Cubic critical, medium range

Average Error: 43.5 → 13.1

Time: 16.2s

Precision: 64

• unused variable d

\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le 572469477.8805368:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \mathbf{elif}\;a \le 437014546157.40485:\\ \;\;\;\;\frac{\frac{\sqrt{\left(c \cdot a\right) \cdot -3 + b \cdot b} \cdot \left(\left(c \cdot a\right) \cdot -3 + b \cdot b\right) - b \cdot \left(b \cdot b\right)}{\left(\left(c \cdot a\right) \cdot -3 + b \cdot b\right) + \left(\sqrt{\left(c \cdot a\right) \cdot -3 + b \cdot b} \cdot b + b \cdot b\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

C

double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r2941554 = b;
        double r2941555 = -r2941554;
        double r2941556 = r2941554 * r2941554;
        double r2941557 = 3.0;
        double r2941558 = a;
        double r2941559 = r2941557 * r2941558;
        double r2941560 = c;
        double r2941561 = r2941559 * r2941560;
        double r2941562 = r2941556 - r2941561;
        double r2941563 = sqrt(r2941562);
        double r2941564 = r2941555 + r2941563;
        double r2941565 = r2941564 / r2941559;
        return r2941565;
}

↓

double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r2941566 = a;
        double r2941567 = 572469477.8805368;
        bool r2941568 = r2941566 <= r2941567;
        double r2941569 = -0.5;
        double r2941570 = c;
        double r2941571 = b;
        double r2941572 = r2941570 / r2941571;
        double r2941573 = r2941569 * r2941572;
        double r2941574 = 437014546157.40485;
        bool r2941575 = r2941566 <= r2941574;
        double r2941576 = r2941570 * r2941566;
        double r2941577 = -3.0;
        double r2941578 = r2941576 * r2941577;
        double r2941579 = r2941571 * r2941571;
        double r2941580 = r2941578 + r2941579;
        double r2941581 = sqrt(r2941580);
        double r2941582 = r2941581 * r2941580;
        double r2941583 = r2941571 * r2941579;
        double r2941584 = r2941582 - r2941583;
        double r2941585 = r2941581 * r2941571;
        double r2941586 = r2941585 + r2941579;
        double r2941587 = r2941580 + r2941586;
        double r2941588 = r2941584 / r2941587;
        double r2941589 = 3.0;
        double r2941590 = r2941589 * r2941566;
        double r2941591 = r2941588 / r2941590;
        double r2941592 = r2941575 ? r2941591 : r2941573;
        double r2941593 = r2941568 ? r2941573 : r2941592;
        return r2941593;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

1. Split input into 2 regimes

2. if a < 572469477.8805368 or 437014546157.40485 < a

   1. Initial program 44.1

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

   2. Simplified 44.1

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

   3. Taylor expanded around inf 11.8

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

   if 572469477.8805368 < a < 437014546157.40485

   1. Initial program 34.4

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

   2. Simplified 34.4

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

   4. Applied flip3-- 34.5

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

   5. Simplified 33.9

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

   6. Simplified 33.9

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

4. Final simplification 13.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le 572469477.8805368:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \mathbf{elif}\;a \le 437014546157.40485:\\ \;\;\;\;\frac{\frac{\sqrt{\left(c \cdot a\right) \cdot -3 + b \cdot b} \cdot \left(\left(c \cdot a\right) \cdot -3 + b \cdot b\right) - b \cdot \left(b \cdot b\right)}{\left(\left(c \cdot a\right) \cdot -3 + b \cdot b\right) + \left(\sqrt{\left(c \cdot a\right) \cdot -3 + b \cdot b} \cdot b + b \cdot b\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 
(FPCore (a b c d)
  :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 a) c)))) (* 3 a)))