fabs fraction 1

Average Accuracy: 97.4% → 99.8%

Time: 10.5s

Precision: binary64

Cost: 13641

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -50000000000 \lor \neg \left(y \leq 2 \cdot 10^{-29}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}} - \frac{x + 4}{y}\right|\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -50000000000.0) (not (<= y 2e-29)))
   (fabs (fma x (/ z y) (/ (- -4.0 x) y)))
   (fabs (- (/ z (/ y x)) (/ (+ x 4.0) 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 ((y <= -50000000000.0) || !(y <= 2e-29)) {
		tmp = fabs(fma(x, (z / y), ((-4.0 - x) / y)));
	} else {
		tmp = fabs(((z / (y / x)) - ((x + 4.0) / 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 ((y <= -50000000000.0) || !(y <= 2e-29))
		tmp = abs(fma(x, Float64(z / y), Float64(Float64(-4.0 - x) / y)));
	else
		tmp = abs(Float64(Float64(z / Float64(y / x)) - Float64(Float64(x + 4.0) / 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[Or[LessEqual[y, -50000000000.0], N[Not[LessEqual[y, 2e-29]], $MachinePrecision]], N[Abs[N[(x * N[(z / y), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5e10 or 1.99999999999999989e-29 < y

   1. Initial program 95.9%

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

   2. Simplified 99.8%

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

      [Start] 95.9	\[ \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
      fabs-sub [=>] 95.9	\[ \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]
      associate-*l/ [=>] 91.9	\[ \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]
      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 -5e10 < y < 1.99999999999999989e-29

   1. Initial program 99.9%

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

   2. Simplified 99.9%

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

      [Start] 99.9	\[ \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
      *-lft-identity [<=] 99.9	\[ \color{blue}{1 \cdot \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]
      metadata-eval [<=] 99.9	\[ \color{blue}{\left|-1\right|} \cdot \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
      fabs-sub [=>] 99.9	\[ \left|-1\right| \cdot \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]
      fabs-mul [<=] 99.9	\[ \color{blue}{\left|-1 \cdot \left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right|} \]
      neg-mul-1 [<=] 99.9	\[ \left|\color{blue}{-\left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)}\right| \]
      sub0-neg [<=] 99.9	\[ \left|\color{blue}{0 - \left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)}\right| \]
      associate-+l- [<=] 99.9	\[ \left|\color{blue}{\left(0 - \frac{x}{y} \cdot z\right) + \frac{x + 4}{y}}\right| \]
      neg-sub0 [<=] 99.9	\[ \left|\color{blue}{\left(-\frac{x}{y} \cdot z\right)} + \frac{x + 4}{y}\right| \]
      +-commutative [<=] 99.9	\[ \left|\color{blue}{\frac{x + 4}{y} + \left(-\frac{x}{y} \cdot z\right)}\right| \]
      sub-neg [<=] 99.9	\[ \left|\color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z}\right| \]
      associate-*l/ [=>] 99.9	\[ \left|\frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}}\right| \]
      *-commutative [=>] 99.9	\[ \left|\frac{x + 4}{y} - \frac{\color{blue}{z \cdot x}}{y}\right| \]
      associate-/l* [=>] 99.9	\[ \left|\frac{x + 4}{y} - \color{blue}{\frac{z}{\frac{y}{x}}}\right| \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -50000000000 \lor \neg \left(y \leq 2 \cdot 10^{-29}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}} - \frac{x + 4}{y}\right|\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	14276

\[\begin{array}{l} t_0 := \left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \mathbf{if}\;t_0 \leq 1.2 \cdot 10^{+53}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Accuracy	99.6%
Cost	8776

\[\begin{array}{l} t_0 := \frac{x + 4}{y}\\ t_1 := t_0 - z \cdot \frac{x}{y}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+85}:\\ \;\;\;\;\left|t_1\right|\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\left|t_0 - \frac{\frac{x}{\frac{1}{z}}}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}} - t_0\right|\\ \end{array} \]

Alternative 3
Accuracy	99.6%
Cost	8648

\[\begin{array}{l} t_0 := \frac{x + 4}{y}\\ t_1 := t_0 - z \cdot \frac{x}{y}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+85}:\\ \;\;\;\;\left|t_1\right|\\ \mathbf{elif}\;t_1 \leq 10^{+40}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}} - t_0\right|\\ \end{array} \]

Alternative 4
Accuracy	78.8%
Cost	7248

\[\begin{array}{l} t_0 := \left|\frac{x + 4}{y}\right|\\ \mathbf{if}\;z \leq -9.5 \cdot 10^{+151}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{+41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -310:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 10^{+200}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \]

Alternative 5
Accuracy	79.0%
Cost	7248

\[\begin{array}{l} t_0 := \left|\frac{x + 4}{y}\right|\\ \mathbf{if}\;z \leq -4 \cdot 10^{+150}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{+41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -310:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{elif}\;z \leq 10^{+200}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \]

Alternative 6
Accuracy	98.5%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -1.6 \lor \neg \left(x \leq 4.3\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 7
Accuracy	96.9%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{+114}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 8
Accuracy	69.9%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;x \leq -1.52 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \end{array} \]

Alternative 9
Accuracy	47.8%
Cost	6592

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

Reproduce

herbie shell --seed 2023122 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))