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

Average Error: 24.5 → 6.1

Time: 4.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -3.48711103202709677 \cdot 10^{146}:\\ \;\;\;\;\left(x \cdot y\right) \cdot -1\\ \mathbf{elif}\;z \le -1.56344281963474788 \cdot 10^{-286}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{elif}\;z \le 1.8216614650284523 \cdot 10^{-230}:\\ \;\;\;\;1 \cdot \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{elif}\;z \le 2.966256665740778 \cdot 10^{111}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a) {
	return (((x * y) * z) / sqrt(((z * z) - (t * a))));
}

↓

double code(double x, double y, double z, double t, double a) {
	double temp;
	if ((z <= -3.4871110320270968e+146)) {
		temp = ((x * y) * -1.0);
	} else {
		double temp_1;
		if ((z <= -1.5634428196347479e-286)) {
			temp_1 = (x * (y * (z / sqrt(((z * z) - (t * a))))));
		} else {
			double temp_2;
			if ((z <= 1.8216614650284523e-230)) {
				temp_2 = (1.0 * (((x * y) * z) / sqrt(((z * z) - (t * a)))));
			} else {
				double temp_3;
				if ((z <= 2.966256665740778e+111)) {
					temp_3 = ((x * y) * (z / sqrt(((z * z) - (t * a)))));
				} else {
					temp_3 = (x * y);
				}
				temp_2 = temp_3;
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.5
Target	7.6
Herbie	6.1

\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.9762681209208942 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

1. Split input into 5 regimes

2. if z < -3.4871110320270968e+146

   1. Initial program 51.7

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

   3. Applied *-un-lft-identity 51.7

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

   4. Applied sqrt-prod 51.7

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

   5. Applied times-frac 50.9

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

   6. Simplified 50.9

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

   7. Taylor expanded around -inf 1.4

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

   if -3.4871110320270968e+146 < z < -1.5634428196347479e-286

   1. Initial program 10.9

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

   3. Applied *-un-lft-identity 10.9

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

   4. Applied sqrt-prod 10.9

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

   5. Applied times-frac 8.6

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

   6. Simplified 8.6

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

   8. Applied associate-*l* 8.4

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

   if -1.5634428196347479e-286 < z < 1.8216614650284523e-230

   1. Initial program 16.5

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

   3. Applied add-sqr-sqrt 16.5

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

   4. Applied sqrt-prod 16.6

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

   5. Applied times-frac 17.6

      \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{\sqrt{z \cdot z - t \cdot a}}} \cdot \frac{z}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 17.6

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

   8. Applied associate-*l* 17.6

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

   9. Simplified 16.5

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

   if 1.8216614650284523e-230 < z < 2.966256665740778e+111

   1. Initial program 8.9

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

   3. Applied *-un-lft-identity 8.9

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

   4. Applied sqrt-prod 8.9

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

   5. Applied times-frac 6.7

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

   6. Simplified 6.7

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

   if 2.966256665740778e+111 < z

   1. Initial program 44.7

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

   2. Taylor expanded around inf 2.5

      \[\leadsto \color{blue}{x \cdot y}\]
3. Recombined 5 regimes into one program.

4. Final simplification 6.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.48711103202709677 \cdot 10^{146}:\\ \;\;\;\;\left(x \cdot y\right) \cdot -1\\ \mathbf{elif}\;z \le -1.56344281963474788 \cdot 10^{-286}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{elif}\;z \le 1.8216614650284523 \cdot 10^{-230}:\\ \;\;\;\;1 \cdot \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{elif}\;z \le 2.966256665740778 \cdot 10^{111}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"
  :precision binary64

  :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)))))