Average Error: 34.0 → 9.1
Time: 21.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.900769547116861 \cdot 10^{+46}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 5.4213851798811764 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b\right)}{2}\\ \mathbf{elif}\;b \le 1.1597179970514171 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\frac{\mathsf{fma}\left(c, -4 \cdot a, 0\right)}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} + b}}{\sqrt[3]{a}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -2.900769547116861 \cdot 10^{+46}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 5.4213851798811764 \cdot 10^{-102}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b\right)}{2}\\

\mathbf{elif}\;b \le 1.1597179970514171 \cdot 10^{+23}:\\
\;\;\;\;\frac{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\frac{\mathsf{fma}\left(c, -4 \cdot a, 0\right)}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} + b}}{\sqrt[3]{a}}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r3084653 = b;
        double r3084654 = -r3084653;
        double r3084655 = r3084653 * r3084653;
        double r3084656 = 4.0;
        double r3084657 = a;
        double r3084658 = r3084656 * r3084657;
        double r3084659 = c;
        double r3084660 = r3084658 * r3084659;
        double r3084661 = r3084655 - r3084660;
        double r3084662 = sqrt(r3084661);
        double r3084663 = r3084654 + r3084662;
        double r3084664 = 2.0;
        double r3084665 = r3084664 * r3084657;
        double r3084666 = r3084663 / r3084665;
        return r3084666;
}
↓
double f(double a, double b, double c) {
        double r3084667 = b;
        double r3084668 = -2.900769547116861e+46;
        bool r3084669 = r3084667 <= r3084668;
        double r3084670 = c;
        double r3084671 = r3084670 / r3084667;
        double r3084672 = a;
        double r3084673 = r3084667 / r3084672;
        double r3084674 = r3084671 - r3084673;
        double r3084675 = 2.0;
        double r3084676 = r3084674 * r3084675;
        double r3084677 = r3084676 / r3084675;
        double r3084678 = 5.4213851798811764e-102;
        bool r3084679 = r3084667 <= r3084678;
        double r3084680 = 1.0;
        double r3084681 = r3084680 / r3084672;
        double r3084682 = -4.0;
        double r3084683 = r3084682 * r3084672;
        double r3084684 = r3084683 * r3084670;
        double r3084685 = fma(r3084667, r3084667, r3084684);
        double r3084686 = sqrt(r3084685);
        double r3084687 = r3084686 - r3084667;
        double r3084688 = r3084681 * r3084687;
        double r3084689 = r3084688 / r3084675;
        double r3084690 = 1.1597179970514171e+23;
        bool r3084691 = r3084667 <= r3084690;
        double r3084692 = cbrt(r3084672);
        double r3084693 = r3084692 * r3084692;
        double r3084694 = r3084680 / r3084693;
        double r3084695 = 0.0;
        double r3084696 = fma(r3084670, r3084683, r3084695);
        double r3084697 = r3084686 + r3084667;
        double r3084698 = r3084696 / r3084697;
        double r3084699 = r3084698 / r3084692;
        double r3084700 = r3084694 * r3084699;
        double r3084701 = r3084700 / r3084675;
        double r3084702 = -2.0;
        double r3084703 = r3084702 * r3084671;
        double r3084704 = r3084703 / r3084675;
        double r3084705 = r3084691 ? r3084701 : r3084704;
        double r3084706 = r3084679 ? r3084689 : r3084705;
        double r3084707 = r3084669 ? r3084677 : r3084706;
        return r3084707;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.0
Target20.7
Herbie9.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -2.900769547116861e+46

    1. Initial program 35.9

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

    2. Simplified35.9

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

    4. Applied div-inv36.0

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

    5. Taylor expanded around -inf 5.3

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]

    6. Simplified5.3

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

    if -2.900769547116861e+46 < b < 5.4213851798811764e-102

    1. Initial program 12.8

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

    2. Simplified12.8

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

    4. Applied div-inv12.9

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

    if 5.4213851798811764e-102 < b < 1.1597179970514171e+23

    1. Initial program 39.1

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

    2. Simplified39.1

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

    4. Applied add-cube-cbrt39.5

      \[\leadsto \frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}}{2}\]

    5. Applied *-un-lft-identity39.5

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b\right)}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}{2}\]

    6. Applied times-frac39.5

      \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b}{\sqrt[3]{a}}}}{2}\]
    7. Using strategy rm

    8. Applied flip--39.6

      \[\leadsto \frac{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b \cdot b}{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} + b}}}{\sqrt[3]{a}}}{2}\]

    9. Simplified17.8

      \[\leadsto \frac{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\frac{\color{blue}{\mathsf{fma}\left(c, -4 \cdot a, 0\right)}}{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} + b}}{\sqrt[3]{a}}}{2}\]

    if 1.1597179970514171e+23 < b

    1. Initial program 55.6

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

    2. Simplified55.6

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

    3. Taylor expanded around inf 4.4

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.900769547116861 \cdot 10^{+46}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 5.4213851798811764 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b\right)}{2}\\ \mathbf{elif}\;b \le 1.1597179970514171 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\frac{\mathsf{fma}\left(c, -4 \cdot a, 0\right)}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} + b}}{\sqrt[3]{a}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))