Average Error: 6.5 → 5.9
Time: 5.7s
Precision: binary64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{1}{\sqrt{1 + z \cdot z}} \cdot \frac{\frac{\frac{1}{x}}{\sqrt{1 + z \cdot z}}}{y}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{1}{\sqrt{1 + z \cdot z}} \cdot \frac{\frac{\frac{1}{x}}{\sqrt{1 + z \cdot z}}}{y}
(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))
↓
(FPCore (x y z)
 :precision binary64
 (* (/ 1.0 (sqrt (+ 1.0 (* z z)))) (/ (/ (/ 1.0 x) (sqrt (+ 1.0 (* z z)))) y)))
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 (1.0 / sqrt(1.0 + (z * z))) * (((1.0 / x) / sqrt(1.0 + (z * z))) / y);
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.5
Target5.9
Herbie5.9
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.5

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

  3. Applied *-un-lft-identity_binary64_109906.5

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

  4. Applied add-sqr-sqrt_binary64_110126.5

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

  5. Applied times-frac_binary64_109966.5

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

  6. Applied times-frac_binary64_109966.1

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

  7. Simplified6.1

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

  8. Simplified6.1

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

  10. Applied associate-*l/_binary64_109336.1

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

  11. Simplified6.1

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

  13. Applied *-un-lft-identity_binary64_109906.1

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

  14. Applied add-sqr-sqrt_binary64_110126.1

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

  15. Applied *-un-lft-identity_binary64_109906.1

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

  16. Applied add-sqr-sqrt_binary64_110126.1

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

  17. Applied times-frac_binary64_109966.1

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

  18. Applied times-frac_binary64_109966.1

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

  19. Applied times-frac_binary64_109965.9

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

  20. Simplified5.9

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

  21. Simplified5.9

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

  22. Final simplification5.9

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

Reproduce

herbie shell --seed 2020353 
(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)))))