Average Error: 34.3 → 8.6
Time: 29.8s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9661478263987.111328125:\\ \;\;\;\;\frac{c \cdot \sqrt[3]{-1}}{b} \cdot \left(\sqrt[3]{-1} \cdot \sqrt[3]{-1}\right)\\ \mathbf{elif}\;b \le -1.244932636718084290671504385697528170065 \cdot 10^{-183}:\\ \;\;\;\;\frac{\frac{a \cdot \left(c \cdot 4\right) + \left({b}^{2} - {b}^{2}\right)}{\sqrt{\mathsf{fma}\left(a, 4 \cdot \left(-c\right), {b}^{2}\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le 2.280923374767716130571300401308257426651 \cdot 10^{83}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{\mathsf{fma}\left(a, 4 \cdot \left(-c\right), {b}^{2}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -9661478263987.111328125:\\
\;\;\;\;\frac{c \cdot \sqrt[3]{-1}}{b} \cdot \left(\sqrt[3]{-1} \cdot \sqrt[3]{-1}\right)\\

\mathbf{elif}\;b \le -1.244932636718084290671504385697528170065 \cdot 10^{-183}:\\
\;\;\;\;\frac{\frac{a \cdot \left(c \cdot 4\right) + \left({b}^{2} - {b}^{2}\right)}{\sqrt{\mathsf{fma}\left(a, 4 \cdot \left(-c\right), {b}^{2}\right)} - b}}{2 \cdot a}\\

\mathbf{elif}\;b \le 2.280923374767716130571300401308257426651 \cdot 10^{83}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{\mathsf{fma}\left(a, 4 \cdot \left(-c\right), {b}^{2}\right)}}{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 r71461 = b;
        double r71462 = -r71461;
        double r71463 = r71461 * r71461;
        double r71464 = 4.0;
        double r71465 = a;
        double r71466 = c;
        double r71467 = r71465 * r71466;
        double r71468 = r71464 * r71467;
        double r71469 = r71463 - r71468;
        double r71470 = sqrt(r71469);
        double r71471 = r71462 - r71470;
        double r71472 = 2.0;
        double r71473 = r71472 * r71465;
        double r71474 = r71471 / r71473;
        return r71474;
}
↓
double f(double a, double b, double c) {
        double r71475 = b;
        double r71476 = -9661478263987.111;
        bool r71477 = r71475 <= r71476;
        double r71478 = c;
        double r71479 = -1.0;
        double r71480 = cbrt(r71479);
        double r71481 = r71478 * r71480;
        double r71482 = r71481 / r71475;
        double r71483 = r71480 * r71480;
        double r71484 = r71482 * r71483;
        double r71485 = -1.2449326367180843e-183;
        bool r71486 = r71475 <= r71485;
        double r71487 = a;
        double r71488 = 4.0;
        double r71489 = r71478 * r71488;
        double r71490 = r71487 * r71489;
        double r71491 = 2.0;
        double r71492 = pow(r71475, r71491);
        double r71493 = r71492 - r71492;
        double r71494 = r71490 + r71493;
        double r71495 = -r71478;
        double r71496 = r71488 * r71495;
        double r71497 = fma(r71487, r71496, r71492);
        double r71498 = sqrt(r71497);
        double r71499 = r71498 - r71475;
        double r71500 = r71494 / r71499;
        double r71501 = 2.0;
        double r71502 = r71501 * r71487;
        double r71503 = r71500 / r71502;
        double r71504 = 2.280923374767716e+83;
        bool r71505 = r71475 <= r71504;
        double r71506 = -r71475;
        double r71507 = r71506 / r71502;
        double r71508 = r71498 / r71502;
        double r71509 = r71507 - r71508;
        double r71510 = 1.0;
        double r71511 = r71478 / r71475;
        double r71512 = r71475 / r71487;
        double r71513 = r71511 - r71512;
        double r71514 = r71510 * r71513;
        double r71515 = r71505 ? r71509 : r71514;
        double r71516 = r71486 ? r71503 : r71515;
        double r71517 = r71477 ? r71484 : r71516;
        return r71517;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.3
Target21.1
Herbie8.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -9661478263987.111

    1. Initial program 56.4

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

    2. Taylor expanded around -inf 5.0

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

    3. Simplified5.9

      \[\leadsto \color{blue}{\frac{-1}{\frac{b}{c}}}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity5.9

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

    6. Applied *-un-lft-identity5.9

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

    7. Applied times-frac5.9

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

    8. Applied add-cube-cbrt5.9

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{-1} \cdot \sqrt[3]{-1}\right) \cdot \sqrt[3]{-1}}}{\frac{1}{1} \cdot \frac{b}{c}}\]

    9. Applied times-frac5.9

      \[\leadsto \color{blue}{\frac{\sqrt[3]{-1} \cdot \sqrt[3]{-1}}{\frac{1}{1}} \cdot \frac{\sqrt[3]{-1}}{\frac{b}{c}}}\]

    10. Simplified5.9

      \[\leadsto \color{blue}{\left(\sqrt[3]{-1} \cdot \sqrt[3]{-1}\right)} \cdot \frac{\sqrt[3]{-1}}{\frac{b}{c}}\]

    11. Simplified5.0

      \[\leadsto \left(\sqrt[3]{-1} \cdot \sqrt[3]{-1}\right) \cdot \color{blue}{\frac{\sqrt[3]{-1} \cdot c}{b}}\]

    if -9661478263987.111 < b < -1.2449326367180843e-183

    1. Initial program 34.1

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

    3. Applied flip--34.1

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

    4. Simplified18.3

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

    5. Simplified18.3

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

    if -1.2449326367180843e-183 < b < 2.280923374767716e+83

    1. Initial program 10.5

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

    3. Applied div-sub10.5

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

    4. Simplified10.5

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

    5. Simplified10.5

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

    if 2.280923374767716e+83 < b

    1. Initial program 43.8

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

    2. Taylor expanded around inf 3.5

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

    3. Simplified3.5

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9661478263987.111328125:\\ \;\;\;\;\frac{c \cdot \sqrt[3]{-1}}{b} \cdot \left(\sqrt[3]{-1} \cdot \sqrt[3]{-1}\right)\\ \mathbf{elif}\;b \le -1.244932636718084290671504385697528170065 \cdot 10^{-183}:\\ \;\;\;\;\frac{\frac{a \cdot \left(c \cdot 4\right) + \left({b}^{2} - {b}^{2}\right)}{\sqrt{\mathsf{fma}\left(a, 4 \cdot \left(-c\right), {b}^{2}\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le 2.280923374767716130571300401308257426651 \cdot 10^{83}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{\mathsf{fma}\left(a, 4 \cdot \left(-c\right), {b}^{2}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"

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

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