Average Error: 6.5 → 2.8
Time: 5.2s
Precision: binary64
\[[x, y]=\mathsf{sort}([x, y])\]
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -2.570544231834083 \cdot 10^{-59}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{1 + z \cdot z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(x + z \cdot \left(x \cdot z\right)\right)}\\ \end{array}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -2.570544231834083 \cdot 10^{-59}:\\
\;\;\;\;\frac{\frac{\frac{1}{y}}{1 + z \cdot z}}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{y \cdot \left(x + z \cdot \left(x \cdot z\right)\right)}\\

\end{array}
(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= x -2.570544231834083e-59)
   (/ (/ (/ 1.0 y) (+ 1.0 (* z z))) x)
   (/ 1.0 (* y (+ x (* z (* x 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 (x <= -2.570544231834083e-59) {
		tmp = ((1.0 / y) / (1.0 + (z * z))) / x;
	} else {
		tmp = 1.0 / (y * (x + (z * (x * 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
Target5.4
Herbie2.8
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -2.570544231834083e-59

    1. Initial program 2.5

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

    3. Applied div-inv_binary642.5

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

    4. Simplified2.5

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

    6. Applied associate-*l/_binary642.5

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

    7. Simplified2.5

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

    9. Applied associate-/r*_binary642.5

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

    if -2.570544231834083e-59 < x

    1. Initial program 11.3

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

    3. Applied div-inv_binary6411.3

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

    4. Simplified11.3

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

    6. Applied associate-*l/_binary6411.3

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

    7. Simplified11.3

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

    8. Taylor expanded around 0 9.0

      \[\leadsto \color{blue}{\frac{1}{y \cdot \left(x + x \cdot {z}^{2}\right)}}\]

    9. Simplified9.0

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

    11. Applied associate-*l*_binary643.1

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

    12. Simplified3.1

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

  4. Final simplification2.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.570544231834083 \cdot 10^{-59}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{1 + z \cdot z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(x + z \cdot \left(x \cdot z\right)\right)}\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021118 
(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)))))