fabs fraction 1

Average Accuracy: 97.5% → 99.8%

Time: 10.2s

Precision: binary64

Cost: 13508

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+23}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))

↓

(FPCore (x y z)
 :precision binary64
 (if (<= x -2e+23)
   (fabs (fma x (/ z y) (/ (- -4.0 x) y)))
   (if (<= x 2e-11)
     (fabs (/ (- (+ x 4.0) (* x z)) y))
     (fabs (- (/ (+ x 4.0) y) (* z (/ x y)))))))

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 tmp;
	if (x <= -2e+23) {
		tmp = fabs(fma(x, (z / y), ((-4.0 - x) / y)));
	} else if (x <= 2e-11) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs((((x + 4.0) / y) - (z * (x / y))));
	}
	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)
	tmp = 0.0
	if (x <= -2e+23)
		tmp = abs(fma(x, Float64(z / y), Float64(Float64(-4.0 - x) / y)));
	elseif (x <= 2e-11)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(z * Float64(x / y))));
	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_] := If[LessEqual[x, -2e+23], N[Abs[N[(x * N[(z / y), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2e-11], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.9999999999999998e23

   1. Initial program 99.8%

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

   2. Simplified 99.8%

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

      [Start] 99.8	\[ \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
      fabs-sub [=>] 99.8	\[ \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]
      associate-*l/ [=>] 85.5	\[ \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]
      associate-*r/ [<=] 99.8	\[ \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]
      *-commutative [<=] 99.8	\[ \left|\color{blue}{\frac{z}{y} \cdot x} - \frac{x + 4}{y}\right| \]
      *-commutative [=>] 99.8	\[ \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]
      fma-neg [=>] 99.8	\[ \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]
      distribute-neg-frac [=>] 99.8	\[ \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]
      neg-sub0 [=>] 99.8	\[ \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{0 - \left(x + 4\right)}}{y}\right)\right| \]
      +-commutative [=>] 99.8	\[ \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{0 - \color{blue}{\left(4 + x\right)}}{y}\right)\right| \]
      associate--r+ [=>] 99.8	\[ \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(0 - 4\right) - x}}{y}\right)\right| \]
      metadata-eval [=>] 99.8	\[ \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   if -1.9999999999999998e23 < x < 1.99999999999999988e-11

   1. Initial program 96.0%

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

   2. Applied egg-rr 99.8%

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

      [Start] 96.0	\[ \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
      associate-*l/ [=>] 99.8	\[ \left|\frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}}\right| \]
      sub-div [=>] 99.8	\[ \left|\color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}}\right| \]

   if 1.99999999999999988e-11 < x

   1. Initial program 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+23}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	67.4%
Cost	7778

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \left|z \cdot \frac{x}{y}\right|\\ \mathbf{if}\;x \leq -2.55 \cdot 10^{+125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-19}:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{elif}\;x \leq 4000000000000 \lor \neg \left(x \leq 4.5 \cdot 10^{+93}\right) \land \left(x \leq 6.8 \cdot 10^{+103} \lor \neg \left(x \leq 2.2 \cdot 10^{+211}\right) \land x \leq 3.8 \cdot 10^{+237}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Accuracy	67.6%
Cost	7777

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \left|z \cdot \frac{x}{y}\right|\\ \mathbf{if}\;x \leq -3.3 \cdot 10^{+128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-19}:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+93}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+104}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+212} \lor \neg \left(x \leq 1.9 \cdot 10^{+234}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	67.3%
Cost	7777

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \left|z \cdot \frac{x}{y}\right|\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{+125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-37}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-19}:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{elif}\;x \leq 22500000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+93}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+107}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+212} \lor \neg \left(x \leq 5.3 \cdot 10^{+241}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	99.8%
Cost	7369

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+28} \lor \neg \left(x \leq 0.5\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 5
Accuracy	99.8%
Cost	7368

\[\begin{array}{l} t_0 := \frac{x + 4}{y}\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{+33}:\\ \;\;\;\;\left|t_0 - \frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t_0 - z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 6
Accuracy	99.7%
Cost	7241

\[\begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+33} \lor \neg \left(x \leq 1.25 \cdot 10^{+24}\right):\\ \;\;\;\;\left|\frac{1 - z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 7
Accuracy	85.3%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{-37} \lor \neg \left(x \leq 3.8 \cdot 10^{-19}\right):\\ \;\;\;\;\left|\frac{1 - z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \end{array} \]

Alternative 8
Accuracy	80.7%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+187}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+90}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 9
Accuracy	69.9%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \end{array} \]

Alternative 10
Accuracy	48.0%
Cost	6592

\[\frac{4}{\left|y\right|} \]

Reproduce

herbie shell --seed 2023135 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))