Cubic critical, medium range

Average Error: 43.6 → 10.1

Time: 23.5s

Precision: 64

\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c}}{a \cdot 3} \le -1.584861931461741305603163620929763055756 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 3} - b \cdot \left(b \cdot b\right)}{b \cdot b + \left(b \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 3} + \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r3037571 = b;
        double r3037572 = -r3037571;
        double r3037573 = r3037571 * r3037571;
        double r3037574 = 3.0;
        double r3037575 = a;
        double r3037576 = r3037574 * r3037575;
        double r3037577 = c;
        double r3037578 = r3037576 * r3037577;
        double r3037579 = r3037573 - r3037578;
        double r3037580 = sqrt(r3037579);
        double r3037581 = r3037572 + r3037580;
        double r3037582 = r3037581 / r3037576;
        return r3037582;
}

↓

double f(double a, double b, double c) {
        double r3037583 = b;
        double r3037584 = -r3037583;
        double r3037585 = r3037583 * r3037583;
        double r3037586 = a;
        double r3037587 = 3.0;
        double r3037588 = r3037586 * r3037587;
        double r3037589 = c;
        double r3037590 = r3037588 * r3037589;
        double r3037591 = r3037585 - r3037590;
        double r3037592 = sqrt(r3037591);
        double r3037593 = r3037584 + r3037592;
        double r3037594 = r3037593 / r3037588;
        double r3037595 = -1.5848619314617413e-05;
        bool r3037596 = r3037594 <= r3037595;
        double r3037597 = r3037589 * r3037586;
        double r3037598 = r3037597 * r3037587;
        double r3037599 = r3037585 - r3037598;
        double r3037600 = sqrt(r3037599);
        double r3037601 = r3037599 * r3037600;
        double r3037602 = r3037583 * r3037585;
        double r3037603 = r3037601 - r3037602;
        double r3037604 = r3037583 * r3037600;
        double r3037605 = r3037604 + r3037599;
        double r3037606 = r3037585 + r3037605;
        double r3037607 = r3037603 / r3037606;
        double r3037608 = r3037607 / r3037588;
        double r3037609 = -0.5;
        double r3037610 = r3037589 / r3037583;
        double r3037611 = r3037609 * r3037610;
        double r3037612 = r3037596 ? r3037608 : r3037611;
        return r3037612;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 2 regimes

2. if (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) < -1.5848619314617413e-05

   1. Initial program 21.1

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

   3. Applied flip3-+ 21.1

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

   4. Simplified 20.4

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

   5. Simplified 20.4

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

   if -1.5848619314617413e-05 < (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a))

   1. Initial program 51.5

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

   2. Taylor expanded around inf 6.5

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 10.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c}}{a \cdot 3} \le -1.584861931461741305603163620929763055756 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 3} - b \cdot \left(b \cdot b\right)}{b \cdot b + \left(b \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 3} + \left(b \cdot b - \left(c \cdot a\right) \cdot 3\right)\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))