Average Error: 33.2 → 9.9
Time: 31.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.8774910265390396 \cdot 10^{-73}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 2.5703497435733685 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{1}{\frac{1}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}}{a}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \end{array}\]
\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 -1.8774910265390396 \cdot 10^{-73}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\mathbf{elif}\;b \le 2.5703497435733685 \cdot 10^{+102}:\\
\;\;\;\;\frac{\frac{1}{\frac{1}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}}{a}}}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r1469290 = b;
        double r1469291 = -r1469290;
        double r1469292 = r1469290 * r1469290;
        double r1469293 = 4.0;
        double r1469294 = a;
        double r1469295 = c;
        double r1469296 = r1469294 * r1469295;
        double r1469297 = r1469293 * r1469296;
        double r1469298 = r1469292 - r1469297;
        double r1469299 = sqrt(r1469298);
        double r1469300 = r1469291 - r1469299;
        double r1469301 = 2.0;
        double r1469302 = r1469301 * r1469294;
        double r1469303 = r1469300 / r1469302;
        return r1469303;
}
↓
double f(double a, double b, double c) {
        double r1469304 = b;
        double r1469305 = -1.8774910265390396e-73;
        bool r1469306 = r1469304 <= r1469305;
        double r1469307 = -2.0;
        double r1469308 = c;
        double r1469309 = r1469308 / r1469304;
        double r1469310 = r1469307 * r1469309;
        double r1469311 = 2.0;
        double r1469312 = r1469310 / r1469311;
        double r1469313 = 2.5703497435733685e+102;
        bool r1469314 = r1469304 <= r1469313;
        double r1469315 = 1.0;
        double r1469316 = -r1469304;
        double r1469317 = a;
        double r1469318 = -4.0;
        double r1469319 = r1469317 * r1469318;
        double r1469320 = r1469304 * r1469304;
        double r1469321 = fma(r1469319, r1469308, r1469320);
        double r1469322 = sqrt(r1469321);
        double r1469323 = r1469316 - r1469322;
        double r1469324 = r1469323 / r1469317;
        double r1469325 = r1469315 / r1469324;
        double r1469326 = r1469315 / r1469325;
        double r1469327 = r1469326 / r1469311;
        double r1469328 = r1469304 / r1469317;
        double r1469329 = r1469309 - r1469328;
        double r1469330 = r1469329 * r1469311;
        double r1469331 = r1469330 / r1469311;
        double r1469332 = r1469314 ? r1469327 : r1469331;
        double r1469333 = r1469306 ? r1469312 : r1469332;
        return r1469333;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.2
Target20.4
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.8774910265390396e-73

    1. Initial program 52.5

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

    2. Simplified52.5

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

    4. Applied *-un-lft-identity52.5

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

    5. Applied *-un-lft-identity52.5

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

    6. Applied distribute-lft-out--52.5

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

    7. Applied associate-/l*52.5

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

    9. Applied *-un-lft-identity52.5

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

    10. Applied associate-/l*52.5

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

    11. Taylor expanded around -inf 8.6

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]

    if -1.8774910265390396e-73 < b < 2.5703497435733685e+102

    1. Initial program 13.1

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

    2. Simplified13.1

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

    4. Applied *-un-lft-identity13.1

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

    5. Applied *-un-lft-identity13.1

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

    6. Applied distribute-lft-out--13.1

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

    7. Applied associate-/l*13.2

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

    9. Applied *-un-lft-identity13.2

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

    10. Applied associate-/l*13.2

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

    if 2.5703497435733685e+102 < b

    1. Initial program 43.9

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

    2. Simplified43.9

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

    3. Taylor expanded around inf 3.0

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]

    4. Simplified3.0

      \[\leadsto \frac{\color{blue}{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}}{2}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.8774910265390396 \cdot 10^{-73}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 2.5703497435733685 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{1}{\frac{1}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}}{a}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +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)))