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

Average Error: 24.7 → 7.1

Time: 18.6s

Precision: binary64

\[[x, y]=\mathsf{sort}([x, y])\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -2.705239368116784 \cdot 10^{+72}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 7.0143619415122286 \cdot 10^{-304}:\\ \;\;\;\;\left(z \cdot \left(x \cdot y\right)\right) \cdot \frac{1}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{elif}\;z \leq 2.8243032036053995 \cdot 10^{+60}:\\ \;\;\;\;x \cdot \left(\left(y \cdot \frac{\sqrt{z}}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}\right) \cdot \frac{\sqrt{z}}{\left|\sqrt[3]{z \cdot z - t \cdot a}\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{z}\\ \end{array}\]

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 -2.705239368116784e+72)
   (- (* x y))
   (if (<= z 7.0143619415122286e-304)
     (* (* z (* x y)) (/ 1.0 (sqrt (- (* z z) (* t a)))))
     (if (<= z 2.8243032036053995e+60)
       (*
        x
        (*
         (* y (/ (sqrt z) (sqrt (cbrt (- (* z z) (* t a))))))
         (/ (sqrt z) (fabs (cbrt (- (* z z) (* t a)))))))
       (* (* x y) (/ z z))))))

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 tmp;
	if (z <= -2.705239368116784e+72) {
		tmp = -(x * y);
	} else if (z <= 7.0143619415122286e-304) {
		tmp = (z * (x * y)) * (1.0 / sqrt((z * z) - (t * a)));
	} else if (z <= 2.8243032036053995e+60) {
		tmp = x * ((y * (sqrt(z) / sqrt(cbrt((z * z) - (t * a))))) * (sqrt(z) / fabs(cbrt((z * z) - (t * a)))));
	} else {
		tmp = (x * y) * (z / z);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.7
Target	8.0
Herbie	7.1

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

Derivation

1. Split input into 4 regimes

2. if z < -2.705239368116784e72

   1. Initial program 40.5

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

   2. Taylor expanded around -inf 2.8

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

   if -2.705239368116784e72 < z < 7.01436194151222859e-304

   1. Initial program 11.5

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

   3. Applied div-inv_binary64_7918 11.5

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

   if 7.01436194151222859e-304 < z < 2.82430320360539952e60

   1. Initial program 11.5

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

   3. Applied *-un-lft-identity_binary64_7921 11.5

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

   4. Applied sqrt-prod_binary64_7937 11.5

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

   5. Applied times-frac_binary64_7927 10.6

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

   6. Simplified 10.6

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

   8. Applied add-cube-cbrt_binary64_7956 11.0

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

   9. Applied sqrt-prod_binary64_7937 11.0

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

   10. Applied add-cube-cbrt_binary64_7956 11.3

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

   11. Applied times-frac_binary64_7927 11.3

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

   12. Simplified 11.3

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

   14. Applied associate-*l*_binary64_7862 10.3

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

   15. Simplified 10.0

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

   17. Applied add-sqr-sqrt_binary64_7943 10.0

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

   18. Applied times-frac_binary64_7927 10.0

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

   19. Applied associate-*r*_binary64_7861 9.7

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

   if 2.82430320360539952e60 < z

   1. Initial program 38.5

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

   3. Applied *-un-lft-identity_binary64_7921 38.5

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

   4. Applied sqrt-prod_binary64_7937 38.5

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

   5. Applied times-frac_binary64_7927 36.0

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

   6. Simplified 36.0

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

   7. Taylor expanded around inf 3.5

      \[\leadsto \left(x \cdot y\right) \cdot \frac{z}{\color{blue}{z}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 7.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.705239368116784 \cdot 10^{+72}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;z \leq 7.0143619415122286 \cdot 10^{-304}:\\ \;\;\;\;\left(z \cdot \left(x \cdot y\right)\right) \cdot \frac{1}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{elif}\;z \leq 2.8243032036053995 \cdot 10^{+60}:\\ \;\;\;\;x \cdot \left(\left(y \cdot \frac{\sqrt{z}}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}\right) \cdot \frac{\sqrt{z}}{\left|\sqrt[3]{z \cdot z - t \cdot a}\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2021098 
(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)))))