Average Error: 15.2 → 3.3
Time: 6.9s
Precision: binary64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{\frac{x}{z}}{\frac{z}{\frac{y}{z + 1}}}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{\frac{x}{z}}{\frac{z}{\frac{y}{z + 1}}}
(FPCore (x y z) :precision binary64 (/ (* x y) (* (* z z) (+ z 1.0))))
↓
(FPCore (x y z) :precision binary64 (/ (/ x z) (/ z (/ y (+ z 1.0)))))
double code(double x, double y, double z) {
	return (((double) (x * y)) / ((double) (((double) (z * z)) * ((double) (z + 1.0)))));
}
↓
double code(double x, double y, double z) {
	return ((x / z) / (z / (y / ((double) (z + 1.0)))));
}

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

Original15.2
Target4.1
Herbie3.3
\[\begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 15.2

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

  3. Applied associate-/l*_binary6413.9

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

  4. Simplified12.2

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

  6. Applied *-un-lft-identity_binary6412.2

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

  7. Applied *-un-lft-identity_binary6412.2

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

  8. Applied times-frac_binary6412.2

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

  9. Applied times-frac_binary648.1

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

  10. Applied associate-/r*_binary643.3

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

  11. Simplified3.3

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

  12. Final simplification3.3

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

Reproduce

herbie shell --seed 2020210 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1.0 z)) x) z))

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