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

Average Error: 24.6 → 6.2

Time: 6.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -9.892107016045583 \cdot 10^{+95}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}\right)\\ \mathbf{elif}\;z \leq 1.2713248643122116 \cdot 10^{+134}:\\ \;\;\;\;x \cdot \left(\left(y \cdot \left(\sqrt[3]{z} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}} \cdot \sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}\right)\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + \left(t \cdot \frac{a}{z}\right) \cdot -0.5}\right)\\ \end{array}\]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (* (* x y) z) (sqrt (- (* z z) (* t a)))))

↓

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -9.892107016045583e+95)
   (* x (* y (/ z (- (* 0.5 (* t (/ a z))) z))))
   (if (<= z 1.2713248643122116e+134)
     (*
      x
      (*
       (*
        y
        (*
         (cbrt z)
         (/
          (cbrt z)
          (*
           (cbrt (sqrt (- (* z z) (* t a))))
           (cbrt (sqrt (- (* z z) (* t a))))))))
       (/ (cbrt z) (cbrt (sqrt (- (* z z) (* t a)))))))
     (* x (* y (/ z (+ z (* (* t (/ a z)) -0.5))))))))

C

double code(double x, double y, double z, double t, double a) {
	return (((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 <= -9.892107016045583e+95)) {
		VAR = ((double) (x * ((double) (y * (z / ((double) (((double) (0.5 * ((double) (t * (a / z))))) - z)))))));
	} else {
		double VAR_1;
		if ((z <= 1.2713248643122116e+134)) {
			VAR_1 = ((double) (x * ((double) (((double) (y * ((double) (((double) cbrt(z)) * (((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) cbrt(z)) / ((double) cbrt(((double) sqrt(((double) (((double) (z * z)) - ((double) (t * a)))))))))))));
		} else {
			VAR_1 = ((double) (x * ((double) (y * (z / ((double) (z + ((double) (((double) (t * (a / z))) * -0.5)))))))));
		}
		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	24.6
Target	7.4
Herbie	6.2

\[\begin{array}{l} \mathbf{if}\;z < -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z < 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if z < -9.8921070160455831e95

   1. Initial program 41.6

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

   2. Simplified 39.2

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

   3. Taylor expanded around -inf 5.3

      \[\leadsto x \cdot \left(y \cdot \frac{z}{\color{blue}{0.5 \cdot \frac{a \cdot t}{z} - z}}\right)\]

   4. Simplified 2.3

      \[\leadsto x \cdot \left(y \cdot \frac{z}{\color{blue}{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}}\right)\]

   if -9.8921070160455831e95 < z < 1.27132486431221162e134

   1. Initial program 11.0

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

   2. Simplified 9.3

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

   4. Applied add-cube-cbrt 10.0

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

   5. Applied add-cube-cbrt 9.6

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

   6. Applied times-frac 9.6

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

   7. Applied associate-*r* 8.9

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

   8. Simplified 9.0

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

   if 1.27132486431221162e134 < z

   1. Initial program 50.8

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

   2. Simplified 49.3

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

   3. Taylor expanded around inf 5.1

      \[\leadsto x \cdot \left(y \cdot \frac{z}{\color{blue}{z - 0.5 \cdot \frac{a \cdot t}{z}}}\right)\]

   4. Simplified 1.2

      \[\leadsto x \cdot \left(y \cdot \frac{z}{\color{blue}{z + \left(t \cdot \frac{a}{z}\right) \cdot -0.5}}\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 6.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.892107016045583 \cdot 10^{+95}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{0.5 \cdot \left(t \cdot \frac{a}{z}\right) - z}\right)\\ \mathbf{elif}\;z \leq 1.2713248643122116 \cdot 10^{+134}:\\ \;\;\;\;x \cdot \left(\left(y \cdot \left(\sqrt[3]{z} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}} \cdot \sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}\right)\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt{z \cdot z - t \cdot a}}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + \left(t \cdot \frac{a}{z}\right) \cdot -0.5}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020198 
(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)))))