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

Average Error: 24.6 → 7.1

Time: 35.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -9.95667684592268629 \cdot 10^{71}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le 1.4708351990208096 \cdot 10^{56}:\\ \;\;\;\;\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

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 <= -9.956676845922686e+71)) {
		temp = (-1.0 * (x * y));
	} else {
		double temp_1;
		if ((z <= 1.4708351990208096e+56)) {
			temp_1 = (y / (sqrt(((z * z) - (t * a))) / (x * z)));
		} else {
			temp_1 = (y * x);
		}
		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	24.6
Target	7.7
Herbie	7.1

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

Derivation

1. Split input into 3 regimes

2. if z < -9.956676845922686e+71

   1. Initial program 40.6

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

   2. Taylor expanded around -inf 2.6

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

   if -9.956676845922686e+71 < z < 1.4708351990208096e+56

   1. Initial program 11.1

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

   3. Applied *-commutative 11.1

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

   4. Applied associate-*l* 11.8

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

   5. Applied associate-/l* 11.0

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

   if 1.4708351990208096e+56 < z

   1. Initial program 37.3

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

   3. Applied *-un-lft-identity 37.3

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

   4. Applied sqrt-prod 37.3

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

   5. Applied associate-*l* 39.0

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

   6. Applied times-frac 37.1

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

   7. Simplified 37.1

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

   8. Taylor expanded around inf 3.4

      \[\leadsto \color{blue}{x \cdot y}\]

   9. Simplified 3.4

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

4. Final simplification 7.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -9.95667684592268629 \cdot 10^{71}:\\ \;\;\;\;-1 \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \le 1.4708351990208096 \cdot 10^{56}:\\ \;\;\;\;\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +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)))))