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

Average Error: 24.8 → 6.4

Time: 4.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.16931718240994252 \cdot 10^{101}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le -1.90424718612399385 \cdot 10^{-51}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \left(\frac{1}{\sqrt{\sqrt{z \cdot z - t \cdot a}}} \cdot \frac{z}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}\right)\\ \mathbf{elif}\;z \le 2.17248773687094426 \cdot 10^{-195}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{elif}\;z \le 2.8132330552631191 \cdot 10^{60}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

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 VAR;
	if ((z <= -1.1693171824099425e+101)) {
		VAR = (-1.0 * (x * y));
	} else {
		double VAR_1;
		if ((z <= -1.9042471861239938e-51)) {
			VAR_1 = ((x * y) * ((1.0 / sqrt(sqrt(((z * z) - (t * a))))) * (z / sqrt(sqrt(((z * z) - (t * a)))))));
		} else {
			double VAR_2;
			if ((z <= 2.1724877368709443e-195)) {
				VAR_2 = ((x * (y * z)) / sqrt(((z * z) - (t * a))));
			} else {
				double VAR_3;
				if ((z <= 2.813233055263119e+60)) {
					VAR_3 = (x * (y * (z / sqrt(((z * z) - (t * a))))));
				} else {
					VAR_3 = (x * y);
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		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	7.8
Herbie	6.4

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

Derivation

1. Split input into 5 regimes

2. if z < -1.1693171824099425e+101

   1. Initial program 44.2

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

   2. Taylor expanded around -inf 2.0

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

   if -1.1693171824099425e+101 < z < -1.9042471861239938e-51

   1. Initial program 7.7

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

   3. Applied *-un-lft-identity 7.7

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

   4. Applied sqrt-prod 7.7

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

   5. Applied times-frac 4.1

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

   6. Simplified 4.1

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

   8. Applied add-sqr-sqrt 4.1

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

   9. Applied sqrt-prod 4.5

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

   10. Applied *-un-lft-identity 4.5

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

   11. Applied times-frac 4.5

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

   if -1.9042471861239938e-51 < z < 2.1724877368709443e-195

   1. Initial program 14.9

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

   3. Applied associate-*l* 14.4

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

   if 2.1724877368709443e-195 < z < 2.813233055263119e+60

   1. Initial program 9.0

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

   3. Applied *-un-lft-identity 9.0

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

   4. Applied sqrt-prod 9.0

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

   5. Applied times-frac 7.2

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

   6. Simplified 7.2

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

   8. Applied associate-*l* 7.1

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

   if 2.813233055263119e+60 < z

   1. Initial program 37.9

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

   2. Taylor expanded around inf 3.1

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

4. Final simplification 6.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.16931718240994252 \cdot 10^{101}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le -1.90424718612399385 \cdot 10^{-51}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \left(\frac{1}{\sqrt{\sqrt{z \cdot z - t \cdot a}}} \cdot \frac{z}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}\right)\\ \mathbf{elif}\;z \le 2.17248773687094426 \cdot 10^{-195}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{elif}\;z \le 2.8132330552631191 \cdot 10^{60}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020075 
(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)))))