Average Error: 6.4 → 1.4
Time: 8.1s
Precision: binary64
\[[x, y] = \mathsf{sort}([x, y]) \\]
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt{\mathsf{hypot}\left(1, z\right)}\\ \frac{1}{t_0} \cdot \left(\frac{\frac{1}{x}}{t_0} \cdot \frac{\frac{1}{\mathsf{hypot}\left(1, z\right)}}{y}\right) \end{array} \]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
t_0 := \sqrt{\mathsf{hypot}\left(1, z\right)}\\
\frac{1}{t_0} \cdot \left(\frac{\frac{1}{x}}{t_0} \cdot \frac{\frac{1}{\mathsf{hypot}\left(1, z\right)}}{y}\right)
\end{array}

(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (sqrt (hypot 1.0 z))))
   (* (/ 1.0 t_0) (* (/ (/ 1.0 x) t_0) (/ (/ 1.0 (hypot 1.0 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) {
	double t_0 = sqrt(hypot(1.0, z));
	return (1.0 / t_0) * (((1.0 / x) / t_0) * ((1.0 / hypot(1.0, 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

Original6.4
Target4.8
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array} \]

Derivation

  1. Initial program 6.4

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

  2. Simplified6.4

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

  3. Applied associate-/r*_binary646.3

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

  4. Applied add-sqr-sqrt_binary646.3

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

  5. Applied div-inv_binary646.3

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

  6. Applied times-frac_binary645.9

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

  7. Simplified5.9

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

  8. Simplified1.6

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

  9. Applied add-sqr-sqrt_binary641.6

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

  10. Applied *-un-lft-identity_binary641.6

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

  11. Applied times-frac_binary641.6

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

  12. Applied associate-*l*_binary641.4

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

  13. Final simplification1.4

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

Reproduce

herbie shell --seed 2022095 
(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)))))