Average Error: 33.9 → 10.3
Time: 18.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.361733299857302083043096878302889042354 \cdot 10^{105}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 3.09136118080059703772253670927164991568 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} + \frac{\frac{-b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.361733299857302083043096878302889042354 \cdot 10^{105}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 3.09136118080059703772253670927164991568 \cdot 10^{-86}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} + \frac{\frac{-b}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r58704 = b;
        double r58705 = -r58704;
        double r58706 = r58704 * r58704;
        double r58707 = 4.0;
        double r58708 = a;
        double r58709 = r58707 * r58708;
        double r58710 = c;
        double r58711 = r58709 * r58710;
        double r58712 = r58706 - r58711;
        double r58713 = sqrt(r58712);
        double r58714 = r58705 + r58713;
        double r58715 = 2.0;
        double r58716 = r58715 * r58708;
        double r58717 = r58714 / r58716;
        return r58717;
}
↓
double f(double a, double b, double c) {
        double r58718 = b;
        double r58719 = -1.361733299857302e+105;
        bool r58720 = r58718 <= r58719;
        double r58721 = 1.0;
        double r58722 = c;
        double r58723 = r58722 / r58718;
        double r58724 = a;
        double r58725 = r58718 / r58724;
        double r58726 = r58723 - r58725;
        double r58727 = r58721 * r58726;
        double r58728 = 3.091361180800597e-86;
        bool r58729 = r58718 <= r58728;
        double r58730 = r58718 * r58718;
        double r58731 = 4.0;
        double r58732 = r58731 * r58724;
        double r58733 = r58732 * r58722;
        double r58734 = r58730 - r58733;
        double r58735 = sqrt(r58734);
        double r58736 = 2.0;
        double r58737 = r58724 * r58736;
        double r58738 = r58735 / r58737;
        double r58739 = -r58718;
        double r58740 = r58739 / r58736;
        double r58741 = r58740 / r58724;
        double r58742 = r58738 + r58741;
        double r58743 = -1.0;
        double r58744 = r58743 * r58723;
        double r58745 = r58729 ? r58742 : r58744;
        double r58746 = r58720 ? r58727 : r58745;
        return r58746;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original33.9
Target21.1
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.361733299857302e+105

    1. Initial program 48.6

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

    2. Simplified48.6

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

    3. Taylor expanded around -inf 3.6

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

    4. Simplified3.6

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

    if -1.361733299857302e+105 < b < 3.091361180800597e-86

    1. Initial program 12.2

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

    2. Simplified12.2

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

    4. Applied clear-num12.3

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

    6. Applied div-inv12.4

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

    7. Applied add-cube-cbrt12.4

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

    8. Applied times-frac12.3

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

    9. Simplified12.3

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

    10. Simplified12.3

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

    12. Applied sub-neg12.3

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

    13. Applied distribute-lft-in12.3

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

    14. Simplified12.2

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

    15. Simplified12.2

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

    if 3.091361180800597e-86 < b

    1. Initial program 51.8

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

    2. Simplified51.8

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

    3. Taylor expanded around inf 10.7

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.361733299857302083043096878302889042354 \cdot 10^{105}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 3.09136118080059703772253670927164991568 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} + \frac{\frac{-b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- 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)))