Average Error: 34.3 → 6.8
Time: 18.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.336444958885988609650625284454613720554 \cdot 10^{152}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.866557489300444694985602640640979475553 \cdot 10^{-301}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le 2.360908374051695590422690701366520544893 \cdot 10^{86}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -3.336444958885988609650625284454613720554 \cdot 10^{152}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 2.866557489300444694985602640640979475553 \cdot 10^{-301}:\\
\;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\

\mathbf{elif}\;b \le 2.360908374051695590422690701366520544893 \cdot 10^{86}:\\
\;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{a \cdot 2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r5189461 = b;
        double r5189462 = -r5189461;
        double r5189463 = r5189461 * r5189461;
        double r5189464 = 4.0;
        double r5189465 = a;
        double r5189466 = c;
        double r5189467 = r5189465 * r5189466;
        double r5189468 = r5189464 * r5189467;
        double r5189469 = r5189463 - r5189468;
        double r5189470 = sqrt(r5189469);
        double r5189471 = r5189462 - r5189470;
        double r5189472 = 2.0;
        double r5189473 = r5189472 * r5189465;
        double r5189474 = r5189471 / r5189473;
        return r5189474;
}
↓
double f(double a, double b, double c) {
        double r5189475 = b;
        double r5189476 = -3.3364449588859886e+152;
        bool r5189477 = r5189475 <= r5189476;
        double r5189478 = -1.0;
        double r5189479 = c;
        double r5189480 = r5189479 / r5189475;
        double r5189481 = r5189478 * r5189480;
        double r5189482 = 2.8665574893004447e-301;
        bool r5189483 = r5189475 <= r5189482;
        double r5189484 = 2.0;
        double r5189485 = r5189479 * r5189484;
        double r5189486 = -r5189475;
        double r5189487 = r5189475 * r5189475;
        double r5189488 = 4.0;
        double r5189489 = a;
        double r5189490 = r5189489 * r5189479;
        double r5189491 = r5189488 * r5189490;
        double r5189492 = r5189487 - r5189491;
        double r5189493 = sqrt(r5189492);
        double r5189494 = r5189486 + r5189493;
        double r5189495 = r5189485 / r5189494;
        double r5189496 = 2.3609083740516956e+86;
        bool r5189497 = r5189475 <= r5189496;
        double r5189498 = r5189486 - r5189493;
        double r5189499 = 1.0;
        double r5189500 = r5189489 * r5189484;
        double r5189501 = r5189499 / r5189500;
        double r5189502 = r5189498 * r5189501;
        double r5189503 = 1.0;
        double r5189504 = r5189475 / r5189489;
        double r5189505 = r5189480 - r5189504;
        double r5189506 = r5189503 * r5189505;
        double r5189507 = r5189497 ? r5189502 : r5189506;
        double r5189508 = r5189483 ? r5189495 : r5189507;
        double r5189509 = r5189477 ? r5189481 : r5189508;
        return r5189509;
}

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

Original34.3
Target21.3
Herbie6.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.3364449588859886e+152

    1. Initial program 63.9

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

    2. Taylor expanded around -inf 1.6

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

    if -3.3364449588859886e+152 < b < 2.8665574893004447e-301

    1. Initial program 35.0

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

    3. Applied div-inv35.0

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

    5. Applied flip--35.0

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

    6. Applied associate-*l/35.0

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

    7. Simplified15.0

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

    8. Taylor expanded around 0 8.9

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

    if 2.8665574893004447e-301 < b < 2.3609083740516956e+86

    1. Initial program 9.0

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

    3. Applied div-inv9.1

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

    if 2.3609083740516956e+86 < b

    1. Initial program 44.1

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

    2. Taylor expanded around inf 3.9

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

    3. Simplified3.9

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.336444958885988609650625284454613720554 \cdot 10^{152}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.866557489300444694985602640640979475553 \cdot 10^{-301}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le 2.360908374051695590422690701366520544893 \cdot 10^{86}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(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)))