fabs fraction 1

Average Error: 1.6 → 0.4

Time: 9.2s

Precision: binary64

Cost: 13704

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + 4}{y}\\ \mathbf{if}\;x \leq -4.0129034545876327 \cdot 10^{-119}:\\ \;\;\;\;\left|t_0 - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 4.0909830144832054 \cdot 10^{-48}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{-x}{y}, z, t_0\right)\right|\\ \end{array} \]

FPCore

(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 (<= x -4.0129034545876327e-119)
     (fabs (- t_0 (* x (/ z y))))
     (if (<= x 4.0909830144832054e-48)
       (fabs (/ (- (+ x 4.0) (* x z)) y))
       (fabs (fma (/ (- x) y) z t_0))))))

C

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 (x <= -4.0129034545876327e-119) {
		tmp = fabs((t_0 - (x * (z / y))));
	} else if (x <= 4.0909830144832054e-48) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs(fma((-x / y), z, t_0));
	}
	return tmp;
}

Julia

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 (x <= -4.0129034545876327e-119)
		tmp = abs(Float64(t_0 - Float64(x * Float64(z / y))));
	elseif (x <= 4.0909830144832054e-48)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(fma(Float64(Float64(-x) / y), z, t_0));
	end
	return tmp
end

Wolfram

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[x, -4.0129034545876327e-119], N[Abs[N[(t$95$0 - N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.0909830144832054e-48], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[((-x) / y), $MachinePrecision] * z + t$95$0), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.0129034545876327e-119

   1. Initial program 0.7

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

   2. Applied egg-rr 1.0

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

   3. Applied egg-rr 1.1

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

   if -4.0129034545876327e-119 < x < 4.0909830144832054e-48

   1. Initial program 3.0

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

   2. Applied egg-rr 6.0

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

   3. Applied egg-rr 6.6

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

   4. Applied egg-rr 0.1

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

   if 4.0909830144832054e-48 < x

   1. Initial program 0.3

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

   2. Applied egg-rr 0.3

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.0129034545876327 \cdot 10^{-119}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 4.0909830144832054 \cdot 10^{-48}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{-x}{y}, z, \frac{x + 4}{y}\right)\right|\\ \end{array} \]

Alternatives

Alternative 1
Error	26.1
Cost	7776

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \left|x \cdot \frac{z}{y}\right|\\ t_2 := \frac{4}{\left|y\right|}\\ \mathbf{if}\;z \leq -3.497344164773292 \cdot 10^{+45}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.0169956265442025 \cdot 10^{-216}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 6.214043346107026 \cdot 10^{-177}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.6208799348167227 \cdot 10^{-126}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.8697493691382045 \cdot 10^{-122}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.75453163950379 \cdot 10^{+19}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.008581866647103 \cdot 10^{+50}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;z \leq 6.329713134161369 \cdot 10^{+67}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	26.2
Cost	7776

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \left|x \cdot \frac{z}{y}\right|\\ t_2 := \frac{4}{\left|y\right|}\\ \mathbf{if}\;z \leq -3.497344164773292 \cdot 10^{+45}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.0169956265442025 \cdot 10^{-216}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 6.214043346107026 \cdot 10^{-177}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.6208799348167227 \cdot 10^{-126}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.8697493691382045 \cdot 10^{-122}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.75453163950379 \cdot 10^{+19}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.008581866647103 \cdot 10^{+50}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 6.329713134161369 \cdot 10^{+67}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	21.0
Cost	7380

\[\begin{array}{l} t_0 := \left|\frac{z}{\frac{y}{x}}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -67035756125322470:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -8.706294842419482 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.8679920730288124 \cdot 10^{-27}:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{elif}\;x \leq 0.44812270334897025:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	21.0
Cost	7380

\[\begin{array}{l} t_0 := \left|z \cdot \frac{x}{y}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -67035756125322470:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -8.706294842419482 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.8679920730288124 \cdot 10^{-27}:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{elif}\;x \leq 0.44812270334897025:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	11.6
Cost	7248

\[\begin{array}{l} t_0 := \left|x \cdot \frac{z}{y}\right|\\ \mathbf{if}\;z \leq -3.497344164773292 \cdot 10^{+45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.75453163950379 \cdot 10^{+19}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{elif}\;z \leq 1.008581866647103 \cdot 10^{+50}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 6.329713134161369 \cdot 10^{+67}:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	11.5
Cost	7248

\[\begin{array}{l} t_0 := \left|x \cdot \frac{z}{y}\right|\\ \mathbf{if}\;z \leq -3.497344164773292 \cdot 10^{+45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.20204895964229214:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{elif}\;z \leq 1.008581866647103 \cdot 10^{+50}:\\ \;\;\;\;\left|\frac{1 - z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;z \leq 6.329713134161369 \cdot 10^{+67}:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	11.5
Cost	7248

\[\begin{array}{l} t_0 := \left|x \cdot \frac{z}{y}\right|\\ \mathbf{if}\;z \leq -3.497344164773292 \cdot 10^{+45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.20204895964229214:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{elif}\;z \leq 1.008581866647103 \cdot 10^{+50}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{elif}\;z \leq 6.329713134161369 \cdot 10^{+67}:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	0.4
Cost	7240

\[\begin{array}{l} t_0 := \left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{if}\;x \leq -2 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 343.3405173301067:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	0.6
Cost	7240

\[\begin{array}{l} \mathbf{if}\;x \leq -4.0129034545876327 \cdot 10^{-119}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 343.3405173301067:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \end{array} \]

Alternative 10
Error	19.5
Cost	6856

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -266784019705.23517:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.44812270334897025:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	46.7
Cost	6592

\[\left|\frac{x}{y}\right| \]

Reproduce

herbie shell --seed 2022317 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))