Average Error: 33.8 → 9.1
Time: 20.5s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.970010565552108757188050455448622102575 \cdot 10^{58}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.241327699195247001117817134883838709979 \cdot 10^{-253}:\\ \;\;\;\;\frac{\left(b \cdot b - b \cdot b\right) + 4 \cdot \left(c \cdot a\right)}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b} \cdot \frac{1}{a \cdot 2}\\ \mathbf{elif}\;b \le 3.628799960716311990444092539387346352569 \cdot 10^{50}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]
\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 -1.970010565552108757188050455448622102575 \cdot 10^{58}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -1.241327699195247001117817134883838709979 \cdot 10^{-253}:\\
\;\;\;\;\frac{\left(b \cdot b - b \cdot b\right) + 4 \cdot \left(c \cdot a\right)}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b} \cdot \frac{1}{a \cdot 2}\\

\mathbf{elif}\;b \le 3.628799960716311990444092539387346352569 \cdot 10^{50}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\end{array}
double f(double a, double b, double c) {
        double r3742667 = b;
        double r3742668 = -r3742667;
        double r3742669 = r3742667 * r3742667;
        double r3742670 = 4.0;
        double r3742671 = a;
        double r3742672 = c;
        double r3742673 = r3742671 * r3742672;
        double r3742674 = r3742670 * r3742673;
        double r3742675 = r3742669 - r3742674;
        double r3742676 = sqrt(r3742675);
        double r3742677 = r3742668 - r3742676;
        double r3742678 = 2.0;
        double r3742679 = r3742678 * r3742671;
        double r3742680 = r3742677 / r3742679;
        return r3742680;
}
↓
double f(double a, double b, double c) {
        double r3742681 = b;
        double r3742682 = -1.9700105655521088e+58;
        bool r3742683 = r3742681 <= r3742682;
        double r3742684 = -1.0;
        double r3742685 = c;
        double r3742686 = r3742685 / r3742681;
        double r3742687 = r3742684 * r3742686;
        double r3742688 = -1.241327699195247e-253;
        bool r3742689 = r3742681 <= r3742688;
        double r3742690 = r3742681 * r3742681;
        double r3742691 = r3742690 - r3742690;
        double r3742692 = 4.0;
        double r3742693 = a;
        double r3742694 = r3742685 * r3742693;
        double r3742695 = r3742692 * r3742694;
        double r3742696 = r3742691 + r3742695;
        double r3742697 = r3742690 - r3742695;
        double r3742698 = sqrt(r3742697);
        double r3742699 = r3742698 - r3742681;
        double r3742700 = r3742696 / r3742699;
        double r3742701 = 1.0;
        double r3742702 = 2.0;
        double r3742703 = r3742693 * r3742702;
        double r3742704 = r3742701 / r3742703;
        double r3742705 = r3742700 * r3742704;
        double r3742706 = 3.628799960716312e+50;
        bool r3742707 = r3742681 <= r3742706;
        double r3742708 = -r3742681;
        double r3742709 = r3742708 - r3742698;
        double r3742710 = r3742709 / r3742703;
        double r3742711 = r3742681 / r3742693;
        double r3742712 = r3742686 - r3742711;
        double r3742713 = 1.0;
        double r3742714 = r3742712 * r3742713;
        double r3742715 = r3742707 ? r3742710 : r3742714;
        double r3742716 = r3742689 ? r3742705 : r3742715;
        double r3742717 = r3742683 ? r3742687 : r3742716;
        return r3742717;
}

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

Original33.8
Target20.7
Herbie9.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.9700105655521088e+58

    1. Initial program 57.5

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

    2. Taylor expanded around -inf 3.4

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

    if -1.9700105655521088e+58 < b < -1.241327699195247e-253

    1. Initial program 31.7

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

    3. Applied div-inv31.7

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

    5. Applied flip--31.8

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

    6. Simplified17.2

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

    7. Simplified17.2

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

    if -1.241327699195247e-253 < b < 3.628799960716312e+50

    1. Initial program 10.1

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

    3. Applied div-inv10.2

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

    5. Applied un-div-inv10.1

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

    if 3.628799960716312e+50 < b

    1. Initial program 38.2

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

    2. Taylor expanded around inf 6.1

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

    3. Simplified6.1

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.970010565552108757188050455448622102575 \cdot 10^{58}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.241327699195247001117817134883838709979 \cdot 10^{-253}:\\ \;\;\;\;\frac{\left(b \cdot b - b \cdot b\right) + 4 \cdot \left(c \cdot a\right)}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b} \cdot \frac{1}{a \cdot 2}\\ \mathbf{elif}\;b \le 3.628799960716311990444092539387346352569 \cdot 10^{50}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +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)))