fabs fraction 1

Average Error: 1.4 → 0.1

Time: 12.4s

Precision: binary64

Cost: 8648

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + 4}{y}\\ t_1 := \frac{x}{y} \cdot z\\ t_2 := t_0 - t_1\\ t_3 := \left|t_1 - t_0\right|\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{-16}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 10^{+14}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

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))
        (t_1 (* (/ x y) z))
        (t_2 (- t_0 t_1))
        (t_3 (fabs (- t_1 t_0))))
   (if (<= t_2 -2e-16)
     t_3
     (if (<= t_2 1e+14) (fabs (/ (- (+ x 4.0) (* x z)) y)) t_3))))

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 t_1 = (x / y) * z;
	double t_2 = t_0 - t_1;
	double t_3 = fabs((t_1 - t_0));
	double tmp;
	if (t_2 <= -2e-16) {
		tmp = t_3;
	} else if (t_2 <= 1e+14) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = t_3;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (x + 4.0d0) / y
    t_1 = (x / y) * z
    t_2 = t_0 - t_1
    t_3 = abs((t_1 - t_0))
    if (t_2 <= (-2d-16)) then
        tmp = t_3
    else if (t_2 <= 1d+14) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = t_3
    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 t_1 = (x / y) * z;
	double t_2 = t_0 - t_1;
	double t_3 = Math.abs((t_1 - t_0));
	double tmp;
	if (t_2 <= -2e-16) {
		tmp = t_3;
	} else if (t_2 <= 1e+14) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = t_3;
	}
	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
	t_1 = (x / y) * z
	t_2 = t_0 - t_1
	t_3 = math.fabs((t_1 - t_0))
	tmp = 0
	if t_2 <= -2e-16:
		tmp = t_3
	elif t_2 <= 1e+14:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = t_3
	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)
	t_1 = Float64(Float64(x / y) * z)
	t_2 = Float64(t_0 - t_1)
	t_3 = abs(Float64(t_1 - t_0))
	tmp = 0.0
	if (t_2 <= -2e-16)
		tmp = t_3;
	elseif (t_2 <= 1e+14)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = t_3;
	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;
	t_1 = (x / y) * z;
	t_2 = t_0 - t_1;
	t_3 = abs((t_1 - t_0));
	tmp = 0.0;
	if (t_2 <= -2e-16)
		tmp = t_3;
	elseif (t_2 <= 1e+14)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = t_3;
	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]}, Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Abs[N[(t$95$1 - t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, -2e-16], t$95$3, If[LessEqual[t$95$2, 1e+14], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z)) < -2e-16 or 1e14 < (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z))

   1. Initial program 0.1

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

   if -2e-16 < (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z)) < 1e14

   1. Initial program 3.5

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

   2. Applied egg-rr 0.1

      \[\leadsto \left|\color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|\\ \mathbf{elif}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \leq 10^{+14}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z - \frac{x + 4}{y}\right|\\ \end{array} \]

Alternatives

Alternative 1
Error	20.5
Cost	7644

\[\begin{array}{l} t_0 := \left|\frac{z}{\frac{y}{x}}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{+155}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.35 \cdot 10^{+17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{-17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.6970903300718654 \cdot 10^{-56}:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{elif}\;x \leq 7200:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+253}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+270}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	20.5
Cost	7644

\[\begin{array}{l} t_0 := \left|\frac{z}{\frac{y}{x}}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{+155}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.35 \cdot 10^{+17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{-17}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \leq 2.6970903300718654 \cdot 10^{-56}:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{elif}\;x \leq 7200:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+253}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+270}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	20.9
Cost	7644

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ t_1 := \left|x \cdot \frac{z}{y}\right|\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{+155}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq -2.35 \cdot 10^{+17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{-17}:\\ \;\;\;\;\left|\frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \leq 2.511117432626301 \cdot 10^{-77}:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{elif}\;x \leq 7200:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+253}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+270}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	20.9
Cost	7644

\[\begin{array}{l} t_0 := \left|x \cdot \frac{z}{y}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{+155}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq -2.35 \cdot 10^{+17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{-17}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;x \leq 2.511117432626301 \cdot 10^{-77}:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{elif}\;x \leq 7200:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+253}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+270}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	0.7
Cost	7240

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+125}:\\ \;\;\;\;\left|\frac{1 - z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq 10^{+70}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \end{array} \]

Alternative 6
Error	9.7
Cost	7112

\[\begin{array}{l} t_0 := \left|\frac{1 - z}{\frac{y}{x}}\right|\\ \mathbf{if}\;x \leq -2.9 \cdot 10^{-27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.6970903300718654 \cdot 10^{-56}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	10.1
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-27}:\\ \;\;\;\;\left|\frac{1 - z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq 2.511117432626301 \cdot 10^{-77}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \end{array} \]

Alternative 8
Error	12.2
Cost	6984

\[\begin{array}{l} t_0 := \left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{if}\;z \leq -3.1 \cdot 10^{+152}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.001959311428655546:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	18.9
Cost	6856

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

Alternative 10
Error	47.2
Cost	6592

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

Reproduce

herbie shell --seed 2022313 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))