Average Error: 32.6 → 9.7
Time: 23.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.90131991727783 \cdot 10^{-39}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 4.012768074517757 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{\frac{1}{2}}{a}}{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]
\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 -6.90131991727783 \cdot 10^{-39}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 4.012768074517757 \cdot 10^{+87}:\\
\;\;\;\;\frac{\frac{\frac{1}{2}}{a}}{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2585534 = b;
        double r2585535 = -r2585534;
        double r2585536 = r2585534 * r2585534;
        double r2585537 = 4.0;
        double r2585538 = a;
        double r2585539 = c;
        double r2585540 = r2585538 * r2585539;
        double r2585541 = r2585537 * r2585540;
        double r2585542 = r2585536 - r2585541;
        double r2585543 = sqrt(r2585542);
        double r2585544 = r2585535 - r2585543;
        double r2585545 = 2.0;
        double r2585546 = r2585545 * r2585538;
        double r2585547 = r2585544 / r2585546;
        return r2585547;
}
↓
double f(double a, double b, double c) {
        double r2585548 = b;
        double r2585549 = -6.90131991727783e-39;
        bool r2585550 = r2585548 <= r2585549;
        double r2585551 = c;
        double r2585552 = r2585551 / r2585548;
        double r2585553 = -r2585552;
        double r2585554 = 4.012768074517757e+87;
        bool r2585555 = r2585548 <= r2585554;
        double r2585556 = 0.5;
        double r2585557 = a;
        double r2585558 = r2585556 / r2585557;
        double r2585559 = 1.0;
        double r2585560 = -r2585548;
        double r2585561 = r2585548 * r2585548;
        double r2585562 = r2585551 * r2585557;
        double r2585563 = 4.0;
        double r2585564 = r2585562 * r2585563;
        double r2585565 = r2585561 - r2585564;
        double r2585566 = sqrt(r2585565);
        double r2585567 = r2585560 - r2585566;
        double r2585568 = r2585559 / r2585567;
        double r2585569 = r2585558 / r2585568;
        double r2585570 = r2585548 / r2585557;
        double r2585571 = -r2585570;
        double r2585572 = r2585555 ? r2585569 : r2585571;
        double r2585573 = r2585550 ? r2585553 : r2585572;
        return r2585573;
}

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

Original32.6
Target20.0
Herbie9.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -6.90131991727783e-39

    1. Initial program 53.0

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

    3. Applied div-inv53.1

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

    4. Simplified53.1

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

    5. Taylor expanded around -inf 7.9

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

    6. Simplified7.9

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

    if -6.90131991727783e-39 < b < 4.012768074517757e+87

    1. Initial program 13.2

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

    3. Applied *-un-lft-identity13.2

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

    4. Applied *-un-lft-identity13.2

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

    5. Applied distribute-rgt-neg-in13.2

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

    6. Applied distribute-lft-out--13.2

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

    7. Applied associate-/l*13.4

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

    9. Applied div-inv13.4

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

    10. Applied associate-/r*13.4

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

    11. Simplified13.4

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

    if 4.012768074517757e+87 < b

    1. Initial program 41.3

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

    3. Applied *-un-lft-identity41.3

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

    4. Applied *-un-lft-identity41.3

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

    5. Applied distribute-rgt-neg-in41.3

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

    6. Applied distribute-lft-out--41.3

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

    7. Applied associate-/l*41.3

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

    8. Taylor expanded around 0 3.3

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

    9. Simplified3.3

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

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.90131991727783 \cdot 10^{-39}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 4.012768074517757 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{\frac{1}{2}}{a}}{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 
(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)))