Average Error: 33.7 → 10.9
Time: 6.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.98276540088900058 \cdot 10^{134}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.1860189201379418 \cdot 10^{-161}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.98276540088900058 \cdot 10^{134}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.1860189201379418 \cdot 10^{-161}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r180379 = b;
        double r180380 = -r180379;
        double r180381 = r180379 * r180379;
        double r180382 = 4.0;
        double r180383 = a;
        double r180384 = r180382 * r180383;
        double r180385 = c;
        double r180386 = r180384 * r180385;
        double r180387 = r180381 - r180386;
        double r180388 = sqrt(r180387);
        double r180389 = r180380 + r180388;
        double r180390 = 2.0;
        double r180391 = r180390 * r180383;
        double r180392 = r180389 / r180391;
        return r180392;
}
↓
double f(double a, double b, double c) {
        double r180393 = b;
        double r180394 = -1.9827654008890006e+134;
        bool r180395 = r180393 <= r180394;
        double r180396 = 1.0;
        double r180397 = c;
        double r180398 = r180397 / r180393;
        double r180399 = a;
        double r180400 = r180393 / r180399;
        double r180401 = r180398 - r180400;
        double r180402 = r180396 * r180401;
        double r180403 = 1.1860189201379418e-161;
        bool r180404 = r180393 <= r180403;
        double r180405 = -r180393;
        double r180406 = r180393 * r180393;
        double r180407 = 4.0;
        double r180408 = r180407 * r180399;
        double r180409 = r180408 * r180397;
        double r180410 = r180406 - r180409;
        double r180411 = sqrt(r180410);
        double r180412 = r180405 + r180411;
        double r180413 = 1.0;
        double r180414 = 2.0;
        double r180415 = r180414 * r180399;
        double r180416 = r180413 / r180415;
        double r180417 = r180412 * r180416;
        double r180418 = -1.0;
        double r180419 = r180418 * r180398;
        double r180420 = r180404 ? r180417 : r180419;
        double r180421 = r180395 ? r180402 : r180420;
        return r180421;
}

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

Derivation

  1. Split input into 3 regimes

  2. if b < -1.9827654008890006e+134

    1. Initial program 56.8

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

    2. Taylor expanded around -inf 3.1

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

    3. Simplified3.1

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

    if -1.9827654008890006e+134 < b < 1.1860189201379418e-161

    1. Initial program 10.3

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

    3. Applied div-inv10.5

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

    if 1.1860189201379418e-161 < b

    1. Initial program 49.7

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

    2. Taylor expanded around inf 13.7

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

  4. Final simplification10.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.98276540088900058 \cdot 10^{134}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.1860189201379418 \cdot 10^{-161}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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)))