Average Error: 1.6 → 0.2
Time: 4.1s
Precision: binary64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
↓
\[\begin{array}{l} t_0 := \left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{if}\;x \leq -6.637804509326453 \cdot 10^{-23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.957230781506295 \cdot 10^{-53}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, 1 - z, 4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
(FPCore (x y z) :precision binary64 (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (- (/ (+ x 4.0) y) (* x (/ z y))))))
   (if (<= x -6.637804509326453e-23)
     t_0
     (if (<= x 2.957230781506295e-53)
       (fabs (/ (fma x (- 1.0 z) 4.0) y))
       t_0))))
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 = fabs((((x + 4.0) / y) - (x * (z / y))));
	double tmp;
	if (x <= -6.637804509326453e-23) {
		tmp = t_0;
	} else if (x <= 2.957230781506295e-53) {
		tmp = fabs((fma(x, (1.0 - z), 4.0) / y));
	} else {
		tmp = t_0;
	}
	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 = abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(x * Float64(z / y))))
	tmp = 0.0
	if (x <= -6.637804509326453e-23)
		tmp = t_0;
	elseif (x <= 2.957230781506295e-53)
		tmp = abs(Float64(fma(x, Float64(1.0 - z), 4.0) / y));
	else
		tmp = t_0;
	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[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -6.637804509326453e-23], t$95$0, If[LessEqual[x, 2.957230781506295e-53], N[Abs[N[(N[(x * N[(1.0 - z), $MachinePrecision] + 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
t_0 := \left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\
\mathbf{if}\;x \leq -6.637804509326453 \cdot 10^{-23}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;x \leq 2.957230781506295 \cdot 10^{-53}:\\
\;\;\;\;\left|\frac{\mathsf{fma}\left(x, 1 - z, 4\right)}{y}\right|\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Split input into 2 regimes

  2. if x < -6.6378045093264531e-23 or 2.9572307815062952e-53 < x

    1. Initial program 0.2

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

    2. Applied egg-rr0.3

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

    3. Applied egg-rr0.3

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

    if -6.6378045093264531e-23 < x < 2.9572307815062952e-53

    1. Initial program 2.7

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

    2. Taylor expanded in x around 0 0.1

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

    3. Simplified0.1

      \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(x, 1 - z, 4\right)}{y}}\right| \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.637804509326453 \cdot 10^{-23}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 2.957230781506295 \cdot 10^{-53}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(x, 1 - z, 4\right)}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))