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

Average Error: 24.8 → 5.8

Time: 8.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.6394538799208849 \cdot 10^{151}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\frac{1}{2} \cdot \left(t \cdot \frac{a}{z}\right) - z}\right)\\ \mathbf{elif}\;z \le 4.27791588576282506 \cdot 10^{70}:\\ \;\;\;\;x \cdot \left(\left(y \cdot \left(\sqrt[3]{z} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}} \cdot \sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}\right)\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z - \frac{1}{2} \cdot \left(t \cdot \frac{a}{z}\right)}\right)\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((z <= -1.639453879920885e+151)) {
		VAR = ((double) (x * ((double) (y * ((double) (z / ((double) (((double) (0.5 * ((double) (t * ((double) (a / z)))))) - z))))))));
	} else {
		double VAR_1;
		if ((z <= 4.277915885762825e+70)) {
			VAR_1 = ((double) (x * ((double) (((double) (y * ((double) (((double) cbrt(z)) * ((double) (((double) cbrt(z)) / ((double) (((double) cbrt(((double) sqrt(((double) (((double) (z * z)) - ((double) (t * a)))))))) * ((double) cbrt(((double) sqrt(((double) (((double) (z * z)) - ((double) (t * a)))))))))))))))) * ((double) (((double) cbrt(z)) / ((double) cbrt(((double) sqrt(((double) (((double) (z * z)) - ((double) (t * a))))))))))))));
		} else {
			VAR_1 = ((double) (x * ((double) (y * ((double) (z / ((double) (z - ((double) (0.5 * ((double) (t * ((double) (a / z))))))))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.8
Target	7.4
Herbie	5.8

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

Derivation

1. Split input into 3 regimes

2. if z < -1.6394538799208849e151

   1. Initial program 53.8

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

   2. Simplified 53.2

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

   3. Taylor expanded around -inf 5.6

      \[\leadsto x \cdot \left(y \cdot \frac{z}{\color{blue}{\frac{1}{2} \cdot \frac{a \cdot t}{z} - z}}\right)\]

   4. Simplified 0.9

      \[\leadsto x \cdot \left(y \cdot \frac{z}{\color{blue}{\frac{1}{2} \cdot \left(t \cdot \frac{a}{z}\right) - z}}\right)\]

   if -1.6394538799208849e151 < z < 4.27791588576282506e70

   1. Initial program 10.8

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

   2. Simplified 8.7

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

   4. Applied add-cube-cbrt 9.5

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

   5. Applied add-cube-cbrt 9.1

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

   6. Applied times-frac 9.1

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

   7. Applied associate-*r* 8.4

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

   8. Simplified 8.5

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

   if 4.27791588576282506e70 < z

   1. Initial program 40.3

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

   2. Simplified 37.5

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

   3. Taylor expanded around inf 5.6

      \[\leadsto x \cdot \left(y \cdot \frac{z}{\color{blue}{z - \frac{1}{2} \cdot \frac{a \cdot t}{z}}}\right)\]

   4. Simplified 2.3

      \[\leadsto x \cdot \left(y \cdot \frac{z}{\color{blue}{z - \frac{1}{2} \cdot \left(t \cdot \frac{a}{z}\right)}}\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 5.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.6394538799208849 \cdot 10^{151}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\frac{1}{2} \cdot \left(t \cdot \frac{a}{z}\right) - z}\right)\\ \mathbf{elif}\;z \le 4.27791588576282506 \cdot 10^{70}:\\ \;\;\;\;x \cdot \left(\left(y \cdot \left(\sqrt[3]{z} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}} \cdot \sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}\right)\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z - \frac{1}{2} \cdot \left(t \cdot \frac{a}{z}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(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) (neg (* 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)))))