quad2m (problem 3.2.1, negative)

Average Error: 33.6 → 10.3

Time: 8.3s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.16908657181932359 \cdot 10^{-104}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.3316184968738608 \cdot 10^{61}:\\ \;\;\;\;\frac{-\left(b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r15792 = b_2;
        double r15793 = -r15792;
        double r15794 = r15792 * r15792;
        double r15795 = a;
        double r15796 = c;
        double r15797 = r15795 * r15796;
        double r15798 = r15794 - r15797;
        double r15799 = sqrt(r15798);
        double r15800 = r15793 - r15799;
        double r15801 = r15800 / r15795;
        return r15801;
}

↓

double f(double a, double b_2, double c) {
        double r15802 = b_2;
        double r15803 = -4.1690865718193236e-104;
        bool r15804 = r15802 <= r15803;
        double r15805 = -0.5;
        double r15806 = c;
        double r15807 = r15806 / r15802;
        double r15808 = r15805 * r15807;
        double r15809 = 1.3316184968738608e+61;
        bool r15810 = r15802 <= r15809;
        double r15811 = r15802 * r15802;
        double r15812 = a;
        double r15813 = r15812 * r15806;
        double r15814 = r15811 - r15813;
        double r15815 = sqrt(r15814);
        double r15816 = r15802 + r15815;
        double r15817 = -r15816;
        double r15818 = r15817 / r15812;
        double r15819 = -2.0;
        double r15820 = r15802 / r15812;
        double r15821 = r15819 * r15820;
        double r15822 = r15810 ? r15818 : r15821;
        double r15823 = r15804 ? r15808 : r15822;
        return r15823;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.1690865718193236e-104

   1. Initial program 51.5

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

   3. Applied clear-num 51.5

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

   5. Applied *-un-lft-identity 51.5

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

   6. Applied *-un-lft-identity 51.5

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

   7. Applied times-frac 51.5

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

   8. Applied add-cube-cbrt 51.5

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{1}{1} \cdot \frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]

   9. Applied times-frac 51.5

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{1}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}\]

   10. Simplified 51.5

       \[\leadsto \color{blue}{1} \cdot \frac{\sqrt[3]{1}}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]

   11. Simplified 51.5

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

   12. Taylor expanded around -inf 11.0

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

   if -4.1690865718193236e-104 < b_2 < 1.3316184968738608e+61

   1. Initial program 12.2

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

   3. Applied clear-num 12.4

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

   5. Applied *-un-lft-identity 12.4

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

   6. Applied *-un-lft-identity 12.4

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

   7. Applied times-frac 12.4

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

   8. Applied add-cube-cbrt 12.4

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{1}{1} \cdot \frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]

   9. Applied times-frac 12.4

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{1}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}\]

   10. Simplified 12.4

       \[\leadsto \color{blue}{1} \cdot \frac{\sqrt[3]{1}}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]

   11. Simplified 12.2

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

   if 1.3316184968738608e+61 < b_2

   1. Initial program 39.5

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

   3. Applied clear-num 39.6

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

   4. Taylor expanded around 0 4.5

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.16908657181932359 \cdot 10^{-104}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.3316184968738608 \cdot 10^{61}:\\ \;\;\;\;\frac{-\left(b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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))