?

Average Error: 9.46% → 9.66%
Time: 23.9s
Precision: binary64

?

\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
↓
\[\frac{\frac{1}{y}}{\mathsf{fma}\left(z, z, 1\right) \cdot x} \]
(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))
↓
(FPCore (x y z) :precision binary64 (/ (/ 1.0 y) (* (fma z z 1.0) x)))
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) {
	return (1.0 / y) / (fma(z, z, 1.0) * x);
}

function code(x, y, z)
	return Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))))
end
↓
function code(x, y, z)
	return Float64(Float64(1.0 / y) / Float64(fma(z, z, 1.0) * x))
end

code[x_, y_, z_] := N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_] := N[(N[(1.0 / y), $MachinePrecision] / N[(N[(z * z + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{1}{y}}{\mathsf{fma}\left(z, z, 1\right) \cdot x}

Error?

Target

Original9.46%
Target8.54%
Herbie9.66%
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array} \]

Derivation?

  1. Initial program 9.46

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

  2. Simplified9.84

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

  3. Taylor expanded in x around 0 10.08

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

  4. Simplified10.08

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

  5. Applied egg-rr9.66

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

Reproduce?

herbie shell --seed 2023136 
(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)))))