Average Error: 6.3 → 6.3
Time: 4.8s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{1}{x} \cdot \frac{\frac{\frac{1}{y}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{1}{x} \cdot \frac{\frac{\frac{1}{y}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}
double code(double x, double y, double z) {
	return ((double) (((double) (1.0 / x)) / ((double) (y * ((double) (1.0 + ((double) (z * z))))))));
}
↓
double code(double x, double y, double z) {
	return ((double) (((double) (1.0 / x)) * ((double) (((double) (((double) (1.0 / y)) / ((double) sqrt(((double) fma(z, z, 1.0)))))) / ((double) sqrt(((double) fma(z, 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

Original6.3
Target5.7
Herbie6.3
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.3

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

  3. Applied div-inv6.4

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

  4. Simplified6.3

    \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{\frac{1}{y}}{\mathsf{fma}\left(z, z, 1\right)}}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt6.3

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

  7. Applied associate-/r*6.3

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

  8. Final simplification6.3

    \[\leadsto \frac{1}{x} \cdot \frac{\frac{\frac{1}{y}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}\]

Reproduce

herbie shell --seed 2020120 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< (* y (+ 1 (* z z))) #f) (/ (/ 1 y) (* (+ 1 (* z z)) x)) (if (< (* y (+ 1 (* z z))) 8.680743250567252e+305) (/ (/ 1 x) (* (+ 1 (* z z)) y)) (/ (/ 1 y) (* (+ 1 (* z z)) x))))

  (/ (/ 1 x) (* y (+ 1 (* z z)))))