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

Average Error: 24.2 → 6.1

Time: 3.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -8.0625792978696209 \cdot 10^{153}:\\ \;\;\;\;\left(x \cdot y\right) \cdot -1\\ \mathbf{elif}\;z \le 3.0412120532343217 \cdot 10^{100}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 1\\ \end{array}\]

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 <= -8.062579297869621e+153)) {
		VAR = ((x * y) * -1.0);
	} else {
		double VAR_1;
		if ((z <= 3.0412120532343217e+100)) {
			VAR_1 = ((x * y) * (z / sqrt(((z * z) - (t * a)))));
		} else {
			VAR_1 = ((x * y) * 1.0);
		}
		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	24.2
Target	7.5
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 3 regimes

2. if z < -8.062579297869621e+153

   1. Initial program 53.7

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

   3. Applied associate-/l* 53.3

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

   5. Applied div-inv 53.3

      \[\leadsto \frac{x \cdot y}{\color{blue}{\sqrt{z \cdot z - t \cdot a} \cdot \frac{1}{z}}}\]
   6. Using strategy rm

   7. Applied div-inv 53.3

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

   8. Simplified 53.3

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

   9. Taylor expanded around -inf 1.2

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

   if -8.062579297869621e+153 < z < 3.0412120532343217e+100

   1. Initial program 10.8

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

   3. Applied associate-/l* 8.9

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

   5. Applied div-inv 8.9

      \[\leadsto \frac{x \cdot y}{\color{blue}{\sqrt{z \cdot z - t \cdot a} \cdot \frac{1}{z}}}\]
   6. Using strategy rm

   7. Applied div-inv 9.0

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

   8. Simplified 8.7

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

   if 3.0412120532343217e+100 < z

   1. Initial program 42.6

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

   3. Applied associate-/l* 40.7

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

   5. Applied div-inv 40.7

      \[\leadsto \frac{x \cdot y}{\color{blue}{\sqrt{z \cdot z - t \cdot a} \cdot \frac{1}{z}}}\]
   6. Using strategy rm

   7. Applied div-inv 40.7

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

   8. Simplified 40.6

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

   9. Taylor expanded around inf 2.0

      \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{1}\]
3. Recombined 3 regimes into one program.

4. Final simplification 6.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -8.0625792978696209 \cdot 10^{153}:\\ \;\;\;\;\left(x \cdot y\right) \cdot -1\\ \mathbf{elif}\;z \le 3.0412120532343217 \cdot 10^{100}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(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)))))