quad2m (problem 3.2.1, negative)

Average Error: 34.1 → 6.6

Time: 21.1s

Precision: 64

Math

\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.513258824878011748257049801344805265531 \cdot 10^{152}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 8.216265756828381163830890149037103205802 \cdot 10^{-276}:\\ \;\;\;\;\frac{c}{\sqrt{{b_2}^{2} - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 5.031608061939102936286074782173578716838 \cdot 10^{53}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2}{a} \cdot -2\right)\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r28578 = b_2;
        double r28579 = -r28578;
        double r28580 = r28578 * r28578;
        double r28581 = a;
        double r28582 = c;
        double r28583 = r28581 * r28582;
        double r28584 = r28580 - r28583;
        double r28585 = sqrt(r28584);
        double r28586 = r28579 - r28585;
        double r28587 = r28586 / r28581;
        return r28587;
}

↓

double f(double a, double b_2, double c) {
        double r28588 = b_2;
        double r28589 = -3.5132588248780117e+152;
        bool r28590 = r28588 <= r28589;
        double r28591 = -0.5;
        double r28592 = c;
        double r28593 = r28592 / r28588;
        double r28594 = r28591 * r28593;
        double r28595 = 8.216265756828381e-276;
        bool r28596 = r28588 <= r28595;
        double r28597 = 2.0;
        double r28598 = pow(r28588, r28597);
        double r28599 = a;
        double r28600 = r28599 * r28592;
        double r28601 = r28598 - r28600;
        double r28602 = sqrt(r28601);
        double r28603 = r28602 - r28588;
        double r28604 = r28592 / r28603;
        double r28605 = 5.031608061939103e+53;
        bool r28606 = r28588 <= r28605;
        double r28607 = 1.0;
        double r28608 = -r28588;
        double r28609 = r28588 * r28588;
        double r28610 = r28609 - r28600;
        double r28611 = sqrt(r28610);
        double r28612 = r28608 - r28611;
        double r28613 = r28599 / r28612;
        double r28614 = r28607 / r28613;
        double r28615 = 0.5;
        double r28616 = r28588 / r28599;
        double r28617 = -2.0;
        double r28618 = r28616 * r28617;
        double r28619 = fma(r28615, r28593, r28618);
        double r28620 = r28606 ? r28614 : r28619;
        double r28621 = r28596 ? r28604 : r28620;
        double r28622 = r28590 ? r28594 : r28621;
        return r28622;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

1. Split input into 4 regimes

2. if b_2 < -3.5132588248780117e+152

   1. Initial program 63.9

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Taylor expanded around -inf 1.4

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]

   if -3.5132588248780117e+152 < b_2 < 8.216265756828381e-276

   1. Initial program 33.1

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
   2. Using strategy rm

   3. Applied flip-- 33.1

      \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

   4. Simplified 15.2

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

   5. Simplified 15.2

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{{b_2}^{2} - a \cdot c} - b_2}}}{a}\]
   6. Using strategy rm

   7. Applied div-inv 15.3

      \[\leadsto \color{blue}{\frac{0 + a \cdot c}{\sqrt{{b_2}^{2} - a \cdot c} - b_2} \cdot \frac{1}{a}}\]
   8. Using strategy rm

   9. Applied pow1 15.3

      \[\leadsto \frac{0 + a \cdot c}{\sqrt{{b_2}^{2} - a \cdot c} - b_2} \cdot \color{blue}{{\left(\frac{1}{a}\right)}^{1}}\]

   10. Applied pow1 15.3

       \[\leadsto \color{blue}{{\left(\frac{0 + a \cdot c}{\sqrt{{b_2}^{2} - a \cdot c} - b_2}\right)}^{1}} \cdot {\left(\frac{1}{a}\right)}^{1}\]

   11. Applied pow-prod-down 15.3

       \[\leadsto \color{blue}{{\left(\frac{0 + a \cdot c}{\sqrt{{b_2}^{2} - a \cdot c} - b_2} \cdot \frac{1}{a}\right)}^{1}}\]

   12. Simplified 7.8

       \[\leadsto {\color{blue}{\left(\frac{c}{\sqrt{{b_2}^{2} - a \cdot c} - b_2}\right)}}^{1}\]

   if 8.216265756828381e-276 < b_2 < 5.031608061939103e+53

   1. Initial program 9.3

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
   2. Using strategy rm

   3. Applied clear-num 9.5

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}\]

   if 5.031608061939103e+53 < b_2

   1. Initial program 39.6

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Taylor expanded around inf 5.7

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]

   3. Simplified 5.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2}{a} \cdot -2\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 6.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.513258824878011748257049801344805265531 \cdot 10^{152}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 8.216265756828381163830890149037103205802 \cdot 10^{-276}:\\ \;\;\;\;\frac{c}{\sqrt{{b_2}^{2} - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 5.031608061939102936286074782173578716838 \cdot 10^{53}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2}{a} \cdot -2\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))