fabs fraction 1

Average Error: 1.6 → 0.6

Time: 9.1s

Precision: binary64

Cost: 13641

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+120} \lor \neg \left(x \leq 2 \cdot 10^{-94}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.5e+120) (not (<= x 2e-94)))
   (fabs (fma x (/ z y) (/ (- -4.0 x) y)))
   (fabs (/ (- (+ x 4.0) (* x z)) 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 <= -1.5e+120) || !(x <= 2e-94)) {
		tmp = fabs(fma(x, (z / y), ((-4.0 - x) / y)));
	} else {
		tmp = fabs((((x + 4.0) - (x * z)) / 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 <= -1.5e+120) || !(x <= 2e-94))
		tmp = abs(fma(x, Float64(z / y), Float64(Float64(-4.0 - x) / y)));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / 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[x, -1.5e+120], N[Not[LessEqual[x, 2e-94]], $MachinePrecision]], N[Abs[N[(x * N[(z / y), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5e120 or 1.9999999999999999e-94 < x

   1. Initial program 0.4

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

   2. Simplified 0.6

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

      [Start] 0.4	\[ \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
      fabs-sub [=>] 0.4	\[ \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]
      associate-*l/ [=>] 7.5	\[ \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]
      associate-*r/ [<=] 0.6	\[ \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]
      *-commutative [<=] 0.6	\[ \left|\color{blue}{\frac{z}{y} \cdot x} - \frac{x + 4}{y}\right| \]
      *-commutative [=>] 0.6	\[ \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]
      fma-neg [=>] 0.6	\[ \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]
      distribute-neg-frac [=>] 0.6	\[ \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]
      neg-sub0 [=>] 0.6	\[ \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{0 - \left(x + 4\right)}}{y}\right)\right| \]
      +-commutative [=>] 0.6	\[ \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{0 - \color{blue}{\left(4 + x\right)}}{y}\right)\right| \]
      associate--r+ [=>] 0.6	\[ \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(0 - 4\right) - x}}{y}\right)\right| \]
      metadata-eval [=>] 0.6	\[ \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

