Average Error: 33.7 → 10.5
Time: 4.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.3295118613703302 \cdot 10^{-13}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.29545095081340793 \cdot 10^{65}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -1.3295118613703302 \cdot 10^{-13}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 3.29545095081340793 \cdot 10^{65}:\\
\;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r80458 = b;
        double r80459 = -r80458;
        double r80460 = r80458 * r80458;
        double r80461 = 4.0;
        double r80462 = a;
        double r80463 = c;
        double r80464 = r80462 * r80463;
        double r80465 = r80461 * r80464;
        double r80466 = r80460 - r80465;
        double r80467 = sqrt(r80466);
        double r80468 = r80459 - r80467;
        double r80469 = 2.0;
        double r80470 = r80469 * r80462;
        double r80471 = r80468 / r80470;
        return r80471;
}
↓
double f(double a, double b, double c) {
        double r80472 = b;
        double r80473 = -1.3295118613703302e-13;
        bool r80474 = r80472 <= r80473;
        double r80475 = -1.0;
        double r80476 = c;
        double r80477 = r80476 / r80472;
        double r80478 = r80475 * r80477;
        double r80479 = 3.295450950813408e+65;
        bool r80480 = r80472 <= r80479;
        double r80481 = -r80472;
        double r80482 = r80472 * r80472;
        double r80483 = 4.0;
        double r80484 = a;
        double r80485 = r80484 * r80476;
        double r80486 = r80483 * r80485;
        double r80487 = r80482 - r80486;
        double r80488 = sqrt(r80487);
        double r80489 = r80481 - r80488;
        double r80490 = 1.0;
        double r80491 = 2.0;
        double r80492 = r80491 * r80484;
        double r80493 = r80490 / r80492;
        double r80494 = r80489 * r80493;
        double r80495 = 1.0;
        double r80496 = r80472 / r80484;
        double r80497 = r80477 - r80496;
        double r80498 = r80495 * r80497;
        double r80499 = r80480 ? r80494 : r80498;
        double r80500 = r80474 ? r80478 : r80499;
        return r80500;
}

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
Herbie10.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.3295118613703302e-13

    1. Initial program 55.0

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

    2. Taylor expanded around -inf 6.9

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

    if -1.3295118613703302e-13 < b < 3.295450950813408e+65

    1. Initial program 15.4

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

    3. Applied div-inv15.5

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

    if 3.295450950813408e+65 < b

    1. Initial program 40.0

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

    2. Taylor expanded around inf 4.7

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

    3. Simplified4.7

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3295118613703302 \cdot 10^{-13}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.29545095081340793 \cdot 10^{65}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ 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)))