fabs fraction 1

Average Error: 1.6 → 0.1

Time: 10.9s

Precision: binary64

Cost: 7240

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-10}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+16}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= x -1e-10)
   (fabs (- (/ (+ x 4.0) y) (/ x (/ y z))))
   (if (<= x 3.6e+16)
     (fabs (/ (- (+ 4.0 x) (* z x)) y))
     (fabs (* (- 1.0 z) (/ 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 <= -1e-10) {
		tmp = fabs((((x + 4.0) / y) - (x / (y / z))));
	} else if (x <= 3.6e+16) {
		tmp = fabs((((4.0 + x) - (z * x)) / y));
	} else {
		tmp = fabs(((1.0 - z) * (x / 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 <= (-1d-10)) then
        tmp = abs((((x + 4.0d0) / y) - (x / (y / z))))
    else if (x <= 3.6d+16) then
        tmp = abs((((4.0d0 + x) - (z * x)) / y))
    else
        tmp = abs(((1.0d0 - z) * (x / 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 <= -1e-10) {
		tmp = Math.abs((((x + 4.0) / y) - (x / (y / z))));
	} else if (x <= 3.6e+16) {
		tmp = Math.abs((((4.0 + x) - (z * x)) / y));
	} else {
		tmp = Math.abs(((1.0 - z) * (x / 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 <= -1e-10:
		tmp = math.fabs((((x + 4.0) / y) - (x / (y / z))))
	elif x <= 3.6e+16:
		tmp = math.fabs((((4.0 + x) - (z * x)) / y))
	else:
		tmp = math.fabs(((1.0 - z) * (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 <= -1e-10)
		tmp = abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(x / Float64(y / z))));
	elseif (x <= 3.6e+16)
		tmp = abs(Float64(Float64(Float64(4.0 + x) - Float64(z * x)) / y));
	else
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / 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 <= -1e-10)
		tmp = abs((((x + 4.0) / y) - (x / (y / z))));
	elseif (x <= 3.6e+16)
		tmp = abs((((4.0 + x) - (z * x)) / y));
	else
		tmp = abs(((1.0 - z) * (x / 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[LessEqual[x, -1e-10], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 3.6e+16], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] - N[(z * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.00000000000000004e-10

   1. Initial program 0.1

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

   2. Applied egg-rr 0.2

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

   if -1.00000000000000004e-10 < x < 3.6e16

   1. Initial program 2.6

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

   2. Taylor expanded in y around 0 0.1

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

   if 3.6e16 < x

   1. Initial program 0.1

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

   2. Taylor expanded in y around 0 9.0

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

   3. Taylor expanded in x around inf 9.0

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

   4. Simplified 0.1

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

      [Start] 9.0	\[ \left|\frac{\left(1 - z\right) \cdot x}{y}\right| \]
      rational.json-simplify-2 [=>] 9.0	\[ \left|\frac{\color{blue}{x \cdot \left(1 - z\right)}}{y}\right| \]
      rational.json-simplify-49 [=>] 0.1	\[ \left|\color{blue}{\left(1 - z\right) \cdot \frac{x}{y}}\right| \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-10}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+16}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	7240

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+80}:\\ \;\;\;\;\left|\frac{x}{y} - \frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 2
Error	0.8
Cost	7240

\[\begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{+174}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{z}{y} \cdot x\right|\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+18}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 3
Error	9.5
Cost	7112

\[\begin{array}{l} t_0 := \left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{if}\;x \leq -680:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-44}:\\ \;\;\;\;\left|\frac{4 + x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	9.4
Cost	7112

\[\begin{array}{l} t_0 := \left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \mathbf{if}\;x \leq -60000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-42}:\\ \;\;\;\;\left|\frac{4 + x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	9.4
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -120000:\\ \;\;\;\;\left|\frac{x}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \leq 6.7 \cdot 10^{-43}:\\ \;\;\;\;\left|\frac{4 + x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 6
Error	13.2
Cost	7048

\[\begin{array}{l} t_0 := \left|-\frac{z \cdot x}{y}\right|\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{+132}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+145}:\\ \;\;\;\;\left|\frac{4 + x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	12.7
Cost	7048

\[\begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+135}:\\ \;\;\;\;\left|-\frac{z \cdot x}{y}\right|\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+113}:\\ \;\;\;\;\left|\frac{4 + x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{-z}{y}\right|\\ \end{array} \]

Alternative 8
Error	11.7
Cost	7048

\[\begin{array}{l} t_0 := \left|z \cdot \frac{x}{-y}\right|\\ \mathbf{if}\;z \leq -6.2 \cdot 10^{+104}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.26 \cdot 10^{+111}:\\ \;\;\;\;\left|\frac{4 + x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	19.0
Cost	6856

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -10.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	18.1
Cost	6720

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

Alternative 11
Error	32.8
Cost	6592

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

Reproduce

herbie shell --seed 2023075 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))