Average Error: 33.1 → 6.7
Time: 3.8m
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.9239515900644342 \cdot 10^{+146}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.0639583319549608 \cdot 10^{-280}:\\ \;\;\;\;\frac{\frac{c}{\frac{1}{2}}}{\sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)} - b}\\ \mathbf{elif}\;b \le 2.0710701119913226 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -1.9239515900644342 \cdot 10^{+146}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 1.0639583319549608 \cdot 10^{-280}:\\
\;\;\;\;\frac{\frac{c}{\frac{1}{2}}}{\sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)} - b}\\

\mathbf{elif}\;b \le 2.0710701119913226 \cdot 10^{+79}:\\
\;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r18339470 = b;
        double r18339471 = -r18339470;
        double r18339472 = r18339470 * r18339470;
        double r18339473 = 4.0;
        double r18339474 = a;
        double r18339475 = c;
        double r18339476 = r18339474 * r18339475;
        double r18339477 = r18339473 * r18339476;
        double r18339478 = r18339472 - r18339477;
        double r18339479 = sqrt(r18339478);
        double r18339480 = r18339471 - r18339479;
        double r18339481 = 2.0;
        double r18339482 = r18339481 * r18339474;
        double r18339483 = r18339480 / r18339482;
        return r18339483;
}
↓
double f(double a, double b, double c) {
        double r18339484 = b;
        double r18339485 = -1.9239515900644342e+146;
        bool r18339486 = r18339484 <= r18339485;
        double r18339487 = c;
        double r18339488 = r18339487 / r18339484;
        double r18339489 = -r18339488;
        double r18339490 = 1.0639583319549608e-280;
        bool r18339491 = r18339484 <= r18339490;
        double r18339492 = 0.5;
        double r18339493 = r18339487 / r18339492;
        double r18339494 = a;
        double r18339495 = r18339487 * r18339494;
        double r18339496 = -4.0;
        double r18339497 = r18339484 * r18339484;
        double r18339498 = fma(r18339495, r18339496, r18339497);
        double r18339499 = sqrt(r18339498);
        double r18339500 = r18339499 - r18339484;
        double r18339501 = r18339493 / r18339500;
        double r18339502 = 2.0710701119913226e+79;
        bool r18339503 = r18339484 <= r18339502;
        double r18339504 = -r18339484;
        double r18339505 = r18339504 - r18339499;
        double r18339506 = 2.0;
        double r18339507 = r18339505 / r18339506;
        double r18339508 = r18339507 / r18339494;
        double r18339509 = r18339484 / r18339494;
        double r18339510 = r18339488 - r18339509;
        double r18339511 = r18339503 ? r18339508 : r18339510;
        double r18339512 = r18339491 ? r18339501 : r18339511;
        double r18339513 = r18339486 ? r18339489 : r18339512;
        return r18339513;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.1
Target20.4
Herbie6.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.9239515900644342e+146

    1. Initial program 62.0

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

    2. Simplified62.0

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

    3. Taylor expanded around -inf 1.7

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

    4. Simplified1.7

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

    if -1.9239515900644342e+146 < b < 1.0639583319549608e-280

    1. Initial program 33.3

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

    2. Simplified33.3

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

    4. Applied flip--33.4

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

    5. Simplified16.0

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

    6. Simplified16.0

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

    8. Applied *-un-lft-identity16.0

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

    9. Applied *-un-lft-identity16.0

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

    10. Applied times-frac16.0

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

    11. Simplified16.0

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

    12. Simplified8.7

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

    if 1.0639583319549608e-280 < b < 2.0710701119913226e+79

    1. Initial program 8.5

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

    2. Simplified8.5

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

    3. Taylor expanded around 0 8.5

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

    4. Simplified8.5

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

    if 2.0710701119913226e+79 < b

    1. Initial program 40.5

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

    2. Simplified40.4

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

    3. Taylor expanded around inf 4.7

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

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.9239515900644342 \cdot 10^{+146}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.0639583319549608 \cdot 10^{-280}:\\ \;\;\;\;\frac{\frac{c}{\frac{1}{2}}}{\sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)} - b}\\ \mathbf{elif}\;b \le 2.0710701119913226 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(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)))