   if -1.5e120 < x < 1.9999999999999999e-94

   1. Initial program 2.3

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

   2. Taylor expanded in x around 0 5.0

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

   3. Simplified 0.6

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

      [Start] 5.0	\[ \left|4 \cdot \frac{1}{y} + \left(\frac{1}{y} - \frac{z}{y}\right) \cdot x\right| \]
      *-commutative [=>] 5.0	\[ \left|4 \cdot \frac{1}{y} + \color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]
      sub-neg [=>] 5.0	\[ \left|4 \cdot \frac{1}{y} + x \cdot \color{blue}{\left(\frac{1}{y} + \left(-\frac{z}{y}\right)\right)}\right| \]
      mul-1-neg [<=] 5.0	\[ \left|4 \cdot \frac{1}{y} + x \cdot \left(\frac{1}{y} + \color{blue}{-1 \cdot \frac{z}{y}}\right)\right| \]
      distribute-rgt-in [=>] 5.0	\[ \left|4 \cdot \frac{1}{y} + \color{blue}{\left(\frac{1}{y} \cdot x + \left(-1 \cdot \frac{z}{y}\right) \cdot x\right)}\right| \]
      *-commutative [<=] 5.0	\[ \left|4 \cdot \frac{1}{y} + \left(\color{blue}{x \cdot \frac{1}{y}} + \left(-1 \cdot \frac{z}{y}\right) \cdot x\right)\right| \]
      associate-*r* [<=] 5.0	\[ \left|4 \cdot \frac{1}{y} + \left(x \cdot \frac{1}{y} + \color{blue}{-1 \cdot \left(\frac{z}{y} \cdot x\right)}\right)\right| \]
      associate-*l/ [=>] 0.6	\[ \left|4 \cdot \frac{1}{y} + \left(x \cdot \frac{1}{y} + -1 \cdot \color{blue}{\frac{z \cdot x}{y}}\right)\right| \]
      mul-1-neg [=>] 0.6	\[ \left|4 \cdot \frac{1}{y} + \left(x \cdot \frac{1}{y} + \color{blue}{\left(-\frac{z \cdot x}{y}\right)}\right)\right| \]
      *-commutative [=>] 0.6	\[ \left|4 \cdot \frac{1}{y} + \left(x \cdot \frac{1}{y} + \left(-\frac{\color{blue}{x \cdot z}}{y}\right)\right)\right| \]
      associate-*l/ [<=] 2.4	\[ \left|4 \cdot \frac{1}{y} + \left(x \cdot \frac{1}{y} + \left(-\color{blue}{\frac{x}{y} \cdot z}\right)\right)\right| \]
      distribute-rgt-neg-out [<=] 2.4	\[ \left|4 \cdot \frac{1}{y} + \left(x \cdot \frac{1}{y} + \color{blue}{\frac{x}{y} \cdot \left(-z\right)}\right)\right| \]
      associate-+r+ [=>] 2.4	\[ \left|\color{blue}{\left(4 \cdot \frac{1}{y} + x \cdot \frac{1}{y}\right) + \frac{x}{y} \cdot \left(-z\right)}\right| \]
      +-commutative [<=] 2.4	\[ \left|\color{blue}{\left(x \cdot \frac{1}{y} + 4 \cdot \frac{1}{y}\right)} + \frac{x}{y} \cdot \left(-z\right)\right| \]
      distribute-rgt-in [<=] 2.4	\[ \left|\color{blue}{\frac{1}{y} \cdot \left(x + 4\right)} + \frac{x}{y} \cdot \left(-z\right)\right| \]
      associate-*l/ [=>] 2.3	\[ \left|\color{blue}{\frac{1 \cdot \left(x + 4\right)}{y}} + \frac{x}{y} \cdot \left(-z\right)\right| \]
      *-lft-identity [=>] 2.3	\[ \left|\frac{\color{blue}{x + 4}}{y} + \frac{x}{y} \cdot \left(-z\right)\right| \]
      distribute-rgt-neg-out [=>] 2.3	\[ \left|\frac{x + 4}{y} + \color{blue}{\left(-\frac{x}{y} \cdot z\right)}\right| \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+120} \lor \neg \left(x \leq 2 \cdot 10^{-94}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	21448

\[\begin{array}{l} t_0 := \frac{x + 4}{y}\\ t_1 := \left|t_0 - z \cdot \frac{x}{y}\right|\\ \mathbf{if}\;t_1 \leq 5 \cdot 10^{-82}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+241}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}} - t_0\right|\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	0.5
Cost	8648

\[\begin{array}{l} t_0 := \frac{x + 4}{y}\\ t_1 := t_0 - z \cdot \frac{x}{y}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+109}:\\ \;\;\;\;\left|t_0 + \frac{-1}{\frac{\frac{y}{x}}{z}}\right|\\ \mathbf{elif}\;t_1 \leq 10^{-127}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}} - t_0\right|\\ \end{array} \]

Alternative 3
Error	0.5
Cost	7369

\[\begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+120} \lor \neg \left(x \leq 240000000\right):\\ \;\;\;\;\left|\frac{x}{y} - \frac{1}{\frac{\frac{y}{x}}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 4
Error	0.4
Cost	7368

\[\begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+120}:\\ \;\;\;\;\left|\frac{x}{y} - \frac{1}{\frac{\frac{y}{x}}{z}}\right|\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-18}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 5
Error	11.5
Cost	6984

\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+49}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+63}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]

Alternative 6
Error	3.3
Cost	6976

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

Alternative 7
Error	18.2
Cost	6857

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

Alternative 8
Error	32.3
Cost	6592

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

Reproduce

herbie shell --seed 2023012 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))