Cubic critical

Average Error: 34.8 → 15.1

Time: 5.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.33791852996056488 \cdot 10^{154}:\\ \;\;\;\;\frac{1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{3 \cdot a}\\ \mathbf{elif}\;b \le 1.8843431932504646 \cdot 10^{-40}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -\left(3 \cdot a\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r86545 = b;
        double r86546 = -r86545;
        double r86547 = r86545 * r86545;
        double r86548 = 3.0;
        double r86549 = a;
        double r86550 = r86548 * r86549;
        double r86551 = c;
        double r86552 = r86550 * r86551;
        double r86553 = r86547 - r86552;
        double r86554 = sqrt(r86553);
        double r86555 = r86546 + r86554;
        double r86556 = r86555 / r86550;
        return r86556;
}

↓

double f(double a, double b, double c) {
        double r86557 = b;
        double r86558 = -1.3379185299605649e+154;
        bool r86559 = r86557 <= r86558;
        double r86560 = 1.5;
        double r86561 = a;
        double r86562 = c;
        double r86563 = r86561 * r86562;
        double r86564 = r86563 / r86557;
        double r86565 = r86560 * r86564;
        double r86566 = 2.0;
        double r86567 = r86566 * r86557;
        double r86568 = r86565 - r86567;
        double r86569 = 3.0;
        double r86570 = r86569 * r86561;
        double r86571 = r86568 / r86570;
        double r86572 = 1.8843431932504646e-40;
        bool r86573 = r86557 <= r86572;
        double r86574 = -r86557;
        double r86575 = r86570 * r86562;
        double r86576 = -r86575;
        double r86577 = fma(r86557, r86557, r86576);
        double r86578 = sqrt(r86577);
        double r86579 = r86574 + r86578;
        double r86580 = r86579 / r86570;
        double r86581 = -1.5;
        double r86582 = r86581 * r86564;
        double r86583 = r86582 / r86570;
        double r86584 = r86573 ? r86580 : r86583;
        double r86585 = r86559 ? r86571 : r86584;
        return r86585;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b < -1.3379185299605649e+154

   1. Initial program 64.0

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

   2. Taylor expanded around -inf 11.6

      \[\leadsto \frac{\color{blue}{1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{3 \cdot a}\]

   if -1.3379185299605649e+154 < b < 1.8843431932504646e-40

   1. Initial program 14.3

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

   3. Applied fma-neg 14.3

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

   if 1.8843431932504646e-40 < b

   1. Initial program 55.1

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

   2. Taylor expanded around inf 17.2

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]
3. Recombined 3 regimes into one program.

4. Final simplification 15.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.33791852996056488 \cdot 10^{154}:\\ \;\;\;\;\frac{1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{3 \cdot a}\\ \mathbf{elif}\;b \le 1.8843431932504646 \cdot 10^{-40}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -\left(3 \cdot a\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))