Average Error: 34.7 → 8.5
Time: 37.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.270528699455007486596308100489334356636 \cdot 10^{152}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 2.1777947371956334507732300386925067972 \cdot 10^{-143}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 123203260287366115360768:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - b \cdot b\right) + c \cdot \left(a \cdot 4\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]
\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 -1.270528699455007486596308100489334356636 \cdot 10^{152}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\mathbf{elif}\;b \le 2.1777947371956334507732300386925067972 \cdot 10^{-143}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} - b}{a \cdot 2}\\

\mathbf{elif}\;b \le 123203260287366115360768:\\
\;\;\;\;\frac{\frac{\left(b \cdot b - b \cdot b\right) + c \cdot \left(a \cdot 4\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{a \cdot 2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3650419 = b;
        double r3650420 = -r3650419;
        double r3650421 = r3650419 * r3650419;
        double r3650422 = 4.0;
        double r3650423 = a;
        double r3650424 = r3650422 * r3650423;
        double r3650425 = c;
        double r3650426 = r3650424 * r3650425;
        double r3650427 = r3650421 - r3650426;
        double r3650428 = sqrt(r3650427);
        double r3650429 = r3650420 + r3650428;
        double r3650430 = 2.0;
        double r3650431 = r3650430 * r3650423;
        double r3650432 = r3650429 / r3650431;
        return r3650432;
}
↓
double f(double a, double b, double c) {
        double r3650433 = b;
        double r3650434 = -1.2705286994550075e+152;
        bool r3650435 = r3650433 <= r3650434;
        double r3650436 = c;
        double r3650437 = r3650436 / r3650433;
        double r3650438 = a;
        double r3650439 = r3650433 / r3650438;
        double r3650440 = r3650437 - r3650439;
        double r3650441 = 1.0;
        double r3650442 = r3650440 * r3650441;
        double r3650443 = 2.1777947371956335e-143;
        bool r3650444 = r3650433 <= r3650443;
        double r3650445 = r3650433 * r3650433;
        double r3650446 = 4.0;
        double r3650447 = r3650436 * r3650446;
        double r3650448 = r3650447 * r3650438;
        double r3650449 = r3650445 - r3650448;
        double r3650450 = sqrt(r3650449);
        double r3650451 = r3650450 - r3650433;
        double r3650452 = 2.0;
        double r3650453 = r3650438 * r3650452;
        double r3650454 = r3650451 / r3650453;
        double r3650455 = 1.2320326028736612e+23;
        bool r3650456 = r3650433 <= r3650455;
        double r3650457 = r3650445 - r3650445;
        double r3650458 = r3650438 * r3650446;
        double r3650459 = r3650436 * r3650458;
        double r3650460 = r3650457 + r3650459;
        double r3650461 = -r3650433;
        double r3650462 = r3650445 - r3650459;
        double r3650463 = sqrt(r3650462);
        double r3650464 = r3650461 - r3650463;
        double r3650465 = r3650460 / r3650464;
        double r3650466 = r3650465 / r3650453;
        double r3650467 = -1.0;
        double r3650468 = r3650437 * r3650467;
        double r3650469 = r3650456 ? r3650466 : r3650468;
        double r3650470 = r3650444 ? r3650454 : r3650469;
        double r3650471 = r3650435 ? r3650442 : r3650470;
        return r3650471;
}

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

Original34.7
Target21.1
Herbie8.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.2705286994550075e+152

    1. Initial program 62.9

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

    2. Taylor expanded around -inf 1.7

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

    3. Simplified1.7

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

    if -1.2705286994550075e+152 < b < 2.1777947371956335e-143

    1. Initial program 10.4

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

    3. Applied div-inv10.5

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

    5. Applied associate-*r/10.4

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

    6. Simplified10.4

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

    if 2.1777947371956335e-143 < b < 1.2320326028736612e+23

    1. Initial program 36.9

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

    3. Applied flip-+37.0

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

    4. Simplified18.3

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

    if 1.2320326028736612e+23 < b

    1. Initial program 56.3

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

    2. Taylor expanded around inf 4.4

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

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.270528699455007486596308100489334356636 \cdot 10^{152}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 2.1777947371956334507732300386925067972 \cdot 10^{-143}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 123203260287366115360768:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - b \cdot b\right) + c \cdot \left(a \cdot 4\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))