Average Error: 33.7 → 10.6
Time: 26.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.363255598823911 \cdot 10^{-15}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -6.936587154412951 \cdot 10^{-28}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{1}{2 \cdot a} \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\\ \mathbf{elif}\;b \le -2.3344326820285623 \cdot 10^{-123}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 1.6691257204922504 \cdot 10^{+85}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -7.363255598823911 \cdot 10^{-15}:\\
\;\;\;\;\frac{-c}{b}\\

\mathbf{elif}\;b \le -6.936587154412951 \cdot 10^{-28}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{1}{2 \cdot a} \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\\

\mathbf{elif}\;b \le -2.3344326820285623 \cdot 10^{-123}:\\
\;\;\;\;\frac{-c}{b}\\

\mathbf{elif}\;b \le 1.6691257204922504 \cdot 10^{+85}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r3317583 = b;
        double r3317584 = -r3317583;
        double r3317585 = r3317583 * r3317583;
        double r3317586 = 4.0;
        double r3317587 = a;
        double r3317588 = c;
        double r3317589 = r3317587 * r3317588;
        double r3317590 = r3317586 * r3317589;
        double r3317591 = r3317585 - r3317590;
        double r3317592 = sqrt(r3317591);
        double r3317593 = r3317584 - r3317592;
        double r3317594 = 2.0;
        double r3317595 = r3317594 * r3317587;
        double r3317596 = r3317593 / r3317595;
        return r3317596;
}
↓
double f(double a, double b, double c) {
        double r3317597 = b;
        double r3317598 = -7.363255598823911e-15;
        bool r3317599 = r3317597 <= r3317598;
        double r3317600 = c;
        double r3317601 = -r3317600;
        double r3317602 = r3317601 / r3317597;
        double r3317603 = -6.936587154412951e-28;
        bool r3317604 = r3317597 <= r3317603;
        double r3317605 = -r3317597;
        double r3317606 = 2.0;
        double r3317607 = a;
        double r3317608 = r3317606 * r3317607;
        double r3317609 = r3317605 / r3317608;
        double r3317610 = 1.0;
        double r3317611 = r3317610 / r3317608;
        double r3317612 = r3317597 * r3317597;
        double r3317613 = r3317607 * r3317600;
        double r3317614 = 4.0;
        double r3317615 = r3317613 * r3317614;
        double r3317616 = r3317612 - r3317615;
        double r3317617 = sqrt(r3317616);
        double r3317618 = r3317611 * r3317617;
        double r3317619 = r3317609 - r3317618;
        double r3317620 = -2.3344326820285623e-123;
        bool r3317621 = r3317597 <= r3317620;
        double r3317622 = 1.6691257204922504e+85;
        bool r3317623 = r3317597 <= r3317622;
        double r3317624 = r3317608 / r3317617;
        double r3317625 = r3317610 / r3317624;
        double r3317626 = r3317609 - r3317625;
        double r3317627 = r3317600 / r3317597;
        double r3317628 = r3317597 / r3317607;
        double r3317629 = r3317627 - r3317628;
        double r3317630 = r3317623 ? r3317626 : r3317629;
        double r3317631 = r3317621 ? r3317602 : r3317630;
        double r3317632 = r3317604 ? r3317619 : r3317631;
        double r3317633 = r3317599 ? r3317602 : r3317632;
        return r3317633;
}

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

Original33.7
Target21.0
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -7.363255598823911e-15 or -6.936587154412951e-28 < b < -2.3344326820285623e-123

    1. Initial program 50.9

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

    3. Applied div-sub51.4

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

    5. Applied clear-num52.2

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

    6. Taylor expanded around -inf 10.6

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

    7. Simplified10.6

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

    if -7.363255598823911e-15 < b < -6.936587154412951e-28

    1. Initial program 35.8

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

    3. Applied div-sub35.8

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

    5. Applied div-inv35.9

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

    if -2.3344326820285623e-123 < b < 1.6691257204922504e+85

    1. Initial program 12.6

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

    3. Applied div-sub12.6

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

    5. Applied clear-num12.7

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

    if 1.6691257204922504e+85 < b

    1. Initial program 42.9

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

    2. Taylor expanded around inf 3.7

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.363255598823911 \cdot 10^{-15}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -6.936587154412951 \cdot 10^{-28}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{1}{2 \cdot a} \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\\ \mathbf{elif}\;b \le -2.3344326820285623 \cdot 10^{-123}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 1.6691257204922504 \cdot 10^{+85}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :herbie-target
  (if (< b 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)))