fabs fraction 1

Average Error: 1.6 → 0.4

Time: 9.0s

Precision: binary64

Cost: 7368

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + 4}{y}\\ \mathbf{if}\;x \leq -4.6541639256961256 \cdot 10^{-110}:\\ \;\;\;\;\left|t_0 - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq 10^{+34}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t_0 - x \cdot \frac{z}{y}\right|\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x 4.0) y)))
   (if (<= x -4.6541639256961256e-110)
     (fabs (- t_0 (/ x (/ y z))))
     (if (<= x 1e+34)
       (fabs (/ (- (+ x 4.0) (* x z)) y))
       (fabs (- t_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 t_0 = (x + 4.0) / y;
	double tmp;
	if (x <= -4.6541639256961256e-110) {
		tmp = fabs((t_0 - (x / (y / z))));
	} else if (x <= 1e+34) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs((t_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) :: t_0
    real(8) :: tmp
    t_0 = (x + 4.0d0) / y
    if (x <= (-4.6541639256961256d-110)) then
        tmp = abs((t_0 - (x / (y / z))))
    else if (x <= 1d+34) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = abs((t_0 - (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 t_0 = (x + 4.0) / y;
	double tmp;
	if (x <= -4.6541639256961256e-110) {
		tmp = Math.abs((t_0 - (x / (y / z))));
	} else if (x <= 1e+34) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = Math.abs((t_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):
	t_0 = (x + 4.0) / y
	tmp = 0
	if x <= -4.6541639256961256e-110:
		tmp = math.fabs((t_0 - (x / (y / z))))
	elif x <= 1e+34:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = math.fabs((t_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)
	t_0 = Float64(Float64(x + 4.0) / y)
	tmp = 0.0
	if (x <= -4.6541639256961256e-110)
		tmp = abs(Float64(t_0 - Float64(x / Float64(y / z))));
	elseif (x <= 1e+34)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(Float64(t_0 - Float64(x * Float64(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)
	t_0 = (x + 4.0) / y;
	tmp = 0.0;
	if (x <= -4.6541639256961256e-110)
		tmp = abs((t_0 - (x / (y / z))));
	elseif (x <= 1e+34)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = abs((t_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_] := Block[{t$95$0 = N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -4.6541639256961256e-110], N[Abs[N[(t$95$0 - N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1e+34], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$0 - N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.6541639256961256e-110

   1. Initial program 0.7

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

   2. Applied egg-rr 1.1

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

   if -4.6541639256961256e-110 < x < 9.99999999999999946e33

   1. Initial program 2.6

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

   2. Applied egg-rr 2.8

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

   3. Applied egg-rr 5.9

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

   4. Applied egg-rr 0.2

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

   if 9.99999999999999946e33 < x

   1. Initial program 0.1

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

   2. Applied egg-rr 0.1

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

   3. Applied egg-rr 0.1

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.6541639256961256 \cdot 10^{-110}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq 10^{+34}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	7368

\[\begin{array}{l} t_0 := \frac{x + 4}{y}\\ \mathbf{if}\;x \leq -4.038698827692165 \cdot 10^{-37}:\\ \;\;\;\;\left|t_0 - \frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq 10^{+34}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t_0 - x \cdot \frac{z}{y}\right|\\ \end{array} \]

Alternative 2
Error	1.6
Cost	7236

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

Alternative 3
Error	19.5
Cost	7120

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{+66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{+28}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -358279.2040577703:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.0183503999642145:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	19.5
Cost	7120

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{+66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{+28}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq -358279.2040577703:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.0183503999642145:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	9.1
Cost	7112

\[\begin{array}{l} t_0 := \left|\frac{1 - z}{\frac{y}{x}}\right|\\ \mathbf{if}\;x \leq -4.038698827692165 \cdot 10^{-37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.0183503999642145:\\ \;\;\;\;\left|\frac{-4}{y} - \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	1.7
Cost	7108

\[\begin{array}{l} \mathbf{if}\;x \leq 10^{+40}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1 - z}{\frac{y}{x}}\right|\\ \end{array} \]

Alternative 7
Error	1.7
Cost	7108

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

Alternative 8
Error	11.6
Cost	6984

\[\begin{array}{l} \mathbf{if}\;z \leq -2.8330409732078132 \cdot 10^{+62}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq 1.524302918986109 \cdot 10^{+54}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 9
Error	19.0
Cost	6856

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -358279.2040577703:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.0183503999642145:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	47.0
Cost	6592

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

Reproduce

herbie shell --seed 2022310 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))