fabs fraction 1

Average Accuracy: 97.7% → 99.5%

Time: 9.6s

Precision: binary64

Cost: 7241

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+50} \lor \neg \left(x \leq 3.3 \cdot 10^{+74}\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.6e+50) (not (<= x 3.3e+74)))
   (fabs (* (/ x y) (- 1.0 z)))
   (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 <= -3.6e+50) || !(x <= 3.3e+74)) {
		tmp = fabs(((x / y) * (1.0 - z)));
	} else {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = abs((((x + 4.0d0) / y) - ((x / y) * z)))
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-3.6d+50)) .or. (.not. (x <= 3.3d+74))) then
        tmp = abs(((x / y) * (1.0d0 - z)))
    else
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	return Math.abs((((x + 4.0) / y) - ((x / y) * z)));
}

↓

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.6e+50) || !(x <= 3.3e+74)) {
		tmp = Math.abs(((x / y) * (1.0 - z)));
	} else {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	return math.fabs((((x + 4.0) / y) - ((x / y) * z)))

↓

def code(x, y, z):
	tmp = 0
	if (x <= -3.6e+50) or not (x <= 3.3e+74):
		tmp = math.fabs(((x / y) * (1.0 - z)))
	else:
		tmp = math.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 <= -3.6e+50) || !(x <= 3.3e+74))
		tmp = abs(Float64(Float64(x / y) * Float64(1.0 - z)));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = abs((((x + 4.0) / y) - ((x / y) * z)));
end

↓

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -3.6e+50) || ~((x <= 3.3e+74)))
		tmp = abs(((x / y) * (1.0 - z)));
	else
		tmp = abs((((x + 4.0) - (x * z)) / y));
	end
	tmp_2 = 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, -3.6e+50], N[Not[LessEqual[x, 3.3e+74]], $MachinePrecision]], N[Abs[N[(N[(x / y), $MachinePrecision] * N[(1.0 - z), $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 < -3.59999999999999986e50 or 3.3000000000000002e74 < x

   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/ [=>] 82.7	\[ \left|\frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}}\right| \]
      *-commutative [=>] 82.7	\[ \left|\frac{x + 4}{y} - \frac{\color{blue}{z \cdot x}}{y}\right| \]
      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/ [=>] 82.5	\[ \left|-1 \cdot \color{blue}{\frac{z \cdot x}{y}} + \frac{1}{y} \cdot x\right| \]
      associate-*l/ [=>] 82.7	\[ \left|-1 \cdot \frac{z \cdot x}{y} + \color{blue}{\frac{1 \cdot x}{y}}\right| \]
      associate-*r/ [<=] 82.7	\[ \left|-1 \cdot \frac{z \cdot x}{y} + \color{blue}{1 \cdot \frac{x}{y}}\right| \]
      *-lft-identity [=>] 82.7	\[ \left|-1 \cdot \frac{z \cdot x}{y} + \color{blue}{\frac{x}{y}}\right| \]
      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 -3.59999999999999986e50 < x < 3.3000000000000002e74

   1. Initial program 96.9%

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

   2. Applied egg-rr 99.3%

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

      [Start] 96.9	\[ \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
      associate-*l/ [=>] 99.3	\[ \left|\frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}}\right| \]
      sub-div [=>] 99.3	\[ \left|\color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}}\right| \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+50} \lor \neg \left(x \leq 3.3 \cdot 10^{+74}\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	69.3%
Cost	7380

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \left|\frac{x}{y} \cdot z\right|\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{+87}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{+70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.52:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-30}:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{+78}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Accuracy	69.3%
Cost	7380

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{+87}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{+70}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \leq -1.52:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-30}:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{+79}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Accuracy	85.1%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-24} \lor \neg \left(x \leq 6.6 \cdot 10^{-60}\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \end{array} \]

Alternative 4
Accuracy	85.2%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-25}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-60}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1 - z}{\frac{y}{x}}\right|\\ \end{array} \]

Alternative 5
Accuracy	85.2%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{-26}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-60}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1 - z}{\frac{y}{x}}\right|\\ \end{array} \]

Alternative 6
Accuracy	97.7%
Cost	7104

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

Alternative 7
Accuracy	81.9%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;z \leq -3.35 \cdot 10^{+65}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+150}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \]

Alternative 8
Accuracy	70.1%
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	48.8%
Cost	6592

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

Reproduce

herbie shell --seed 2023137 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))