fabs fraction 1

Average Accuracy: 97.4% → 99.7%

Time: 10.7s

Precision: binary64

Cost: 13640

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+66}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-43}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.5e+66)
   (fabs (* (/ x y) (- 1.0 z)))
   (if (<= x 5e-43)
     (fabs (/ (- (+ x 4.0) (* x z)) y))
     (fabs (fma x (/ z y) (/ (- -4.0 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 <= -1.5e+66) {
		tmp = fabs(((x / y) * (1.0 - z)));
	} else if (x <= 5e-43) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs(fma(x, (z / y), ((-4.0 - 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 <= -1.5e+66)
		tmp = abs(Float64(Float64(x / y) * Float64(1.0 - z)));
	elseif (x <= 5e-43)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(fma(x, Float64(z / y), Float64(Float64(-4.0 - 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, -1.5e+66], N[Abs[N[(N[(x / y), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 5e-43], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.50000000000000001e66

   1. Initial program 99.8%

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

   2. Simplified 99.8%

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

      [Start] 99.8	\[ \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
      *-lft-identity [<=] 99.8	\[ \color{blue}{1 \cdot \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|} \]
      metadata-eval [<=] 99.8	\[ \color{blue}{\left|-1\right|} \cdot \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
      fabs-sub [=>] 99.8	\[ \left|-1\right| \cdot \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]
      fabs-mul [<=] 99.8	\[ \color{blue}{\left|-1 \cdot \left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)\right|} \]
      neg-mul-1 [<=] 99.8	\[ \left|\color{blue}{-\left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)}\right| \]
      sub0-neg [<=] 99.8	\[ \left|\color{blue}{0 - \left(\frac{x}{y} \cdot z - \frac{x + 4}{y}\right)}\right| \]
      associate-+l- [<=] 99.8	\[ \left|\color{blue}{\left(0 - \frac{x}{y} \cdot z\right) + \frac{x + 4}{y}}\right| \]
      neg-sub0 [<=] 99.8	\[ \left|\color{blue}{\left(-\frac{x}{y} \cdot z\right)} + \frac{x + 4}{y}\right| \]
      +-commutative [<=] 99.8	\[ \left|\color{blue}{\frac{x + 4}{y} + \left(-\frac{x}{y} \cdot z\right)}\right| \]
      sub-neg [<=] 99.8	\[ \left|\color{blue}{\frac{x + 4}{y} - \frac{x}{y} \cdot z}\right| \]
      associate-*l/ [=>] 81.4	\[ \left|\frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}}\right| \]
      *-commutative [=>] 81.4	\[ \left|\frac{x + 4}{y} - \frac{\color{blue}{z \cdot x}}{y}\right| \]
      associate-/l* [=>] 99.8	\[ \left|\frac{x + 4}{y} - \color{blue}{\frac{z}{\frac{y}{x}}}\right| \]

   3. Taylor expanded in x around inf 99.6%

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

   4. Simplified 99.8%

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

      [Start] 99.6	\[ \left|\left(\frac{1}{y} - \frac{z}{y}\right) \cdot x\right| \]
      *-commutative [=>] 99.6	\[ \left|\color{blue}{x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)}\right| \]
      sub-neg [=>] 99.6	\[ \left|x \cdot \color{blue}{\left(\frac{1}{y} + \left(-\frac{z}{y}\right)\right)}\right| \]
      mul-1-neg [<=] 99.6	\[ \left|x \cdot \left(\frac{1}{y} + \color{blue}{-1 \cdot \frac{z}{y}}\right)\right| \]
      +-commutative [=>] 99.6	\[ \left|x \cdot \color{blue}{\left(-1 \cdot \frac{z}{y} + \frac{1}{y}\right)}\right| \]
      associate-*r/ [=>] 99.6	\[ \left|x \cdot \left(\color{blue}{\frac{-1 \cdot z}{y}} + \frac{1}{y}\right)\right| \]
      neg-mul-1 [<=] 99.6	\[ \left|x \cdot \left(\frac{\color{blue}{-z}}{y} + \frac{1}{y}\right)\right| \]
      remove-double-neg [<=] 99.6	\[ \left|x \cdot \left(\frac{-z}{\color{blue}{-\left(-y\right)}} + \frac{1}{y}\right)\right| \]
      distribute-rgt-in [=>] 99.6	\[ \left|\color{blue}{\frac{-z}{-\left(-y\right)} \cdot x + \frac{1}{y} \cdot x}\right| \]
      neg-mul-1 [=>] 99.6	\[ \left|\frac{\color{blue}{-1 \cdot z}}{-\left(-y\right)} \cdot x + \frac{1}{y} \cdot x\right| \]
      remove-double-neg [=>] 99.6	\[ \left|\frac{-1 \cdot z}{\color{blue}{y}} \cdot x + \frac{1}{y} \cdot x\right| \]
      associate-*r/ [<=] 99.6	\[ \left|\color{blue}{\left(-1 \cdot \frac{z}{y}\right)} \cdot x + \frac{1}{y} \cdot x\right| \]
      associate-*r* [<=] 99.6	\[ \left|\color{blue}{-1 \cdot \left(\frac{z}{y} \cdot x\right)} + \frac{1}{y} \cdot x\right| \]
      associate-*l/ [=>] 81.2	\[ \left|-1 \cdot \color{blue}{\frac{z \cdot x}{y}} + \frac{1}{y} \cdot x\right| \]
      associate-*l/ [=>] 81.4	\[ \left|-1 \cdot \frac{z \cdot x}{y} + \color{blue}{\frac{1 \cdot x}{y}}\right| \]
      associate-*r/ [<=] 81.4	\[ \left|-1 \cdot \frac{z \cdot x}{y} + \color{blue}{1 \cdot \frac{x}{y}}\right| \]
      *-lft-identity [=>] 81.4	\[ \left|-1 \cdot \frac{z \cdot x}{y} + \color{blue}{\frac{x}{y}}\right| \]
      associate-*r/ [<=] 99.8	\[ \left|-1 \cdot \color{blue}{\left(z \cdot \frac{x}{y}\right)} + \frac{x}{y}\right| \]
      associate-*r* [=>] 99.8	\[ \left|\color{blue}{\left(-1 \cdot z\right) \cdot \frac{x}{y}} + \frac{x}{y}\right| \]
      neg-mul-1 [<=] 99.8	\[ \left|\color{blue}{\left(-z\right)} \cdot \frac{x}{y} + \frac{x}{y}\right| \]
      distribute-lft1-in [=>] 99.8	\[ \left|\color{blue}{\left(\left(-z\right) + 1\right) \cdot \frac{x}{y}}\right| \]

