Average Error: 1.6 → 0.5
Time: 3.5s
Precision: binary64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
↓
\[\begin{array}{l} t_0 := \frac{x + 4}{y}\\ \mathbf{if}\;\left|t_0 - \frac{x}{y} \cdot z\right| \leq 5 \cdot 10^{-163}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, 1 - z, 4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t_0 - \frac{z}{\frac{y}{x}}\right|\\ \end{array} \]
(FPCore (x y z) :precision binary64 (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x 4.0) y)))
   (if (<= (fabs (- t_0 (* (/ x y) z))) 5e-163)
     (fabs (/ (fma x (- 1.0 z) 4.0) y))
     (fabs (- t_0 (/ z (/ y x)))))))
double code(double x, double y, double z) {
	return fabs((((x + 4.0) / y) - ((x / y) * z)));
}
↓
double code(double x, double y, double z) {
	double t_0 = (x + 4.0) / y;
	double tmp;
	if (fabs((t_0 - ((x / y) * z))) <= 5e-163) {
		tmp = fabs((fma(x, (1.0 - z), 4.0) / y));
	} else {
		tmp = fabs((t_0 - (z / (y / x))));
	}
	return tmp;
}

function code(x, y, z)
	return abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(Float64(x / y) * z)))
end
↓
function code(x, y, z)
	t_0 = Float64(Float64(x + 4.0) / y)
	tmp = 0.0
	if (abs(Float64(t_0 - Float64(Float64(x / y) * z))) <= 5e-163)
		tmp = abs(Float64(fma(x, Float64(1.0 - z), 4.0) / y));
	else
		tmp = abs(Float64(t_0 - Float64(z / Float64(y / x))));
	end
	return tmp
end

code[x_, y_, z_] := N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]
↓
code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[N[Abs[N[(t$95$0 - N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 5e-163], N[Abs[N[(N[(x * N[(1.0 - z), $MachinePrecision] + 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$0 - N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
t_0 := \frac{x + 4}{y}\\
\mathbf{if}\;\left|t_0 - \frac{x}{y} \cdot z\right| \leq 5 \cdot 10^{-163}:\\
\;\;\;\;\left|\frac{\mathsf{fma}\left(x, 1 - z, 4\right)}{y}\right|\\

\mathbf{else}:\\
\;\;\;\;\left|t_0 - \frac{z}{\frac{y}{x}}\right|\\


\end{array}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Split input into 2 regimes

  2. if (fabs.f64 (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z))) < 4.99999999999999977e-163

    1. Initial program 8.3

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

    2. Applied egg-rr8.7

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\frac{z}{\frac{y}{x}}}\right| \]

    3. Taylor expanded in x around 0 0.1

      \[\leadsto \left|\color{blue}{\left(\frac{x}{y} + 4 \cdot \frac{1}{y}\right) - \frac{z \cdot x}{y}}\right| \]

    4. Simplified0.1

      \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(x, 1 - z, 4\right)}{y}}\right| \]

    if 4.99999999999999977e-163 < (fabs.f64 (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z)))

    1. Initial program 0.4

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]

    2. Applied egg-rr0.5

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\frac{z}{\frac{y}{x}}}\right| \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \leq 5 \cdot 10^{-163}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, 1 - z, 4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{z}{\frac{y}{x}}\right|\\ \end{array} \]

Reproduce

herbie shell --seed 2022156 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))