 \cdot 10^{153}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -2.695863739873928877277501764874065503226 \cdot 10^{-295}:\\
\;\;\;\;\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}\\

\mathbf{elif}\;b \le 1.191203142513164639216663918436976252985 \cdot 10^{117}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r134702 = b;
        double r134703 = -r134702;
        double r134704 = r134702 * r134702;
        double r134705 = 4.0;
        double r134706 = a;
        double r134707 = c;
        double r134708 = r134706 * r134707;
        double r134709 = r134705 * r134708;
        double r134710 = r134704 - r134709;
        double r134711 = sqrt(r134710);
        double r134712 = r134703 - r134711;
        double r134713 = 2.0;
        double r134714 = r134713 * r134706;
        double r134715 = r134712 / r134714;
        return r134715;
}
↓
double f(double a, double b, double c) {
        double r134716 = b;
        double r134717 = -1.457738542065717e+153;
        bool r134718 = r134716 <= r134717;
        double r134719 = -1.0;
        double r134720 = c;
        double r134721 = r134720 / r134716;
        double r134722 = r134719 * r134721;
        double r134723 = -2.695863739873929e-295;
        bool r134724 = r134716 <= r134723;
        double r134725 = 2.0;
        double r134726 = r134725 * r134720;
        double r134727 = 4.0;
        double r134728 = a;
        double r134729 = r134728 * r134720;
        double r134730 = r134727 * r134729;
        double r134731 = -r134730;
        double r134732 = fma(r134716, r134716, r134731);
        double r134733 = sqrt(r134732);
        double r134734 = r134733 - r134716;
        double r134735 = r134726 / r134734;
        double r134736 = 1.1912031425131646e+117;
        bool r134737 = r134716 <= r134736;
        double r134738 = 1.0;
        double r134739 = r134725 * r134728;
        double r134740 = -r134716;
        double r134741 = r134716 * r134716;
        double r134742 = r134741 - r134730;
        double r134743 = sqrt(r134742);
        double r134744 = r134740 - r134743;
        double r134745 = r134739 / r134744;
        double r134746 = r134738 / r134745;
        double r134747 = r134716 / r134728;
        double r134748 = r134719 * r134747;
        double r134749 = r134737 ? r134746 : r134748;
        double r134750 = r134724 ? r134735 : r134749;
        double r134751 = r134718 ? r134722 : r134750;
        return r134751;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.4
Target21.2
Herbie7.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.457738542065717e+153

    1. Initial program 63.9

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

    2. Taylor expanded around -inf 1.3

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

    if -1.457738542065717e+153 < b < -2.695863739873929e-295

    1. Initial program 35.5

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

    3. Applied flip--35.5

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

    4. Simplified16.5

      \[\leadsto \frac{\frac{\color{blue}{0 + \left(a \cdot c\right) \cdot 4}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]

    5. Simplified16.5

      \[\leadsto \frac{\frac{0 + \left(a \cdot c\right) \cdot 4}{\color{blue}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}}}{2 \cdot a}\]
    6. Using strategy rm

    7. Applied div-inv16.6

      \[\leadsto \color{blue}{\frac{0 + \left(a \cdot c\right) \cdot 4}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b} \cdot \frac{1}{2 \cdot a}}\]
    8. Using strategy rm

    9. Applied associate-*l/15.1

      \[\leadsto \color{blue}{\frac{\left(0 + \left(a \cdot c\right) \cdot 4\right) \cdot \frac{1}{2 \cdot a}}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}}\]

    10. Simplified15.0

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

    11. Taylor expanded around 0 8.3

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

    if -2.695863739873929e-295 < b < 1.1912031425131646e+117

    1. Initial program 9.8

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

    3. Applied clear-num9.9

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

    if 1.1912031425131646e+117 < b

    1. Initial program 50.7

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

    3. Applied flip--63.7

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

    4. Simplified62.7

      \[\leadsto \frac{\frac{\color{blue}{0 + \left(a \cdot c\right) \cdot 4}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]

    5. Simplified62.7

      \[\leadsto \frac{\frac{0 + \left(a \cdot c\right) \cdot 4}{\color{blue}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}}}{2 \cdot a}\]

    6. Taylor expanded around 0 4.0

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

  4. Final simplification7.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.457738542065716919858398723449020930628 \cdot 10^{153}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -2.695863739873928877277501764874065503226 \cdot 10^{-295}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}\\ \mathbf{elif}\;b \le 1.191203142513164639216663918436976252985 \cdot 10^{117}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :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)))