Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2

Average Error: 6.4 → 5.8

Time: 40.8s

Precision: binary64

Math

\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]

↓

\[\frac{\sqrt[3]{\frac{1}{x}}}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\frac{\sqrt[3]{\sqrt[3]{1} \cdot \sqrt[3]{1}}}{\sqrt{1 + z \cdot z}}}} \cdot \frac{\sqrt[3]{\frac{1}{x}}}{\frac{\sqrt[3]{y}}{\frac{\sqrt[3]{\frac{\sqrt[3]{1}}{x}}}{\sqrt{1 + z \cdot z}}}}\]

FPCore

(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))

↓

(FPCore (x y z)
 :precision binary64
 (*
  (/
   (cbrt (/ 1.0 x))
   (/
    (* (cbrt y) (cbrt y))
    (/ (cbrt (* (cbrt 1.0) (cbrt 1.0))) (sqrt (+ 1.0 (* z z))))))
  (/
   (cbrt (/ 1.0 x))
   (/ (cbrt y) (/ (cbrt (/ (cbrt 1.0) x)) (sqrt (+ 1.0 (* z z))))))))

C

double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * z)));
}

↓

double code(double x, double y, double z) {
	return (cbrt(1.0 / x) / ((cbrt(y) * cbrt(y)) / (cbrt(cbrt(1.0) * cbrt(1.0)) / sqrt(1.0 + (z * z))))) * (cbrt(1.0 / x) / (cbrt(y) / (cbrt(cbrt(1.0) / x) / sqrt(1.0 + (z * z)))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.4
Target	5.6
Herbie	5.8

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

1. Initial program 6.4

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
2. Using strategy rm

3. Applied add-cube-cbrt_binary64 7.0

   \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{1}{x}} \cdot \sqrt[3]{\frac{1}{x}}\right) \cdot \sqrt[3]{\frac{1}{x}}}}{y \cdot \left(1 + z \cdot z\right)}\]

4. Applied associate-/l*_binary64 7.0

   \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{1}{x}} \cdot \sqrt[3]{\frac{1}{x}}}{\frac{y \cdot \left(1 + z \cdot z\right)}{\sqrt[3]{\frac{1}{x}}}}}\]

5. Simplified 6.7

   \[\leadsto \frac{\sqrt[3]{\frac{1}{x}} \cdot \sqrt[3]{\frac{1}{x}}}{\color{blue}{\frac{y}{\frac{\sqrt[3]{\frac{1}{x}}}{1 + z \cdot z}}}}\]
6. Using strategy rm

7. Applied add-sqr-sqrt_binary64 6.7

   \[\leadsto \frac{\sqrt[3]{\frac{1}{x}} \cdot \sqrt[3]{\frac{1}{x}}}{\frac{y}{\frac{\sqrt[3]{\frac{1}{x}}}{\color{blue}{\sqrt{1 + z \cdot z} \cdot \sqrt{1 + z \cdot z}}}}}\]

8. Applied *-un-lft-identity_binary64 6.7

   \[\leadsto \frac{\sqrt[3]{\frac{1}{x}} \cdot \sqrt[3]{\frac{1}{x}}}{\frac{y}{\frac{\sqrt[3]{\frac{1}{\color{blue}{1 \cdot x}}}}{\sqrt{1 + z \cdot z} \cdot \sqrt{1 + z \cdot z}}}}\]

9. Applied add-cube-cbrt_binary64 6.7

   \[\leadsto \frac{\sqrt[3]{\frac{1}{x}} \cdot \sqrt[3]{\frac{1}{x}}}{\frac{y}{\frac{\sqrt[3]{\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot x}}}{\sqrt{1 + z \cdot z} \cdot \sqrt{1 + z \cdot z}}}}\]

10. Applied times-frac_binary64 6.7

    \[\leadsto \frac{\sqrt[3]{\frac{1}{x}} \cdot \sqrt[3]{\frac{1}{x}}}{\frac{y}{\frac{\sqrt[3]{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{x}}}}{\sqrt{1 + z \cdot z} \cdot \sqrt{1 + z \cdot z}}}}\]

11. Applied cbrt-prod_binary64 6.7

    \[\leadsto \frac{\sqrt[3]{\frac{1}{x}} \cdot \sqrt[3]{\frac{1}{x}}}{\frac{y}{\frac{\color{blue}{\sqrt[3]{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}} \cdot \sqrt[3]{\frac{\sqrt[3]{1}}{x}}}}{\sqrt{1 + z \cdot z} \cdot \sqrt{1 + z \cdot z}}}}\]

12. Applied times-frac_binary64 6.7

    \[\leadsto \frac{\sqrt[3]{\frac{1}{x}} \cdot \sqrt[3]{\frac{1}{x}}}{\frac{y}{\color{blue}{\frac{\sqrt[3]{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}}}{\sqrt{1 + z \cdot z}} \cdot \frac{\sqrt[3]{\frac{\sqrt[3]{1}}{x}}}{\sqrt{1 + z \cdot z}}}}}\]

13. Applied add-cube-cbrt_binary64 6.9

    \[\leadsto \frac{\sqrt[3]{\frac{1}{x}} \cdot \sqrt[3]{\frac{1}{x}}}{\frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{\frac{\sqrt[3]{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}}}{\sqrt{1 + z \cdot z}} \cdot \frac{\sqrt[3]{\frac{\sqrt[3]{1}}{x}}}{\sqrt{1 + z \cdot z}}}}\]

14. Applied times-frac_binary64 6.6

    \[\leadsto \frac{\sqrt[3]{\frac{1}{x}} \cdot \sqrt[3]{\frac{1}{x}}}{\color{blue}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\frac{\sqrt[3]{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}}}{\sqrt{1 + z \cdot z}}} \cdot \frac{\sqrt[3]{y}}{\frac{\sqrt[3]{\frac{\sqrt[3]{1}}{x}}}{\sqrt{1 + z \cdot z}}}}}\]

15. Applied times-frac_binary64 5.8

    \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{1}{x}}}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\frac{\sqrt[3]{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}}}{\sqrt{1 + z \cdot z}}}} \cdot \frac{\sqrt[3]{\frac{1}{x}}}{\frac{\sqrt[3]{y}}{\frac{\sqrt[3]{\frac{\sqrt[3]{1}}{x}}}{\sqrt{1 + z \cdot z}}}}}\]

16. Simplified 5.8

    \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{1}{x}}}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\frac{\sqrt[3]{\sqrt[3]{1} \cdot \sqrt[3]{1}}}{\sqrt{1 + z \cdot z}}}}} \cdot \frac{\sqrt[3]{\frac{1}{x}}}{\frac{\sqrt[3]{y}}{\frac{\sqrt[3]{\frac{\sqrt[3]{1}}{x}}}{\sqrt{1 + z \cdot z}}}}\]

17. Final simplification 5.8

    \[\leadsto \frac{\sqrt[3]{\frac{1}{x}}}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\frac{\sqrt[3]{\sqrt[3]{1} \cdot \sqrt[3]{1}}}{\sqrt{1 + z \cdot z}}}} \cdot \frac{\sqrt[3]{\frac{1}{x}}}{\frac{\sqrt[3]{y}}{\frac{\sqrt[3]{\frac{\sqrt[3]{1}}{x}}}{\sqrt{1 + z \cdot z}}}}\]

Reproduce

herbie shell --seed 2020220 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< (* y (+ 1.0 (* z z))) (- INFINITY)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x)) (if (< (* y (+ 1.0 (* z z))) 8.680743250567252e+305) (/ (/ 1.0 x) (* (+ 1.0 (* z z)) y)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x))))

  (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))