   if -1.50000000000000001e66 < x < 5.00000000000000019e-43

   1. Initial program 96.0%

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

   2. Applied egg-rr 99.7%

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

      [Start] 96.0	\[ \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
      associate-*l/ [=>] 99.7	\[ \left|\frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}}\right| \]
      sub-div [=>] 99.7	\[ \left|\color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}}\right| \]

   if 5.00000000000000019e-43 < x

   1. Initial program 99.5%

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

   2. Simplified 99.7%

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

      [Start] 99.5	\[ \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
      fabs-sub [=>] 99.5	\[ \color{blue}{\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|} \]
      associate-*l/ [=>] 88.2	\[ \left|\color{blue}{\frac{x \cdot z}{y}} - \frac{x + 4}{y}\right| \]
      associate-*r/ [<=] 99.7	\[ \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]
      *-commutative [<=] 99.7	\[ \left|\color{blue}{\frac{z}{y} \cdot x} - \frac{x + 4}{y}\right| \]
      *-commutative [=>] 99.7	\[ \left|\color{blue}{x \cdot \frac{z}{y}} - \frac{x + 4}{y}\right| \]
      fma-neg [=>] 99.7	\[ \left|\color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, -\frac{x + 4}{y}\right)}\right| \]
      distribute-neg-frac [=>] 99.7	\[ \left|\mathsf{fma}\left(x, \frac{z}{y}, \color{blue}{\frac{-\left(x + 4\right)}{y}}\right)\right| \]
      neg-sub0 [=>] 99.7	\[ \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{0 - \left(x + 4\right)}}{y}\right)\right| \]
      +-commutative [=>] 99.7	\[ \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{0 - \color{blue}{\left(4 + x\right)}}{y}\right)\right| \]
      associate--r+ [=>] 99.7	\[ \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{\left(0 - 4\right) - x}}{y}\right)\right| \]
      metadata-eval [=>] 99.7	\[ \left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{\color{blue}{-4} - x}{y}\right)\right| \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+66}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-43}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	70.3%
Cost	7513

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \left|z \cdot \frac{x}{y}\right|\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{+23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-30}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-12}:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+14} \lor \neg \left(x \leq 5.5 \cdot 10^{+204}\right) \land x \leq 1.2 \cdot 10^{+228}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Accuracy	70.3%
Cost	7513

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{+23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-30}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-9}:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+14} \lor \neg \left(x \leq 5.6 \cdot 10^{+204}\right) \land x \leq 6 \cdot 10^{+224}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Accuracy	70.2%
Cost	7513

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{+23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-30}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq 1.52 \cdot 10^{-9}:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+17} \lor \neg \left(x \leq 4.2 \cdot 10^{+204}\right) \land x \leq 3.8 \cdot 10^{+224}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Accuracy	70.3%
Cost	7513

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{+23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.75 \cdot 10^{-31}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-10}:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{+14} \lor \neg \left(x \leq 5 \cdot 10^{+204}\right) \land x \leq 7.5 \cdot 10^{+224}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Accuracy	99.7%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+66}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-34}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{z}{\frac{y}{x}}\right|\\ \end{array} \]

Alternative 6
Accuracy	99.7%
Cost	7241

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+66} \lor \neg \left(x \leq 1.45 \cdot 10^{+14}\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 7
Accuracy	85.7%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-30} \lor \neg \left(x \leq 4.3 \cdot 10^{-9}\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \end{array} \]

Alternative 8
Accuracy	85.7%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-30} \lor \neg \left(x \leq 3.3 \cdot 10^{-10}\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} + \frac{4}{y}\right|\\ \end{array} \]

Alternative 9
Accuracy	85.8%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-30}:\\ \;\;\;\;\left|\frac{1 - z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-9}:\\ \;\;\;\;\left|\frac{x}{y} + \frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \end{array} \]

Alternative 10
Accuracy	85.7%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-30}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-12}:\\ \;\;\;\;\left|\frac{x}{y} + \frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \end{array} \]

Alternative 11
Accuracy	81.9%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+59}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;z \leq 1.38 \cdot 10^{+107}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{-z}}\right|\\ \end{array} \]

Alternative 12
Accuracy	81.9%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{+58}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+103}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]

Alternative 13
Accuracy	70.5%
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 14
Accuracy	49.4%
Cost	6592

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

Reproduce

herbie shell --seed 2023143 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))