Average Error: 6.5 → 0.8
Time: 6.6s
Precision: binary64
\[[x, y] = \mathsf{sort}([x, y]) \\]
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq 2.383641413898529 \cdot 10^{-235}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right) \cdot \left(y \cdot \mathsf{hypot}\left(1, z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)}}{y}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array} \]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;y \leq 2.383641413898529 \cdot 10^{-235}:\\
\;\;\;\;\frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right) \cdot \left(y \cdot \mathsf{hypot}\left(1, z\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)}}{y}}{\mathsf{hypot}\left(1, z\right)}\\


\end{array}

(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.383641413898529e-235)
   (/ (/ 1.0 x) (* (hypot 1.0 z) (* y (hypot 1.0 z))))
   (/ (/ (/ (/ 1.0 x) (hypot 1.0 z)) y) (hypot 1.0 z))))
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 tmp;
	if (y <= 2.383641413898529e-235) {
		tmp = (1.0 / x) / (hypot(1.0, z) * (y * hypot(1.0, z)));
	} else {
		tmp = (((1.0 / x) / hypot(1.0, z)) / y) / hypot(1.0, z);
	}
	return tmp;
}

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
Target5.2
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if y < 2.383641413898529e-235

    1. Initial program 6.7

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

    2. Simplified6.7

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

    3. Applied *-un-lft-identity_binary646.7

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

    4. Applied times-frac_binary649.4

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

    5. Applied add-sqr-sqrt_binary649.4

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

    6. Applied *-un-lft-identity_binary649.4

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

    7. Applied add-cube-cbrt_binary649.4

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

    8. Applied times-frac_binary649.4

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

    9. Applied times-frac_binary649.4

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

    10. Simplified9.4

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

    11. Simplified7.5

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

    12. Applied associate-*l/_binary647.5

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

    13. Applied associate-*r/_binary643.1

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

    14. Applied associate-*r/_binary643.1

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

    15. Applied frac-times_binary641.6

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

    16. Applied associate-/l/_binary640.9

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

    if 2.383641413898529e-235 < y

    1. Initial program 6.5

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

    2. Simplified6.5

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

    3. Applied *-un-lft-identity_binary646.5

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

    4. Applied times-frac_binary645.3

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

    5. Applied add-sqr-sqrt_binary645.3

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

    6. Applied *-un-lft-identity_binary645.3

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

    7. Applied add-cube-cbrt_binary645.3

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

    8. Applied times-frac_binary645.3

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

    9. Applied times-frac_binary645.3

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

    10. Simplified5.3

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

    11. Simplified2.1

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

    12. Applied associate-*l/_binary642.0

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

    13. Applied associate-*r/_binary640.9

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

    14. Applied associate-*l/_binary640.8

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

    15. Simplified0.8

      \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)}}}{y}}{\mathsf{hypot}\left(1, z\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.383641413898529 \cdot 10^{-235}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right) \cdot \left(y \cdot \mathsf{hypot}\left(1, z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)}}{y}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(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)))))