Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2

Average Error: 24.7 → 5.8

Time: 19.6s

Precision: 64

Math

\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -2.099203142812841057267611063629875049796 \cdot 10^{122}:\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{elif}\;z \le 8.801249261765559740247641081984816423417 \cdot 10^{98}:\\ \;\;\;\;\frac{\sqrt[3]{z}}{\sqrt{\sqrt[3]{z \cdot z - a \cdot t}}} \cdot \left(y \cdot \left(\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\left|\sqrt[3]{\sqrt{z \cdot z - a \cdot t}} \cdot \sqrt[3]{\sqrt{z \cdot z - a \cdot t}}\right|} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r11242066 = x;
        double r11242067 = y;
        double r11242068 = r11242066 * r11242067;
        double r11242069 = z;
        double r11242070 = r11242068 * r11242069;
        double r11242071 = r11242069 * r11242069;
        double r11242072 = t;
        double r11242073 = a;
        double r11242074 = r11242072 * r11242073;
        double r11242075 = r11242071 - r11242074;
        double r11242076 = sqrt(r11242075);
        double r11242077 = r11242070 / r11242076;
        return r11242077;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r11242078 = z;
        double r11242079 = -2.099203142812841e+122;
        bool r11242080 = r11242078 <= r11242079;
        double r11242081 = x;
        double r11242082 = -r11242081;
        double r11242083 = y;
        double r11242084 = r11242082 * r11242083;
        double r11242085 = 8.80124926176556e+98;
        bool r11242086 = r11242078 <= r11242085;
        double r11242087 = cbrt(r11242078);
        double r11242088 = r11242078 * r11242078;
        double r11242089 = a;
        double r11242090 = t;
        double r11242091 = r11242089 * r11242090;
        double r11242092 = r11242088 - r11242091;
        double r11242093 = cbrt(r11242092);
        double r11242094 = sqrt(r11242093);
        double r11242095 = r11242087 / r11242094;
        double r11242096 = r11242087 * r11242087;
        double r11242097 = sqrt(r11242092);
        double r11242098 = cbrt(r11242097);
        double r11242099 = r11242098 * r11242098;
        double r11242100 = fabs(r11242099);
        double r11242101 = r11242096 / r11242100;
        double r11242102 = r11242101 * r11242081;
        double r11242103 = r11242083 * r11242102;
        double r11242104 = r11242095 * r11242103;
        double r11242105 = r11242081 * r11242083;
        double r11242106 = r11242086 ? r11242104 : r11242105;
        double r11242107 = r11242080 ? r11242084 : r11242106;
        return r11242107;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.7
Target	7.6
Herbie	5.8

\[\begin{array}{l} \mathbf{if}\;z \lt -3.192130590385276419686361646843883646209 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.976268120920894210257945708950453212935 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if z < -2.099203142812841e+122

   1. Initial program 48.1

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]

   2. Taylor expanded around -inf 1.6

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right)}\]

   3. Simplified 1.6

      \[\leadsto \color{blue}{\left(-x\right) \cdot y}\]

   if -2.099203142812841e+122 < z < 8.80124926176556e+98

   1. Initial program 11.0

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 11.0

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{1 \cdot \left(z \cdot z - t \cdot a\right)}}}\]

   4. Applied sqrt-prod 11.0

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{\sqrt{1} \cdot \sqrt{z \cdot z - t \cdot a}}}\]

   5. Applied times-frac 9.6

      \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{1}} \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}}\]

   6. Simplified 9.6

      \[\leadsto \color{blue}{\left(x \cdot y\right)} \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 10.0

      \[\leadsto \left(x \cdot y\right) \cdot \frac{z}{\sqrt{\color{blue}{\left(\sqrt[3]{z \cdot z - t \cdot a} \cdot \sqrt[3]{z \cdot z - t \cdot a}\right) \cdot \sqrt[3]{z \cdot z - t \cdot a}}}}\]

   9. Applied sqrt-prod 10.0

      \[\leadsto \left(x \cdot y\right) \cdot \frac{z}{\color{blue}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a} \cdot \sqrt[3]{z \cdot z - t \cdot a}} \cdot \sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}}\]

   10. Applied add-cube-cbrt 10.3

       \[\leadsto \left(x \cdot y\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a} \cdot \sqrt[3]{z \cdot z - t \cdot a}} \cdot \sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}\]

   11. Applied times-frac 10.3

       \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a} \cdot \sqrt[3]{z \cdot z - t \cdot a}}} \cdot \frac{\sqrt[3]{z}}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}\right)}\]

   12. Applied associate-*r* 9.5

       \[\leadsto \color{blue}{\left(\left(x \cdot y\right) \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a} \cdot \sqrt[3]{z \cdot z - t \cdot a}}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}}\]

   13. Simplified 8.4

       \[\leadsto \color{blue}{\left(\left(\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\left|\sqrt[3]{z \cdot z - t \cdot a}\right|} \cdot x\right) \cdot y\right)} \cdot \frac{\sqrt[3]{z}}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}\]
   14. Using strategy rm

   15. Applied add-sqr-sqrt 8.4

       \[\leadsto \left(\left(\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\left|\sqrt[3]{\color{blue}{\sqrt{z \cdot z - t \cdot a} \cdot \sqrt{z \cdot z - t \cdot a}}}\right|} \cdot x\right) \cdot y\right) \cdot \frac{\sqrt[3]{z}}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}\]

   16. Applied cbrt-prod 8.2

       \[\leadsto \left(\left(\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\left|\color{blue}{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}} \cdot \sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}\right|} \cdot x\right) \cdot y\right) \cdot \frac{\sqrt[3]{z}}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}\]

   if 8.80124926176556e+98 < z

   1. Initial program 43.5

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]

   2. Taylor expanded around inf 2.4

      \[\leadsto \color{blue}{x \cdot y}\]
3. Recombined 3 regimes into one program.

4. Final simplification 5.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.099203142812841057267611063629875049796 \cdot 10^{122}:\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{elif}\;z \le 8.801249261765559740247641081984816423417 \cdot 10^{98}:\\ \;\;\;\;\frac{\sqrt[3]{z}}{\sqrt{\sqrt[3]{z \cdot z - a \cdot t}}} \cdot \left(y \cdot \left(\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\left|\sqrt[3]{\sqrt{z \cdot z - a \cdot t}} \cdot \sqrt[3]{\sqrt{z \cdot z - a \cdot t}}\right|} \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"

  :herbie-target
  (if (< z -3.1921305903852764e+46) (- (* y x)) (if (< z 5.976268120920894e+90) (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y)) (* y x)))

  (/ (* (* x y) z) (sqrt (- (* z z) (* t a)))))