Average Error: 28.9 → 16.2
Time: 34.5s
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 1083.7283358723973:\\ \;\;\;\;\frac{\frac{\left(\left(-3 \cdot c\right) \cdot a + b \cdot b\right) \cdot \sqrt{\left(-3 \cdot c\right) \cdot a + b \cdot b} - b \cdot \left(b \cdot b\right)}{b \cdot \sqrt{\left(-3 \cdot c\right) \cdot a + b \cdot b} + \left(b \cdot b + \left(\left(-3 \cdot c\right) \cdot a + b \cdot b\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le 1083.7283358723973:\\
\;\;\;\;\frac{\frac{\left(\left(-3 \cdot c\right) \cdot a + b \cdot b\right) \cdot \sqrt{\left(-3 \cdot c\right) \cdot a + b \cdot b} - b \cdot \left(b \cdot b\right)}{b \cdot \sqrt{\left(-3 \cdot c\right) \cdot a + b \cdot b} + \left(b \cdot b + \left(\left(-3 \cdot c\right) \cdot a + b \cdot b\right)\right)}}{3 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r4038550 = b;
        double r4038551 = -r4038550;
        double r4038552 = r4038550 * r4038550;
        double r4038553 = 3.0;
        double r4038554 = a;
        double r4038555 = r4038553 * r4038554;
        double r4038556 = c;
        double r4038557 = r4038555 * r4038556;
        double r4038558 = r4038552 - r4038557;
        double r4038559 = sqrt(r4038558);
        double r4038560 = r4038551 + r4038559;
        double r4038561 = r4038560 / r4038555;
        return r4038561;
}
↓
double f(double a, double b, double c) {
        double r4038562 = b;
        double r4038563 = 1083.7283358723973;
        bool r4038564 = r4038562 <= r4038563;
        double r4038565 = -3.0;
        double r4038566 = c;
        double r4038567 = r4038565 * r4038566;
        double r4038568 = a;
        double r4038569 = r4038567 * r4038568;
        double r4038570 = r4038562 * r4038562;
        double r4038571 = r4038569 + r4038570;
        double r4038572 = sqrt(r4038571);
        double r4038573 = r4038571 * r4038572;
        double r4038574 = r4038562 * r4038570;
        double r4038575 = r4038573 - r4038574;
        double r4038576 = r4038562 * r4038572;
        double r4038577 = r4038570 + r4038571;
        double r4038578 = r4038576 + r4038577;
        double r4038579 = r4038575 / r4038578;
        double r4038580 = 3.0;
        double r4038581 = r4038580 * r4038568;
        double r4038582 = r4038579 / r4038581;
        double r4038583 = -0.5;
        double r4038584 = r4038566 / r4038562;
        double r4038585 = r4038583 * r4038584;
        double r4038586 = r4038564 ? r4038582 : r4038585;
        return r4038586;
}

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

Derivation

  1. Split input into 2 regimes

  2. if b < 1083.7283358723973

    1. Initial program 17.5

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

    2. Simplified17.5

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

    4. Applied flip3--17.6

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

    5. Simplified16.9

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

    6. Simplified16.9

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

    if 1083.7283358723973 < b

    1. Initial program 36.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

    2. Simplified36.9

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

    3. Taylor expanded around inf 15.8

      \[\leadsto \frac{\color{blue}{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]
    4. Using strategy rm

    5. Applied clear-num15.8

      \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}}}\]

    6. Simplified15.7

      \[\leadsto \frac{1}{\color{blue}{\frac{b}{\frac{-1}{2} \cdot c}}}\]

    7. Taylor expanded around -inf 15.7

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1083.7283358723973:\\ \;\;\;\;\frac{\frac{\left(\left(-3 \cdot c\right) \cdot a + b \cdot b\right) \cdot \sqrt{\left(-3 \cdot c\right) \cdot a + b \cdot b} - b \cdot \left(b \cdot b\right)}{b \cdot \sqrt{\left(-3 \cdot c\right) \cdot a + b \cdot b} + \left(b \cdot b + \left(\left(-3 \cdot c\right) \cdot a + b \cdot b\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))